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All Textbook Solutions for University Physics Volume 1
Competitive divers pull their limbs in and curl up their bodies when they do flips. Just before entering the water, they fully extend their limbs to enter straight down (see below). Explain the effect of both actions on their angular velocities. Also explain the effect on their angular momentum.Gyroscopes used in guidance systems to indicate directions in space must have an angular momentum that does not change in direction. When placed in the vehicle, they are put in a compartment that is separated from the main fuselage, such that changes in the orientation of the fuselage does not affect the orientation of the gyroscope. If the space vehicle Is subjected to large forces and accelerations how can the direction of the gyroscopes angular momentum be constant at all times?Earth precesses about its vertical axis with a period of 26,000 years. Discuss whether Equation 11.12 can be used to calculate the precessional angular velocity of Earth.What is the angular velocity of a 75.0-cm-diameter tire on an automobile traveling at 90.0 km/h?A boy rides his bicycle 2.00 km. The wheels have radius 30.0 cm. What is the total angle the tires rotate through during his trip?If the boy on the bicycle in the preceding problem accelerates from rest to a speed of 10.0 m/s in 10.0 s, what is the angular acceleration of the tires?Formula One race cars have 66-cm-diameter tires. If a Formula One averages a speed of 300 km/h during a race, what is the angular displacement in revolutions of the wheels if the race car maintains this speed for 1.5 hours?A marble rolls down an incline at 30 from rest. (a) What is its acceleration? (b) How far does it go in 3.0 s?Repeat the preceding problem replacing the marble with a solid cylinder. Explain the new result.A rigid body with a cylindrical cross-section is released from the top of a 30 incline. It rolls 10.0 m to the bottom in 2.60 s. Find the moment of inertia of the body In terms of its mass m and radius r.A yo-yo can be thought of a solid cylinder of mass m and radius r that has a light string wrapped around its circumference (see below). One end of the string is held fixed in space. If the cylinder falls as the string unwinds without slipping, what is the acceleration of the cylinder?A solid cylinder of radius 10.0 cm rolls down an incline with slipping. The angle of the incline is 30 . The coefficient of kinetic friction on the surface is 0.400. What is the angular acceleration of the solid cylinder? What is the linear acceleration?A bowling ball rolls up a ramp 0.5 m high without slipping to storage. It has an initial velocity of its center of mass of 3.0 m/s. (a) What is its velocity at the top of the ramp? (b) If the ramp is 1 m high does it make it to the top?A 40.0-kg solid cylinder is rolling across a horizontal surface at a speed of 6.0 m/s. How much work is required to stop it?A 40.0-kg solid sphere is rolling across a horizontal surface with a speed of 6.0 m/s. How much work is required to stop it? Compare results with the preceding problem.A solid cylinder rolls up an incline at an angle of 20 . If it starts at the bottom with a speed of 10 m/s how far up the incline does it travel?A solid cylindrical wheel of mass M and radius R is pulled by a force applied to the center of the wheel at 37 to the horizontal (see the following figure). If the wheel is to roll without slipping, what is the maximum value of ? The coefficients of static and kinetic friction are s=0.40 and k=0.30 .A hollow cylinder that is rolling without slipping is given a velocity of 5.0 m/s and rolls up an incline to a vertical height of 1.0 m. If a hollow sphere of the same mass and radius is given the same initial velocity how high vertically does it roll up the incline?A 0.2-kg particle is travelling along the line y=2.0m with a velocity 5.0 m/s. What is the angular momentum of the particle about the origin?A bird flies overhead from where you stand at an altitude of 300.0 m and at a speed horizontal to the ground of 20.0 m/s. The bird has a mass of 2.0 kg. The radius vector to the bird makes an angle with respect to the ground. The radius vector to the bird and its momentum vector lie in the xy-plane. What is the bird’s angular momentum about the point where you are standing?A Formula One race car with mass 750.0 kg is speeding through a course in Monaco and enters a circular turn at 220.0 km/h in the counterclockwise direction about the origin of the circle. At another part of the course, the car enters a second circular turn at 180 km/h also in the counterclockwise direction. If the radius of curvature of the first turn is 130.0 m and that of the second is 100.0 m, compare the angular momenta of the race car in each turn taken about the origin of the circular turn.A particle of mass 5.0 kg has position vector at a particular instant of time when its velocity is with respect to the origin. (a) What is the angular momentum of the particle? (b) If a force acts on the particle at this instant, what is the torque about the origin?Use the right-hand rule to determine the directions of the angular momenta about the origin of the particles as shown below. The z-axis is out of the page.Suppose the particles in the preceding problem have masses m1=0.10kg , m2=0.20kg , m3=0.30kg , m4=0.40kg . The velocities of the particles are . (a) Calculate the angular momentum of each particle about the origin. (b) What is the total angular momentum of the tour-particle system about the origin?Two particles of equal mass travel with the same speed in opposite directions along parallel lines separated by a distance d Show that the angular momentum of this two- particle system is the same no matter what point is used as the reference for calculating the angular momentum.An airplane of mass 4.0104kg flies horizontally at an altitude of 10 km with a constant speed of 250 m/s relative to Earth. (a) What is the magnitude of the airplane’s angular momentum relative to a ground observer directly below the plane? (b) Does the angular momentum change as the airplane flies along a constant altitude?At a particular instant, a 1.0-kg particle’s position is , its velocity is , and the force on it is . (a) What is the angular momentum of the particle about the origin? (b) What is the torque on the particle about the origin? (c) What is the time rate of change of the particle’s angular momentum at this instant?43P(a) Calculate the angular momentum of Earth in its orbit around the Sun. (b) Compare this angular momentum with the angular momentum of Earth about its axis.A boulder of mass 20 kg and radius 20 cm rolls down a hill 15 m high from rest. What is its angular momentum when it is half way down the hill? (b) At the bottom?A satellite is spinning at 6.0 rev/s. The satellite consists of a main body in the shape of a sphere of radius 2.0 m and mass 10,000 kg, and two antennas projecting out from the center of mass of the main body that can be approximated with rods of length 3.0 m each and mass 10 kg. The antenna’s lie in the plane of rotation. What is the angular momentum of the satellite?A propeller consists of two blades each 3.0 m in length and mass 120 kg each. The propeller can be approximated by a single rod rotating about its center of mass. The propeller starts from rest and rotates up to 1200 rpm in 30 seconds at a constant rate. (a) What is the angular momentum of the propeller at t=10s ; t=20s ? (b) What is the torque on the propeller?A pulsar is a rapidly rotating neutron star. The Crab nebula pulsar in the constellation Taurus has a period of 33.510-3s , radius 10.0 km, and mass 2.81030kg . The pulsar’s rotational period will increase over time due to the release of electromagnetic radiation, which doesn’t change its radius but reduces its rotational energy. (a) What is the angular momentum of the pulsar? (b) Suppose the angular velocity decreases at a rate of 1014rad/s2 . What is the torque on the pulsar?The blades of a wind turbine are 30 m in length and rotate at a maximum rotation rate of 20 rev/min. (a) If the blades are 6000 kg each and the rotor assembly has three blades, calculate the angular momentum of the turbine at this rotation rate. (b) What Is the torque require to rotate the blades up to the maximum rotation rate in 5 minutes?A roller coaster has mass 3000.0 kg and needs to make it safely through a vertical circular loop of radius 50.0 m. What is the minimum angular momentum of the coaster at the boom of the loop to make it safely through? Neglect friction on the track. Take the coaster to be a point particle.A mountain biker takes a jump in a race and goes airborne. The mountain bike is travelling at 10.0 m/s before It goes airborne. If the mass of the from wheel on the bike is 750 g and has radius 35 cm, what is the angular momentum of the spinning wheel in the air the moment the bike leaves the ground?Conservation of Angular Momentum 52.A disk of mass 2.0 kg and radius 60 cm with a small mass of 0.05 kg attached at the edge is rotating at 2.0 rev/s. The small mass suddenly separates from the disk. What is the disk’s final rotation rate?The Sun’s mass is 2.01030kg , its radius is 7.0105km , and it has a rotational period of approximately 28 days. If the Sun should collapse into a white dwarf of radius 3.5103km , what would its period be if no mass were ejected and a sphere of uniform density can model the Sun both before and after?A cylinder with rotational inertia I1=2.0kgm2 rotates clockwise about a vertical axis through its center with angular speed 1=5.0rad/s . A second cylinder with rotational inertia I2=1.0kgm2 rotates counterclockwise about the same axis with angular speed 2=8.0rad/s . If the cylinders couple so they have the same rotational axis what is the angular speed of the combination? What percentage of the original kinetic energy is lost to friction?A diver off the high board imparts an initial rotation with his body fully extended before going into a tuck and executing three back somersaults before hitting the water. If his moment of inertia before the tuck is 16.9kgm2 and after the tuck during the somersaults is 4.2kgm2 , what rotation rate must he impart to his body directly off the board and before the tuck if he takes 1.4 s to execute the somersaults before hitting the water?An Earth satellite has its apogee at 2500 km above the surface of Earth and perigee at 500 km above the surface of Earth. At apogee its speed is 730 m/s. What is its speed at perigee? Earth’s radius is 6370 km (see below).A Molniya orbit is a highly eccentric orbit of a communication satellite so as to provide continuous communications coverage for Scandinavian countries and adjacent Russia. The orbit is positioned so that these countries have the satellite in view for extended periods in time (see below). If a satellite in such an orbit has an apogee at 40,000.0 km as measured from the center of Earth and a velocity of 3.0 km/s, what would be its velocity at perigee measured at 200.0 km altitude?Shown below is a small particle of mass 20 g that is moving at a speed of 10.0 m/s when it collides and sticks to the edge of a uniform solid cylinder. The cylinder is free to rotate about its axis through its center and is perpendicular to the page. The cylinder has a mass of 0.5 kg and a radius of 10 cm, and is initially at rest. (a) What is the angular velocity of the system after the collision? (b) How much kinetic energy is lost in the collision?A bug of mass 0.020 kg is at rest on the edge of a solid cylindrical disk (M=0.10kg,R=0.10m) rotating in a horizontal plane around the vertical axis through its center. The disk is rotating at 10.0 rad/s. The bug crawls to the center of the disk. (a) What is the new angular velocity of the disk? (b) What is the change in the kinetic energy of the system? (c) If the bug crawls back to the outer edge of the disk, what is the angular velocity of the disk then? (d) What is the new kinetic energy of the system? (e) What is the cause of the increase and decrease of kinetic energy?A uniform rod of mass 200 g and length 100 cm is free to rotate in a horizontal plane around a fixed vertical axis through its center, perpendicular to its length. Two small beads, each of mass 20 g, are mounted in grooves along the rod. Initially, the two beads are held by catches on opposite sides of the rod’s center, 10 cm from the axis of rotation. With the beads in this position, the rod is rotating with an angular velocity of 10.0 rad/s. When the catches are released, the beads slide outward along the rod. (a) What is the rod’s angular velocity when the beads reach the ends of the rod? (b) What is the rod’s angular velocity if the beads fly off the rod?A merry-go-round has a radius of 2.0 m and a moment of inertia 300kgm2 . A boy of mass 50 kg runs tangent to the rim at a speed of 4.0 m/s ad jump on. If the merry-go-round is initially at rest, what is the angular velocity after the boy jumps on?A playground merry-go-round has a mass of 120 kg and a radius of 1.80 m and it is rotating with an angular velocity of 0.500 rev/s. What is its angular velocity after a 22.0-kg child gets onto it by grabbing its outer edge? The child is initially at rest.Three children are riding on the edge of a merry-go-round that is 100 kg, has a 1.60-m radius, and is spinning at 20.0 rpm. The children have masses of 22.0, 28.0, and 33.0 kg. If the child who has a mass of 28.0 kg moves to the center of the merry-go-round, what is the new angular velocity in rpm?(a) Calculate the angular momentum of an ice skater spinning at 6.00 rev/s given his moment of inertia is 0.400kgm2 . (b) He reduces his rate of spin (his angular velocity) by extending his arms and increasing his moment of inertia. Find the value of his moment of inertia if his angular velocity decreases to 1.25 rev/s. (c) Suppose instead he keeps his arms in and allows friction of the ice to slow him to 3.00 rev/s. What average torque was exerted if this takes 15.0 s?Twin skaters approach one another as shown below and lock hands. (a) Calculate their final angular velocity given each had an initial speed of 2.50 m/s relative to the ice. Each has a mass of 70.0 kg and each has a center of mass located 0.800 m from their locked hands. You may approximate their moments of inertia to be that of point masses at this radius. (b) Compare the initial kinetic energy and final kinetic energy.A baseball catcher extends his arm straight up to catch a fast ball with a speed of 40 m/s. The baseball is 0.145 kg and the catcher’s arm length is 0.5 m and mass 4.0 kg. (a) What is the angular velocity of the arm immediately after catching the ball as measured from the arm socket? (b) What is the torque applied if the catcher stops the rotation of his arm 0.3 s after catching the ball?In 2015, in Warsaw, Poland, Olivia Oliver of Nova Scotia broke the world record for being the fastest spinner on ice skates. She achieved a record 342 rev/min, beating the existing Guinness World Record by 34 rotations. If an ice skater extends her aims at that rotation rate, what would be her new rotation rate? Assume she can be approximated by a 45-kg rod that is 1.7 m tall with a radius of 15 cm in the record spin. With her aims stretched take the approximation of a rod of length 130 cm with 10 of her body mass aligned perpendicular to the spin axis. Neglect frictional forces.A satellite in a geosynchronous circular orbit is 42,164.0 km from the center of Earth. A small asteroid collides with the satellite sending it into an elliptical orbit of apogee 45,000.0 km. What is the speed of the satellite at apogee? Assume its angular momentum is conserved.A gymnast does cartwheels along the floor and then launches herself into the air and executes several flips in a tuck while she is airborne. If her moment of inertia when executing the cartwheels is 13.5kgm2 and her spin rate is 0.5 rev/s, how many revolutions does she do in the air if her moment of inertia in the tuck is 3.4kgm2 and she has 2.0 s to do the flips in the air?The centrifuge at NASA Ames Research Center has a radius of 8.8 m and can produce farces on its payload of 20 gs or 20 times the force of gravity on Earth. (a) What is the angular momentum of a 20-kg payload that experiences 10 gs in the centrifuge? (b) If the driver motor was turned off in (a) and the payload lost 10 kg, what would be its new spin rate, taking into account there are no frictional forces present?A ride at a carnival has four spokes to which pods are attached that can hold two people. The spokes are each 15 m long and are attached to a central axis. Each spoke has mass 200.0 kg, and the pods each have mass 100.0 kg. If the ride spins at 0.2 rev/s with each pod containing two 50.0-kg children, what is the new spin rate if all the children jump off the ride?An ice skater is preparing for a jump with turns and has his arms extended. His moment of inertia is 1.8kgm2 while his arms are extended, and he is spinning at 0.5 rev/s. If he launches himself into the air at 9.0 m/s at an angle of 45 with respect to the ice, how many revolutions can he execute while airborne if his moment of inertia in the air is 0.5kgm2 ?A space station consists of a giant rotating hollow cylinder of mass 106kg including people on the station and a radius of 100.00 m. It is rotating in space at 3.30 rev/min in order to produce artificial gravity. If 100 people of an average mass of 65.00 kg spacewalk to an awaiting spaceship, what is the new rotation rate when all the people are off the station?Neptune has a mass of 1.01026kg and is 4.5109km from the Sun with an orbital period of 165 years. Planetesimals in the outer primordial solar system 4.5 billion years ago coalesced into Neptune over hundreds of millions of years. If the primordial disk that evolved into our present day solar system had a radius of 1011km and if the matter that made up these planetesimals that later became Neptune was spread out evenly on the edges of it, what was the orbital period of the outer edges of the primordial disk?A gyroscope has a 0.5-kg disk that spins at 40 rev/s. The center of mass of the disk is 10 cm from a pivot which is also the radius of the disk. What is the precession angular velocity?The precession angular velocity of a gyroscope is 1.0 rad/s. If the mass of the rotating disk is 0.4 kg and its radius is 30 cm, as well as the distance from the center of mass to the pivot, what is the rotation rate in rev/s of the disk?The axis of Earth makes a 23.5 angle with a direction perpendicular to the plane of Earth’s orbit. As shown below, this axis precesses, making one complete rotation in 25,780 y. (a) Calculate the change in angular momentum in half this time. (b) What is the average torque producing this change in angular momentum? (c) If this torque were created by a pair of forces acting at the most effective point on the equator, what would the magnitude of each force be?A marble is rolling across the floor at a speed of 7.0 m/s when it starts up a plane inclined at 30 to the horizontal. (a) How far along the plane does the marble travel before coming to a rest? (b) How much time elapses while the marble moves up the plane?Repeat the preceding problem replacing the marble with a hollow sphere. Explain the new results.The mass of a hoop of radius 1.0 m is 6.0 kg. It rolls across a horizontal surface with a speed of 10.0 m/s. (a) How much work is required to stop the hoop? (b) If the hoop starts up a surface at 30 to the horizontal with a speed of 10.0 m/s, how far along the incline will it travel before stopping and rolling back down?Repeat the preceding problem for a hollow sphere of the same radius and mass and initial speed. Explain the differences in the results.A particle has mass 0.5 kg and is traveling along the line x=5.0m at 2.0 m/s In the positive y-direction. What is the particle’s angular momentum about the origin?A 4.0-kg particle moves in a circle of radius 2.0 m. The angular momentum of the particle varies in time according to l=5.0t2 , (a) What is the torque on the particle about the center of the circle at t=3.4s ? (b) What is the angular velocity of the particle at t=3.4s ?A proton is accelerated in a cyclotron to 5.0106m/s in 0.01 s. The proton follows a circular path. If the radius of the cyclotron is 0.5 km, (a) What is the angular momentum of the proton about the center at its maximum speed? (b) What is the torque on the proton about the center as it accelerates to maximum speed?(a) What is the angular momentum of the Moon in its orbit around Earth? (b) How does this angular momentum compare with the angular momentum of the Moon on its axis? Remember that the Moon keeps one side toward Earth at all times.A DVD is rotating at 500 rpm. What is the angular momentum of the DVD if has a radius of 6.0 cm and mass 20.0 g?A potter’s disk spins from rest up to 10 rev/s in 15 s. The disk has a mass 3.0 kg and radius 30.0 cm. What is the angular momentum of the disk at t=5s , t=10s ?Suppose you start an antique car by exerting a force of 300 N on its crank for 0.250 s. What is the angular momentum given to the engine if the handle of the crank is 0.300 m from the pivot and the force is exerted to create maximum torque the entire time?A solid cylinder of mass 2.0 kg and radius 20 cm is rotating counterclockwise around a vertical axis through its center at 600 rev/min. A second solid cylinder of the same mass and radius is rotating clockwise around the same vertical axis at 900 rev/min. If the cylinders couple so that they rotate about the same vertical axis, what is the angular velocity of the combination?A boy stands at the center of a platform that is rotating without friction at 1.0 rev/s. The boy holds weights as far from his body as possible. At this position the total moment of inertia of the boy, platform, and weights is 5.0kgm2 . The boy draws the weights in close to his body, thereby decreasing the total moment of inertia to 1.5kgm2 . (a) What Is the final angular velocity of the platform? (b) By how much does the rotational kinetic energy increase?Eight children each of mass 40 kg, climb on a small merry-go-round. They position themselves evenly on the outer edge and join hands. The merry-go-round has a radius of 4.0 m and a moment of inertia 1000.0kgm2 . After the merry-go-round is given an angular velocity of 6.0 rev/min, the children walk inward and stop when they are 0.75 m from the axis of rotation. What is the new angular velocity of the merry-go-round? Assume there Is negligible frictional torque on the structure.A thin meter stick of mass 150 g rotates around an axis perpendicular to the stick’s long axis at an angular velocity of 240 rev/min. What is the angular momentum of the stick if the rotation axis (a) passes through the center of the stick? (b) Passes through one end of the stick?A satellite in the shape of a sphere of mass 20,000 kg and radius 5.0 m is spinning about an axis through its center of mass. It has a rotation rate of 8.0 rev/s. Two antennas deploy in the plane of rotation extending from the center of mass of the satellite. Each antenna can be approximated as a rod has mass 200.0 kg and length 7.0 m. What is the new rotation rate of the satellite?A top has moment of inertia 3.2104kgm2 and radius 4.0 cm from the center of mass to the pivot point. If it spins at 20.0 rev/s and is precessing, how many revolutions does it precess in 10.0 s?The truck shown below is initially at rest with solid cylindrical roll of paper sitting on its bed. If the truck moves forward with a uniform acceleration a , what distance s does it move before the paper rolls off its back end? (Hint: If the roll accelerates forward with a , then is acceleration aa . Also, R=aa .)A bowling ball of radius 8.5 cm is tossed onto a bowling lane with speed 9.0 m/s. The direction of the toss is to the left, as viewed by the observer, so the bowing ball starts to rotate counterclockwise when in contact with the floor. The coefficient of kinetic friction on the lane is 0.3. (a) What is the time required for the ball to come to the point where It is no slipping? What is the distance d to the point where the ball is rolling without slipping?A small ball of mass 0.50 kg is attached by a massless sting to a vertical rod that Is spinning as shown below. When the rod has an angular velocity of 6.0 rad/s, the sting makes an angle of 30 with respect to the vertical. (a) If the angular velocity is increased to 10.0 rad/s, what is the new angle of the string? (b) Calculate the initial and final angular momenta of the ball. (c) Can the rod spin fast enough so that the ball is horizontal?A bug flying horizontally at 1.0 m/s collides and sticks to the end of a uniform stick hanging vertically. After the impact, the stick swings out to a maximum angle of 5.0 from the vertical before rotating back. If the mass of the stick is 10 times that of the bug, calculate the length of the stick.Check Your Understanding Solve Example 12.1 by choosing the pivot at the location of the rear axle.Check your Understanding Explain which one of the following satisfies both equilibrium conditions: (a) a tennis ball that does not spin as it travels in the air, (b) a pelican that is gliding in the air at a constant velocity at one altitude; or (c) a crankshaft in the engine of a parked car.Check your Understanding Repeat Example 12.3 using the left end of the meter stick to calculate the torques; that is by placing the pivot at the left end of the meter stick.Check Your understanding Repeat ExampIe12.4 assuming that the forearm is object of uniform density that weighs 8.896 N.Check Your Understanding For the situation in described in Example 12.5, determine the values of the coefficient s of static friction for which the ladder starts slipping, given that is the angle that the ladder makes with the floor.Check Your Understanding Solve the problem in Example 12.6 by taking the pivot position at the center of mass.Check Your Understanding A 50-kg person stands 1.5 m away from one end of a uniform 6.0-m-long scaffold of mass 70.0 kg. Find the tensions in the two vertical ropes supporting the scaffold.Check Your Understanding A 400.0-N sign hangs from the end of a uniform strut. The strut is 4.0 m long and weighs 600.0 N. The strut is supported by a hinge at the wall and by a cable whose other end is tied to the wall at a point 3.0 m above the left end of the strut. Find the tension in the supporting cable and the force of the hinge on the strut.Check Your Understanding Find the compressive stress and strain at the base of Nelson’s column.Check Your Understanding A 2.0-m-long wire stretches 1.0 mm when subjected to a load. What is the tensile strain in the wire?Check Your Understanding If the normal force acting on each face of a cubical 1.0m3 piece of steel is changed by 1.0107 N. find the resulting change in the volume of the piece of steel.Check Your Understanding Explain why the concepts of Young’s modulus and shear modulus do not apply to fluids.What can you say about the velocity of a moving body that is in dynamic equilibrium?Under what conditions can a rotating body be in equilibrium? Give an example.What three factors affect the torque created by a force relative to a specific pivot point?Mechanics sometimes put a length of pipe over the handle of a wrench when trying to remove a very tight boll How does this help?If there is only one external force (or toe) acting on object. It cannot be on equilibrium.If an object is in equilibrium there must be an even number of forces acting on it.If an odd number of forces act on an object, the object cannot be in equilibrium.A body moving in a circle with a constant seed is in rotational equilibrium.What purpose is served by a long and flexible pole carried by wire-walkers?Is it possible to rest a ladder against a rough wall when the floor is frictionless?Show how a spring scale and a simple fulcrum can be used to weigh an object whose weight is larger than the maximum reading on the scale.A painter climbs a ladder. Is the ladder more likely to slip when the painter Is near the bottom or near the top?Note: Unless stated otherwise, the weights of the wires, rods, and other elements are assumed to be negligible. Elastic moduli of selected materials are given In Table 12.1 13.Why can a squirrel jump from a tree branch to the ground and run away undamaged, while a human could break a bone In such a fall?When a glass bottle full of vinegar warms up, both the vinegar and the glass expand, but the vinegar expands significantly more with temperature than does the glass. The bottle will break if it is filled up to its very tight cap. Explain why and how a pocket of air above the vinegar prevents the bottle from breaking.A thin wire strung between two nails in the wall is used to support a large picture. Is the wire likely to snap if it is strung tightly or if it is strung so that it sags considerably?Review the relationship between stress and strain. Can you find any similarities between the two quantities?What type of stress are you applying when you press on the ends of a wooden rod? When you pull on its ends?Can compress stress be applied to a rubber band?Can Young’s modulus have a negative value? What about the bulk modulus?If a hypothetical material has a negative bulk modulus, with happens when you squeeze a piece of it?Discuss how you might measure the bulk modulus of a liquid.Note: Unless stated othen.ise the weights of the wires, rods, and other elements are assumed to be negligible. Elastic moduli of selected materials are given In Table 12.1 22.What is meant when a fishing line is designated as “a 10-lb test?”Steel rods are commonly placed in concrete before it sets. What Is the purpose of se rods?sWhen tightening a bolt, you push perpendicularly on a wrench with a force of 165N at a distance of 0.140m from the center of the bolt. How much torque are you exerting relative to the center of the bolt?When opening do you push on it perpendicularly with a force of 55.0 N at a distance of 0.850 m from the hinges. What torque are you exerting relative to the hinges?Find the magnitude of the tension in each supporting cable shown below. In each case, the weight of the suspended body is 100.0 N and the masses of the cables are negligible.What force must be applied at point P to keep the structure shown in equilibrium? The weight of the structure is negligible.Is it possible to apply a force at P to keep in equilibrium the structure shown? The weight of the structure is negligible.Two children push on opposite of a door during ply. Both push horizontally and perpendicular to the door one child pushes with a force of 17.5 N at a distance of 0.600 m from the hinges, and the second child pushes at a distance of 0.450 m. What force must the second child exert to keep the door from moving? Assume friction is negligible.A small 1000-kg SUV has a wheel base of 3.0 m. If 60 if its weight rests on the front wheels is the wagon’s cents of mass?The uniform seesaw is balanced at its center of mass, as seen below. The smaller boy on the right has a mass of 40.0 kg. What is mass of his friend?A uniform plank rests on a level surface as shown below. The plank has a mass of 30 kg and is 6.0 m long. How much mass can be placed at its right end before itps? (Hint: When the board is about to tip over, it makes contact with the surfsce only along the edge that becomes a momentary axis of rotation.)The uniform seesaw shown below is balanced on a fulcrum located 3.0 m from the left end. The smaller boy on the right has a mass of mass of 40 kg and the bigger boy on the left has a mass 80 kg. What is the mass of the board?In order to get his car out of the mud, a man ties one end of a rope to the front bumper and the other end to a tee 15 m away as shown below. He then pulls on the center of the rope with a force of 400 N, which causes its center to be displaced 0.30 m, as shown. What is the force of the rope on the car?A uniform 40.0-kg scaffold of length 60 m is supported by tow light cables below. An 80.0 kg painter stands 1.0 m from the left end of the scaffold, and his painting equipment is 1.5 m from the right end. If the tension in the cable is twice that in the right cable, find the cables and mass of the equipment.When the structure shown below is supported at point P, it is in equilibrium. Find the magnitude .of force F and the force applied at P. the weight of the structure is negligible.To get up on the roof, a person (mass 70.0 kg) places 6.00-m aluminum ladder (mass 10.0 kg) against the house on a concrete pad with the base of ladder 2.00 m from the house. The ladder rests against a plastic rain gutter, which we can assume to frictionless. The center of ladder is 2.00 m from the bottom. The person is standing 3.00 m from the bottom. Find the normal reaction and friction forces on the ladder at its base.A uniform horizontal strut weighs 400.0 N. One end of the strut is attached to a hinged support the wall and the other end of the strut is attached to a sign that weighs 200.0 N The strut is also supported by a cable attached between the end of the strut and the wall. Assuming that the entire weight of the sign is attached at the very end of the s find the tension in the cable and the force at the hinge of the strut.The forearm shown below is positioned at an angle with respect to the upper arm , and a 5.0- kg mass is held in the hand. The total mass of the forearm and hand is 3.0 kg, and their center of mass is 15.0 cm from the elbow. (a) What is the magnitude of the force that the biceps muscle exerts on the forearm for =60 ? (b) What is the magnitude of the force on the elbow joint for the same angle? (c) How do these forces depend on the angle ?The uniform boom shown below weighs 3000N . It is supported by the horizontal guy wire and by the hinged support at point A. What are the forces on the boom due to the wire and due to the support at A? Does the force at A act along the boom?The uniform boom shown below weighs 700N , and the object hanging from its right end weighs 400N . The boom is supported by a light cable and by a hinge at the wall. Calculate the tension in the cable and the force on the hinge on the boom. Does the force on the hinge act along the boom?A 12.0m boom, of a crane lifting a 3000kg load is shown below. The center of mass of the boom is at its geometric center, and the mass of the boom is 1000kg . For the position shown, calculate tension T in the cable and the force at the axle A.A uniform trapdoor shown below is 1.0m by 1.5m and weighs 300N . It is supported by a single hinge (H) , and by a light rope tied between the middle of the door and the floor. The door is held at the position shown, where its slab makes a 30 angle with the horizontal floor and the rope makes a 20 angle with the floor. Find the tension in the rope and the force at the hinge.A 90kg man walks on a sawhorse, as shown below. The sawhorse is 2.0m long and 1.0m high, and its mass is 25.0kg . Calculate the normal reaction force on each leg at the contact point with the floor when the man is 0.5m from the far end of the sawforce. (Hint: At each end, find the total reaction forces first. This reaction force is the vector sum of two reaction forces, each acting along one leg. The normal reaction force at the contact point with the floor is the normal (with respect to the floor) component of this force.)The “lead” in pencils is a graphite composition with a Young’s modulus of approximately 1.0109N/m2 . Calculate the change in length of the lead in an automatic pencil if you tap it straight into the pencil with a force of 4.0N . The lead is 0.50mm in diameter and 60mm long.TV broadcast antennas are the tallest artificial structure on Earth. In 1987 , a 72.0kg physicist placed himself and 400kg of equipment at the top of a 610mhigh antenna to perform gravity experiments. By how much was the antenna compressed, if we consider it to be equivalent to a steel cylinder 0.150m in radius?By how much does a 65.0kg mountain climber stretch her 0.800cm diameter nylon rope when she hangs 35.0m below a rock outcropping? (For nylon, Y=1.35109Pa. )When water freezes, its volume increases by 9.05 . What force per unit area is water capable of exerting on a container when it freezes?A farmer making grape juice fills a glass bottle to the brim and caps it tightly. The juice expands more than the glass when it warms up, in such a way that the volume increases by 0.2 . Calculate the force exerted by the juice per square centimeter if its bulk modulus is 1.8109N/m2 , assuming the bottle does not break.A disk between vertebrae in the spine is subjected to a shearing force of 600.0N . Find its shear deformation, using the shear modulus of 1.0109N/m2 . The disk is equivalent to a solid cylinder 0.700cm high and 4.00cm in diameter.A vertebrae is subjected to a shearing force of 500.0N . Find the shear deformation, taking the vertebrae to be a cylinder 3.00cm high and 4.00cm in diameter. How does your result compare with the result obtained in the preceding problem? Are spinal problems more comman in disks than in vertebrae?Calculate the force a piano tuner applies to stretch a steel piano wire by 8.00mm , if the wire is originally 1.35m long and its diameter is 0.850mm .A 20.0m -tall hollow aluminium flagpole is equivalent in strength to a solid cylinder 4.00cm in diameter. A strong wind bends the pole as much as a horizontal 900.0N force on the top would do. How far to the side does the top of the pole flex?A copper wire of diameter 1.0cm stretches 1.0 when it is used to lift a load upward with an acceleration of 2.0m/s2 . What is the weight of the load?As an oil well is drilled, each new section of drill pipe support its own weight and the weight of the pipe and the drill bit beneath it. Calculate the stretch in a new 6.00m -long steel pipe that supports a 100kg drill bit and a 3.00km length of pipe with a linear mass density of 20.0kg/m . Treat the pipe as a solid cylinder with a 5.00cm diameter.Alarge uniform cylindrical steel rod of density =7.8g/cm3 is 2.0 m long and has a diameter of 5.0 cm. the rod is fastened to a concrete floor with its long axis vertical.what is the normal stress in the rod at the cross-section located at (a)1.0 mfrom its lower end?(b)1.5 m from the lower end?A 90-kg mountain climber bangs from a nylon rope and stretches it by 25.0 cm. If the rope was originally 30.0 m bog and its diameter is 1.0 cm, is Young’s modulus for the nylon?A suspender rod of a suspension bridge is 25.0 m long. If the rod is made of steel, what must its diameter be so that it does not stretch more than 1.0 cm when a 2.5104kg tuck passes by it? Assume that the rod supports all of the weight of the truck.A copper wire is 1.0 m long and it diameter is 1.0 mm. if the wire hangs vertically how much weight must be added to its free end in order to stretch it 3.0mm?A 100-N weight is attached to a free end of a metallic wire that hangs from the ceiling. When a second 100-N weight is added to the wire, it stretches 3.0 mm. The diameter and the length of the wire are 1.0 mm and2.0 m, respectively. What is Young’s modulus of the metal used to manufacture the wire?The bulk modulus of a material is 1.01011N/m2 . What fractional change in volume does a piece of this material undergo when it is subjected to a bulk stress increase of 107N/m2 ? Assume that the force is applied uniformly over the surface.Normal forces of magnitude 1.0106N are applied uniformly to a spherical surface enclosing a volume of a liquid. This causes the radius of the surface to decrease from 50.000 cm to 49.995 cm. What is the bulk modulus of the liquid?During a walk on a rope, a tightrope walker creates a tension of 3.94103N in a wire that are 15.0m apart. The wire has a diameter of 0.50cm when it is not stretched. When the walker is on the wire in the middle between the poles the wire makes an angle of 5.0 below the horizontal. How much does this tension stretch the steel wire when the walker is this position?When using a pencil eraser, you exert a vertical force of 6.00N at a distance of 2.00cm from the hardwood-eraser joint. The pencil is 6.00mm in diameter and is held at an angle of 20.0 to the horizontal. (a) By how much does the wood flex perpendicular to its length? (b) How much is it compressed lengthwise?Normal forces are applied uniformly over the surface of a spherical volume of water whose radius is 20.0cm . If the pressure on the surface is increased by 200MPa , by how much does the radius of the sphere decrease?A uniform rope of cross-sectional area 0.50cm2 breaks when the tensile stress in it reaches 6.00106N/m2 . (a) What is the maximum load that can be lifted slowly at a constant speed by the rope? (b)What is the maximum load that can be lifted by the rope with an acceleration of 4.00m/s2 ?One end of a vertical metallic wire of length 2.0m and diameter 1.0mm is attached to a ceiling, and the other end is attached to a 5.0N weight pan, as shown below. The position of the pointer before the pan is 4.000cm . Different weights are then added to the pan area, and the position of the pointer is recorded in the table shown. Plot stess versus strain for this wire, then the use the resulting curve to determine Young’s modulus and the proportionality limit of the metal. What metal is this most likely to be?An aluminium (=2.7g/cm3) wire is suspended from the ceiling and hangs vertically. How long must the wire be before the stress at its upper end reaches the proportionality limit, which is 8.0107N/m2 ?The coefficient of static friction between the rubber eraser of the pencil and the tabletop is s=0.80 . If the force F is applied along the axis of the pencil, as shown below, what is the minimum angle at which the pencil can stand without slipping? Ignore the weight of the pencil.A pencil rests against a corner, as shown below. The sharpened end of the pencil touches a rough horizontal floor. The coefficient of static friction between the eraser and the floor is s=0.80 . The center of mass the pencil is located 9.0cm from the tip of the eraser and 11.0cm from the tip of the pencil lead. Find the minimum angle for which the pencil does not slip.A uniform 4.0m plank weighing 200.0N rests against the corner of a wall, as shown below. There is no friction at the point where the plank meets the corner. (a) Find the forces that the corner and the floor exert on the plank. (b) What is the minimum coefficient of static friction between the floor and the plank to prevent the plank from slipping?A 40kg boy jumps from a height of 3.0m , lands on one foot and comes to rest in 0.10s after he hits the ground. Assume that he comes to rest with a constant deceleration. If the total cross-sectional area of the bones in his legs just above his ankles is 3.0cm2 , what is the compression stress in these bones? Leg bones can be fractured when they are subjected to stress greater than 1.7108Pa . Is the boy in danger of breaking his leg?Two thin rods, one made of steel and the other of aluminium, are joined end to end. Each rod is 2.0m long and has cross-sectional area 9.1mm2 . If a 10,000N tensile force is applied at each end of the combination, find: (a) stress in each rod; (b) strain in each rod; and, (c) elongation of each rod.Two rods, one made of copper and the other of steel, have the same dimensions. If the copper rod stretches by 0.15mm under some stress, how much does the steel rod stretch under the same stress?A horizontal force F is applied to a uniform sphere in direction exact toward the center of the sphere, as shown below. Find the magnitude of this forcee so that the sphere remains in static equilibrium. What is the frictional force of the incline on the sphere?When a motor is set on a pivoted mount seen below, its weight can be used to maintain tension in the drive belt. When the motor is not running the tensions T1 and T2 are equal. The total mass of the platform and the motor is 100.0 kg, and the diameter of the drive belt pulley is 16.0 cm. When the motor is off, find: (a) the tension in the belt, and (b) the force at trhe hinged platform support at point C. Assume that the center of mass of the motor plus platform is at the center of the motor.Two wheels A and B with weights w and 2w , respectively,are connected by a uniform rod with weight w/2 , as shown below. The wheels are free to roll on the sloped surfaces. Deternine the angle that the rod forms with the horizontal when the system is in equilibrium. Hint: There are five forces acting on the rod, which is two weights of the wheels, two normal reaction forces at points where the wheels make contacts with the wedge, and the weight of the rod.Weights are gradually added to a pan until a wheel of mass M and radius R is pulled over an obstacle of height d, as shown below. What is the minimum mass of the weights plus the pan needed to accomplish this?In order to lift a shovelful of dirt, a gardener pushes downward on the end of the shovel and pulls upward at distance l2 from the end, as shown below. The weight of the shovel is mg and acts at the point os application of F2 . Calculate the magnitudes of the forces F1 and F2 as functions of l1 , l2 , mg, and the weight W of the load. Why do your answers not depend on the angle that shovel makes with the horizontal?A uniform rod of length 2R and mass M is attached to a small collar C and rests on a cylindrical surface of radius R ,as shown bolow. If the collar can slide without friction along the vertical guide, find the angle for which the rod is in static equilibrium.The pole shown below is at a 90.0 bend in a power line and is therefore subjected to more shear force than poles in strsight part of the line. The tension in each line is 4.00104N , at the angles shown. The pole is 15.0 m tall, has an 18.0 cm diameter, and can be considered to have half the strength of hardwood. (a) Calculate the compression of the pole. (b) Find how much it bends and in what direction. (c) Find the tension in a guy wire used to keep the pole straight if it attached to the top of the pole at an angle of 30.0 with the vertical. The guy wire is in the opposite direction of the bend.Check Your Understanding What happens to force and accleration as the vehicles fall together? What wil our estimate of the velocity at a collision higher or lower than the speed actually be? And finally, what would happen if the masses were not identical? Would the force on each be the same or different? How about their accelerations?Check Your Understanding How does your weight at the top of a tall building compare with that on the first floor? Do you think engineers need to take into account the change in the value of g when designing structural support for a very tall building?Check Your Understanding Why not use the simpler expression U=mg(y2y1) ? How significant would the error be? (Recall the previous result, in Example 13.4, that the value g at 400 km above the Earth is 8.67m/s2 .)Check Your Understanding If we send a probe out of the solar system starting form Earth’s surface, do we only have to escape the Sun?Check Your Understanding Assume you are in a spacecraft in orbit about the Sun at Earth’s orbit, but far away from Earth (so that it can be ignored). How could you redirect your tangential velocity to the radial direction such that you could then pass by Mars’s orbit? What would be required to change just the direction of the velocity?Check Your Understanding By what factor must the radius change to reduce the orbital velocity of a satellite by one-half? By what factor would this change the period?Check Your Understanding There is another consideration to this last calculation of ME. We derived Equation 13.8 assuming that the satellite orbits around the center of the astronomical body at the same radius used in the expression for the gravitational force between them. What assumption is made to justify this? Earth is about 81 times more massive than the Moon. Does the Moon orbit about the exact center of Earth? Which is about 17,000 mph. Using Equation 13.8, the period is T=2r3GME=2( 6.37 10 6+4.00 10 5m)3(6.67 10 11N m 2 /kg 2)(5.96 10 24kg)=5.55103sCheck Your Understanding Galaxies are not single objects. How does the gravitiational force of one galaxy exerted on the “closer” stars of the other galaxy compare to those farther away? What effect would this have on the shape of the galaxies themselves?Check Your Understanding The nearly circular orbit of Saturn has an average radius of about 9.5 AU and has a period of 30 years, whereas Uranus averages about 19 AU and has a period of 84 years. Is this consistent with our results for Halley’s comet?Check Your Understanding Earth exerts a tidal force on the Moon. Is it greater than, the same as, or less than that of the Moon on Earth? Be careful in your response, as tidal forces arise from the difference in gravitational forces between one side and the other. Look at the calculations we performed for the tidal force on Earth and consider the values that would change significantly for the Moon. The diameter of the Moon is one-fourth that of Earth. Tidal forces on the Moon are not easy to detect, since there is no liquid on the surface.Check Your Understanding Consider the density required to make Earth a black hole compared to that required for the Sun. What conclusion can you draw from this comparison abut what would be required to create a black hole? Would you expect the Universe to have many black holes with small mass?Action at a distance, such as is the case for gravity, was once thought to be illogical and therefore untrue. What is the ultimate determinant of the truth in science, and why was this action at a distance ultimately accepted?In the law of universal gravitation, Newton assumed that the force was proportional to the product of the two masses (m1m2) . While all scientific conjectures must be experimentally verified, can you provided arguments as to why this must be? (You may wish to consider simple examples in which any other form would lead to contradictory results.)Must engineers take Earth’s rotation into account when constructing very tall buildings at any location other than the equator or very near the poles?It was stated that a satellite with negative total energy is in a bound orbit, whereas one with zero or positive total energy is in an unbounded orbit. Why zero or positive total energy is in an unbounded orbit. Why is this true? What choice for gravitational potential energy was made such that this is true?It was shown that the energy required to lift a satellite into a low Earth orbit (the change in potential energy) is only a small fraction of the kinetic energy needed to keep it in orbit. Is this true for larger orbits? Is there a trend to the ratio of kinetic energy to change in potential energy as the size of the orbit increase?One student argues that a satellite in orbit is in free fall because the satellite keeps falling toward Earth. Another says a satellite in orbit is not in free fall because the acceleration due to gravity is not 9.80m/s2 . With whom do you agree with and why?Many satellites are placed in geosynchronous orbits. What is special about these orbits? For a global communication netword, how many of these satellites would be needed?Are Kepler’s laws purely descriptive, or do they contain causal information?In the diagram below for a satellite in an elliptical orbit about a much larger mass, indicate where its speed is the greatest and where it is the least. What conservatrion law dictates this behavior? Indicate the directions of the force, acceleratin, and velocity at these points. Draw vectors fo these same three quantities at the two points where the y-axis intersects (along the semi-minor axis) and from this determine whether the speed is increasing decreasing, or at a max/min.As an object falls into a black hole, tidal forces increase. Will these tidal forces always tear the object apart as it approaches the Schwarzschild radius? How does the mass of the black hole and size of the object affect your answer?The principle of equivalence states that all experiments done in a lab in a uniform gravitational field cannot be distinguished from those done in a lab that is not in a gravitational field but in uniformly accelerating. For the latter case, consider what happens to a laser beam at some height shot perfectly horizontally to the floor, across the accelerating lab. (View this from a nonaccelerating frame outside the lab.) Relative to the height of the laser, where will the laser beam hit the far wall? What does this say about the effect of a gravitational field on light? Does the fact that light has no mass make any difference to the argument?As a person approaches the Schwarzschild radius fo a black hole, outside observers see all the processes of that person (their clocks, their heart rate, etc.) slowing down, and coming to a halst as they reach the Schwarzschild radius. (The person falling into the black hole sees their own processes unaffected.) But the speed of light is the same everywhere for all observers. What does this say about space as you approach the black hole?Evaluate the magnitude of gravitational force between two 5-kg spherical steel balls separated by a center-to-center distance of 15 cm.Estimate the gravitational force between two sumo wrestlers, with masses 220 kg and 240 kg, when they are embraced and their centers are 1.2 m apart.Astrology makes much of the position of the planets at the moment of one’s birth. The only known force a planet exerts on Earth is gravitational. (a) Calculate the gravitational force exerted on a 4.20-kg baby by a 100-kg father 0.200 m away at birth (he is assisting so he is close to the child). (b) Calculate the force on the baby due to jupiter if it is at its closest distance to Earth, some 6.291011m away. How does the force of Jupiter on the baby compare to the force of the father on the baby? Other objects in the room and the hospital building also exert similar gravitational forces. (Of course, there could be an unknown force acting, but scientists first need to be convinced that there is even an effect, much less that an unknown force causes it.)A mountain 10.0 km from a person exerts a gravitational force on him equal to 2.00 of his weight. (a) Calculate the mass of the mountain. (b) Compare the mountain’s mass with that of Earth. (c) What is unreasonable or inconsistent? (Note that accurate gravitational measurements can easily detect the effect of nearby mountains and variations in local geology.)The International Space Station has a mass of approximately 370,000 kg. (a) What is the force on a 150-kg suited astronaut if she is 20 m from the center of mass of the station? (b) How accurate do you think your answer would be?Asteroid Toutatis passed near Earth in 2006 at four times the distance to our Moon. This was the closest approach we will have until 2060. If if has mass of 5.01013kg , what force did it exert on Earth at its closest approach?(a) What was the acceleration of Earth caused by asteroid Toutatis (see previous problem) at its closest approach? (b) What was the acceleration of Toutatis at this point?(a) Calculate Earth’s mass given the acceleratioln due to gravity at the North Pole is measured to be 9.832m/s2 and the radius of the Earth at the pole is 6356 km. (b) Compare this with the NASA’s Earth Fact Sheet value of 5.97261024kg .(a) What is the acceleration due to gravity on the surface of the Moon? (b) On the surface of Mars? The mass of Mars is SW 6.4181023kg and its radius is 3.38106m .(a) Calculate the acceleration due to gravity on the surface of the Sun. (b) By what factor would your weight increase if you could stand on the Sun? (Never mind that you cannot.)The mass of a particle is 15 kg. (a) What is its weight on Earth? (b) What is its weight on the Moon? (c) What is its mass on the Moon? (d) What is its weight in outer space far from any celestial body? (e) What is its mass at this point?On a planet whose radius is 1.2107m , the acceleration due to gravity is 18m/s2 . What is the mass of the planet?The mean diameter of the planet Saturn is 1.2108m , and its mean mass density is 0.69g/cm3 . Find the acceleratin due to gravity at Saturn’s surface.The mean diameter of the planet Mercury is 4.88106m , and the acceleration due to gravity at its surface is 3.78m/s2 . Estimate the mass of this planet.The acceleration due to gravity on the surface of a planet is three times as large as it is on the surface of Earth. The mass density of the planet is known to be twice that of Earth. What is the radius of this planet in terms of Earth’s radius?A body on the surface of a planet with the same radius as Earth’s weighs 10 times more than it does on Earth. What is the mass of this planet in terms of Earth’s mass?Find the escape speed of a projectile from the surface of Mars.Find the escape speed of a projectile from the surface of Jupiter.What is the escape speed of a satellite located at the Moon’s orbit about Earth? Assume the Moon is not nearby.(a) Evaluate the gravitational potential energy between two 5.00-kg spherical steel balls separated by a center-to-center distance of 15.0 cm. (b) Assuming that they are both initially at rest relative to each other in deep space, use conservation of energy to find how fast will they be traveling upon impact. Each shpere has a radius of 5.10 cm.An average-sized asteroid located 5.0107km from Earth with mass 2.01013kg is detected headed directly toward Earth with speed of 2.0km/s . What will its speed be just before it hits our atmosphere? (You may ignore the size of the asteroid.)(a) What will be the kinetic energy of the asteroid in the previous problem just before it hits Earth? (b) Compare this energy to the output of the largest fission bomb, 2100 TJ. What impact would this have on Earth?(a) What is the change in energy of a 1000-kg payload taken from rest at the surface of Earth and placed at rest on the surface of the Moon? (b) What would be the answer if the payload were taken from the Moon’s surface to Earth? Is this a reasonable calculation of the energy needed to move a payload back and forth?If a planet with 1.5 times the mass of Earth was traveling in Earth’s orbit, what would its period be?Two planets in circular orbits around a star have speed of v and 2v . (a) What is the ratio of the orbital radii of the planets? (b) What is the ratio of their periods?Using the average distance of Earth from the Sun, and the orbital peirod of Earth, (a) find the centripetal acceleration of Earth in its motion about the Sun. (b) Compare this value to that of the centripetal acceleration at the equator due to Earth’s rotation.What is the orbital radius of an Earth satellite having a period of 1.00 h? (b) What is unreasonable about this result?Calculate the mass of the Sun based on data for Earth’s orbit and compare the value obtained with the Sun’s actual mass.Find the mass of Jupiter based on the fact that I0 , its innermost moon, has an average orbital radius of 421,700 km and a period of 1.77 days.Astronomical observatrions of our Milky Way galaxy indicate that it has a mass of about 8.01011 solar masses. A star orbiting on the galaxy’s periphery is about 6.0104 light-years from its center. (a) What should the orbital period of that star be? (b) If its period is 6.0107 years instead, what is the mass of the galaxy? Such calculations are used to imply the existence of other matter, such as a very massive black hole at the center of the Milky Way.(a) In order to keep a small satellite from drifting into a nearby asteroid, it is placed in orbit with a period of 3.02 hours and radius of 2.0 km. What is the mass of the asteroid? (b) Does this mass seem reasonable for the size of the orbit?The Moon and Earth rotate about their common center of mass, which is located about 4700 km from the center of Earth. (This is 1690 km below the sufrace.) (a) Calculate the acceleration due to the Moon’s gravity at that point. (b) Calculate the centripetal accelereation of he center of Earth a sit rotates about that point once each lunar month (bout 27.3 d) and compare it with the acceleration found in part (a). Comment on whether or not they are equal and why they should or should not be.The Sun orbits the Milky Way galaxy once each 2.60108 years, with a roughly circular orbit averaging a radius of 3.00104 light-years. (A light-year is the distance traveled by light in 1 year.) Calculate the centripetal accleration of the Sun in its galactic orbit. Does yur result support the contention that a nearly inertial frame of reference can be located at the Sun? (b) Calculate the average speed of the Sun in its galactic orbit. Does the answer surprise you?A geosynchronous Earth satellite is one that has an orbital period of precisely 1 day. Such orbits are sueful for communication and weather observation because the satellite remains above the same point on Earth (provided it orbits in the equatorial plane in the same direction as Earth’s rotation). Calculate the radius of such an orbit based on the data for Earth in Appendis D.Calculate the mass of the Sun based on data for average Earth’s orbit and compare the value obtained with the Sun’s commonly listed value of 1.9891030kg .I0 orbits Jupiter with an average radius of 421,700 km and a period of 1.769 days. Based upon these data, what is tha mass of Jupiter?The “mean” orbital radius listed for astronomical objects orbiting the Sun is typically not an integrated average but is calculated such that it gives the correct period when applied to the equation for circular orbits. Given that, what is the mean orbital radius in terms of aphelion and perihelion?The perihelion of Halley’s comet is 0.586 AU and the aphelion is 17.8 AU. Given that its speed at perihelion is 55 km/s, what is the speed at aphelion ( IAU=1.4961011m )? (Hint: You may use either conservation of energy or angular momentum, but the latter is much easier.)The perihelion of the comet Legerkvist is 2.61 AU and it has a period of 7.36 years. Show that the aphelion for this comet is 4.95 AU.What is the ratio of the speed at perihelion to that at aphelion for the comet Lagerkvist in the previous problem?Eros has an elliptical orbit about the Sun, with a perihelion distance of 1.13 AU and aphelion distance of 1.78 AU. What is the period of its orbit?What is the difference between the force on a 1.0-kg mass on the near side of I0 has mean radius of 1821 km and a mean orbital radius about Jupiter of 421,700 km. (b) Compare this difference to that calculated for the difference for Earth due to the Moon calculated in Example 13.14. Tidal forces are the cause of I0 ’s volcanic activity.If the Sun were to collapse into a black hole, the point of no return for an investigator would be approximately 3 km from the center singularity. Would the investingator be able to survive visiting even 300 km from the center? Answer this by finding the difference in the gravitatoinal attraction the black holes exerts on a 1.0-kg mass at the head and at the feet of the investigator.Consider Figure 13.23 in Tidal Forces. This diagram represents the tidal forces for spring tides. Sketch a similar diagram for neap tides. (Hint: For simplicity, imagine that the Sun and the Moon contribute equally. Your diagram would be the vector sum of two force fields (as in Figure 13.23), reduced by a factor of two, and superimposed at right angles.)What is the Schwarzschild radius for the black hole at the center of our galaxy if it has the mass of 4 million solar masses?What would be the Schwarzschild radius, in light years, if our Milky Way galaxy of 100 billion stars collapsed into a black hole? Compare this to our distance from the center, about 13,000 light years.A neutron star is a cold, collapsed star with nuclear density. A particular neutron star has a mass twice that of our Sun with a radius of 12.0 km. (a) What would be the weight of a 100-kg astronaut on standing on its surface? (b) What does this tell us about landing on a neutron star?(a) How far from the center of Earth would the net gravitational force of Earth and the Moon on an object be zero? (b) Setting the magnitudes of the forces equal should result in two answers from the quadratic. Do you understand why there are two positions, but only one where the net force is zero?How far from the center of the Sun would the net gravitational force of Earth and the Sun on a spaceship be zero?Calculate the values of g at Earth’s surface for the following changes in Earth’s properties: (a) its mass is doubled and its radius is halved; (b) its mass density is doubled and its radius is unchanged; (c) its mass density is halved and its mass is unchanged.Suppose you can communicate with the inhabitants of a planet in another solar system. They tell you that on their planet, whose diameter and mass are 5.0103km and 3.61023kg , respectively, the record for the high jump is 2.0 m. Given that this record is close to 2.4 m on Earth, what would you conclude about your extraterrestrial friends’ jumping ability?(a) Suppose that your measured weight at the equator is one-half your measured weight at the pole on a planet whose mass and diameter are equal to those of Earth. What is the rotational period of the planet? (b) Would you need to take the shape of this planet into account?A body of mass 100 kg is weighed at the North Pole and at the equator with a spring scale. What is the scale reading at these two points? Assume that g=9.83m/s2 at the pole.Find the speed needed to escape from the solar system starting from the surface of Earth. Assume there are no other bodies involved and do not account for the fact that Earth is moving in its orbit. [Hint: Equation 13.6 does not apply. Use Equation 13.5 and include the potential energy of both Earth and the Sun. Substituting the values for Earth’s mass and radius directly into Equation 13.6, we obtain vesc=2GMR=2(6.67 10 11Nm2/kg2)(5.96 10 24kg)(6.37 106m)=1.12104m/s That is about 11 km/s or 25,000 mph. To escape the Sun, starting from Earth’s orbit, we use R=RES=1.501011m and MSum=1.991030kg . The result is vesc=4.21104m/s or about 42 km/s. We have 12mvesc2GMmR=12m02GMm=0 Solving for the escape velocity,Consider the previous problem and include the fact that Earth has an orbital speed about the Sun of 29.8km/s . (a) What speed relative to Earth would be needed and in what direction should you leave Earth? (b) What will be the shape of the trajectory?A comet is observed 1.50 AU from the Sun with a speed of 24.3km/s . Is this comet in a bound or unbound orbit?An asteroid has speed 15.5km/s when it is located 2.00 AU from the sun. At its closest approach, it is 0.400 AU from the Sun. What is its speed at that point?Space debris left from old satellites and their launchers is becoming a hazard to other satellites. (a) Calculate the speed of a satellite in an orbit 900 km above Earth’s surface. (b) Suppose a loose rivet is in an orbit of the same radius that intersects the satellite’s orbit at an angle of 90 . What is the velocity of the rivet relative to the satellite just before striking it? (c) If its mass is 0.500 g, and it comes to rest inside the satellite, how much energy in joules is generated by the collision? (Assume the satellite’s velocity does not change appreciably, because it mass is much greater than the rivets’s.)A satellite of mass 1000 kg is in circular orbit about Earth. The radius of the orbit of the satellite is equal to two times the radius of Earth. (a) How far away is the satellite? (b) Find the kinetic, potential, and total energies of the satellite.After Cares was promoted to a dwarf planet, we now recognize the largest known asteroid to be Vesta, with a mass of 2.671020kg and a diameter ranging from 578 km to 458 km. Assuming that Vesta is spherical with radius 520 km, find the approximate escape velocity from its surface.(a) Using the data in the previous problem for the asteroid Vesta which has a diameter of 520 km and mass of 2.671020kg , what would be the orbital period for a space probe in a circular orbit of 10.0 km from its surface? (b) Why is this calculation marginally useful at best?What is the orbital velocity of our solar system about the center of the Milky Way? Assume that the mass within a sphere of radius equal to our distance away from the center is about a 100 billion solar masses. Our distance from the center is 27,000 light years.(a) Using the information in the previous problem, what velocity do you need to escape the Milky Way galaxy from our present position? (b) Would you need to accelerate a spaceship to this speed relative to Earth?Circular orbits in Equation 13.10 for conic sections must have eccentricity zero. From this, and using Newton’s second law applied to centripeta acceleration, show that the value of in Equation 13.10 is given by Where is the angular momentum of the orbiting body. The value of is constant and given by this expression regardless of the type of orbit.Show that for eccentricity equal to one in Equation 13.10 for conic sections, the path is a parabola. Do this by substituting Cartersian coordinates, x and y, for the polar coordinates, r and , and showing that it has the general form for a parabola, x=ay2+by+c .Using the technique shown in Satellite Orbits and Energy, show that two masses m1 and m2 in circular orbits about their common center of mass, will have total energy E=K+E=K1+K2Gm1m2r=Gm1m22r . We have shown the kinetic energy of both masses explicitly. (Hint: The masses orbit at radii r1 and r2 , respectively, where r=r1+r2 . Be sure not to confuse the radius needed for centripetal acceleration with that for the gravitational force.)Given the perihelion distance, p , and aphelion distance, q , for an elliptical orbit, show that the velocity at perihelion, vp , is given by vp=2GMSun(q+p)qp . (Hint: Use conservation of angular momentum to relate vp and vq , and then substitute into the conservation fo energy equation.)Comet P/1999 R1 has a perihelion of 0.0570 AU and aphelion of 4.99 AU. Using the results of the previous problem, find its speed at aphelion. (Hint: The expression is for the perihelion. Use symmetry ot rewrite the expression for aphelion.)A tunnel is dug through the center of a perfectly spherical and airless planet fo radius R. Using the expression for g derived in Gravitation Near Earth’s Surface for a uniform density, show that a particle of mass m dropped in the tunnel will execute simple harmonic motion. Deduce the period of oscillation of m and show that it has the same period as an orbit the surface.Following the technique used in Gravitation Near Earth’s Surface, find the value of g as a function of the radius r from the center of a spherical shell planet of constant density with inner and outer radii Rin and Rout . Find g for both eq and for RinrRout . Assuming the inside of the shell is kept airless, describe travel inside the spherical shell planet.Show that the areal velocity for a circular orbit of radius r about a mass M is At=12GMr . Does your expression give the correct value for Earth’s areal vilocity about the Sun?Show that the period of orbit for two masses, m1 and m2 , in circular orbits of radii r1 and r2 , respectively, about their common center-of mass, is given by T=2r3G(m1+m2) where r=r1+r2 . (Hint: The masses orbit at radii r1 and r2 , respectively where r=r1+r2 . Use the expression for the center-of-mass to relate the two radii and note that the two masses must have equal but opposite momenta. Start with the relationship of the period to the circumference and speed of orbit for one of the masses. Use the result of the previous problem using momenta in the expression for the kinetic energy.)Show that for small changes in height h, such that hRE , Equation 13.4 reduces to the expression U=mgh .Using Figure 13.9, carefull sketch a free body diagram for the case of a simple pendulum hanging at latitude lambda, labeling all forces acting on the point mass,m. Set up the equations of motion for equilibrium, setting one coordinate in the direction of the centripetal accleration (toward P in the diagram), the other perpendicular to that. Show that the deflection angle , defined as the angle between the pendulum string and the radial direction toward the center of Earth, is given by the expression below. What is the deflection angle at latitude 45 degrees? Assume that Earth is a perfect sphere. tan(+)=gg2REtan , where is the angular velocity of Earth.(a) Show that tidal force on a small object of mass m, defined as the difference in the gravitational force that would be exerted on m at a distance at the near and the far side of the object, due to the gravitational at a distance R from M, is given by Ftidal=2GMmR3r where r is the distance between the near and far side and rR .(b) Assume you are fallijng feet first into the black hole at the center of our galaxy. It has mass of 4 million solar masses. What would be the difference between the force at your head and your feet at the Schwarzschild radius (event horizon)? Assume your feet and head each have mass 5.0 kg and are 2.0 m apart. Would you survive passing through the event horizon?Find the Hohmann transfer velocities, vEllipseEarth and vEllipseMars ,needed for a trip to Mars. Use Equation 13.7 to find the circular orbital velocities for Earth and Mars. Using Equation 13.4 and the total energy of the ellips (with semi-major asix a), given by E=GmMs2a , find the velocities at Earth (perihelion) and at Mars (aphelion) required to be on the transfer ellipse. The difference, v , at each point is the velocity boost or transfer velocity needed.Check Your Understanding If the reservoir in Example 14.1 covered twice the area, but was kept to the same depth, would the dam need to be redesigned?Check Your Understanding Mercury is a hazardous substance. Why do you suppose mercury is typically used in barometers instead of a safer fluid such as water?Check Your Understanding Would a hydraulic press still operate properly if a gas is used instead of a liquid?Which of the following substances are fluids at room temperature and atmospheric pressure: air, mercury, water, glass?Why are gases easier to compress tan liquids and solids?Explain how the density of air varies with altitude.The images show a glass of ice water filled to the brim. Will the water overflow when the ice melts? Explain answer.How is pressure related to sharpness of a knife its ability to cut?Why is a force exerted by a static fluid on a surface always perpendicular to the surface?Imagine a remote location near the Nott Pole, a chunk of Ice floats a lake. Next to lake, a glacier with the same volume as ice sits on land. If both chunks of ice should melt due to rising global temperatures, and the melted ice all goes into the lake, which one would cause the level of the lake to rise the most? Explain.In ballet, dancing en pointe (on the tips of the toes) is much harder on the toes normal dancing or walking. Explain why, in terms of pressure.Atmospheric pressure exerts a large force (equal to the weight of the atmosphere above your body—about 10 tons) on the top of your body when you are lying on the beach sunbathing. Why are you able to get up?Why does atmospheric pressure decrease more rapidly than linearly with altitude?The image shows how sandbags placed around a leak outside a river levee can effectively stop the flow of water under the levee. Explain how the small amount of water inside the column of sandbags is able to balance the much larger body of behind the levee.Is there a net force on a dam due to atmospheric pressure? Explain your answer.Does atmospheric pressure add to the gas pressure in a rigid tank? In a toy balloon? When, in general, does atmospheric pressure not affect the total pressure in a fluid?You can break a strong wine bottle by pounding a cork into it with your fist, but the cork must press directly against the liquid filling the bottle—there can be no air between the cork and liquid. Explain why bottle breaks only if is no air between the cork and liquid.Explain why the fluid reaches equal levels on either side of a manometer if both sides are open to the atmosphere, even if the tubes are of different diameters.Suppose the master cylinder in a hydraulic system is at a greater height than the cylinder it is controlling. Explain how this affect the force produced at be cylinder that is being controlled.More force is required to pull the plug in a full bathtub than when it is empty. Does this contradict Archimedes' principle? Explain your answer.Do fluids exert buoyant forces in a “weightless" environment, such as in the space shuttle? Explain your answer.Will the same ship float higher in salt water than in freshwater? Explain you answer.Marbles dropped into a partially filled bathtub sink to the bottom. Part of their weight is supported by buoyant force, yet the downward force on the bottom of the tub increases by exactly the weight of the marbles. Explain why.Mary figures in the show streamlines. Explain why fluid velocity is greatest where streamlines are closest together. (Hint: Consider the relationship between fluid velocity and the cross-sectional area through which the fluid flows.)You can squirt water from a garden hose a considerably greater distance by partially covering the opening with your thumb. Explain this works.