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All Textbook Solutions for University Physics Volume 1

Shown below is a small ball of mass m attached to a string of length a. A small peg is located a distance h below the point where the string is supported. If the ball is released when the string is horizontal, show that h must be greater than 3 a /5 if the ball is to swing completely around the peg.A block leaves a frictionless inclined surface horizontally after dropping off by a height h. Find the horizontal distance D where it will land on the, in terms of h, and h .A block of mass m, after sliding down a frictionless incline, strikes another block of mass M that is attached to a spring of spring constant k (see below). The blocks stick together upon impact and travel together. (a) Find the compression of the spring in terms of m, M, h, g, and k when the combination comes to rest. Hint: The speed of the combined blocks m + M ( v2 ,) is based on the speed of block in just prior to the collision with the block M ( v1 ) based on the equation v2=(m/m)=M(v1).< This will be discussed further in the chapter on Linear Momentum and Collisions. (b) The loss of kinetic energy as a result of the bonding of the two masses upon impact is stored in the so-called binding energy of the two masses. Calculate the binding energy.A block of mass 300 g is attached to a spring of spring constant 100 N/m. The other end of the spring is attached to a support while the block rests on a smooth horizontal table and can slide freely without any friction. The block is pushed horizontally till the spring compresses by 12 cm, and then the block is released from rest. (a) How much potential energy was stored in the block-spring support system when the block was just released? (b) Determine the speed of the block when it crosses the point when the spring is neither compressed nor stretched. (c) Determine the speed of the block when it has traveled a distance of 20 cm from where it was released.Consider a block of mass 0.200 kg attached to a spring of spring constant 100 N/m. The block is placed on a frictionless table, and the other end of the spring is attached to the wall so that the spring is level with the table. The block is then pushed in so that the spring is compressed by 10.0 cm. Find the speed of the block as it crosses (a) the point when the spring is not stretched, (b) 5.00 cm to the left of point in (a), and (c) 5.00 cm to the right of point in (a).A skier starts from rest and slides downhill. What will be the speed of the skier if he drops by 20 meters in vertical height? Ignore any air resistance (which will, in reality, be quite a lot), and any friction between the skis and the snow.Repeat the preceding problem, but this time, suppose that the work done by air resistance cannot be ignored. Let the work done by the air resistance when the skier goes from A to B along the given hilly path be —2000 J. The work done by air resistance is negative since the air resistance acts in the opposite direction to the displacement. Supposing the mass of the skier is 50 kg, what is the speed of the skier at point B ?Two bodies are interacting by a conservative force Show that the mechanical energy of an isolated system consisting of two bodies interacting with a conservative force is conserved. (Hint: Start by using Newton’s third law and the definition of work to find the work done on each body by the conservative force.)In an amusement park, a car rolls in a track as shown below. Find the speed of the car at A, B, and C. Note that the work done by the rolling friction is zero since the displacement of the point at which the rolling friction acts on the tires is momentarily at rest and therefore has a zero displacement.A 200-g steel ball is tied to a 2.00m “massless” string and hung from the ceiling to make a pendulum, and then, the ball is brought to a position making a 30 angle with the vertical direction and released from rest. Ignoring the effects of the air resistance, find the speed of the ball when the string (a) is vertically down, (b) makes an angle of 20 with the vertical and (c) makes an angle of 10 with the vertical.A 300 g hockey puck is shot across an ice-covered pond. Before the hockey puck was hit, the puck was at rest. After the hit, the puck has a speed of 40 m/s. The puck comes to rest after going a distance of 30 m. (a) Describe how the energy of the puck changes over time, giving the numerical values of any work or energy involved. (b) Find the magnitude of the net friction force.A projectile of mass 2 kg is fired with a speed of 20 m/s at an angle of 30 with respect to the horizontal. (a) Calculate the initial total energy of the projectile given that the reference point of zero gravitational potential energy at the launch position. (b) Calculate the kinetic energy at the highest vertical position of the projectile. (c) Calculate the gravitational potential energy at the highest vertical position. (d) Calculate the maximum height that the projectile reaches. Compare this result by solving the same problem using your knowledge of projectile motion.An artillery shell is fired at a target 200 m above the ground. When the shell is 100 m in the air, it has a speed of 100 m/s. What is its speed when it hits its target? Neglect air friction.How much energy is lost to a dissipative drag force if a 60-kg person falls at a constant speed for 15 meters?A box slides on a frictionless surface with a total energy of 50 J. It hits a spring and compresses the spring a distance of 25 cm from equilibrium. If the same box with the same initial energy slides on a rough surface, it only compresses the spring a distance of 15 cm, how much energy must have been lost by sliding on the rough surface?Check Your Understanding The U.S. Air Force uses “10gs” (an acceleration equal to 109.8m/s2 ) as the maximum acceleration a human can withstand (but only tot several seconds) aid survive. How much time must the Enterprise spend accelerating if the humans on board are to experience an average at most 10gs of acceleration? (Assume the inertial dampeners are offline.)Check Your Understanding What if we had assumed the phone did bounce on impact? Would this have increased the force on the iPhone, decreased it, or made no difference?Check Your Understanding Suppose the second, smaller cart had been initially moving to the left. What would the sign of the final velocity have been in this case?Check Your Understanding Would the ball’s change of momentum have been larger, smaller, or the same, if it had collided with the floor and stopped (without bouncing)? Would the ball’s change of momentum have been larger, smaller, or the same, if it had collided with the floor and stopped (without bouncing)?Check Your Understanding Even if there were some friction on the ice, it is still possible to use conservation of momentum to solve this problem, but you would need to imposed an additional condition on the problem. What is that additional condition?Check Your Understanding The changes of momentum for Philae and for Comet 67/P were equal (in magnitude). Were the impulses experienced by Philae and the comet equal? How about the forces? How about the changes of kinetic energies?Check Your Understanding There is a second solution to the system of equations solved in this example (because the energy equation is quadratic): v1.f=-2.5m/s , v2.f=0 . This solution is unacceptable on physical grounds; what’s with it?Check Your Understanding Suppose there had been no friction (the collision happened on ice); that would make k zero, and thus vc.f=2kgd=0 , which is obviously wrong. What is the mistake in this conclusion?Check Your Understanding Suppose the initial velocities were not at right angles to each other. How would this change both the physical result and the mathematical analysis of the collision?Check Your Understanding Notice that the mass of the air in the tank was neglected in the analysis and solution. How would the solution method changed if the air was included? How large difference do you think if would make in the final answer?Check Your Understanding Suppose we included the sun in the system. Approximately where would the center of mass of the Earth-moon-sun system be located? (Feel free to actually calculate it.)Check Your Understanding Suppose you have a macroscopic salt crystal (that is, a crystal that is large enough to be visible with your unaided eye). It is made up of a huge number of unit cells. Is the center of mass of this crystal necessarily at the geometric center of the crystal?Check Your Understanding How would the firework display change in deep space, far away from any source of gravity?Check Your Understanding What is the physical difference ( or relationship) between dmdt and dmgdt in this example?An object that has a small mass and an object that has a large mass have the same momentum. Which object has the largest kinetic energy?An object that has a small mass and an object that has a large mass have the same kinetic energy. Which mass has the largest momentum?Is it possible for a small force to produce a larger impulse on a given object than a large force? Explain.Why is a 10-m fall onto concrete far more dangerous than a 10-m fall onto water?What external force is responsible for changing the momentum of a car moving along a horizontal road?A piece of putty and a tennis ball with the same mass are thrown against a wall with the same velocity. Which object experience a greater impulse from the wall or are the impulses equal? Explain.Under what circumstances is momentum conserved?Can momentum be conserved for a system if there are external forces acting on the system? If so, under what conditions? If not, why not?Explain in terms of momentum and Newton’s laws how a car’s air resistance is due in part to the fact that it pushes air in its direction of motion.Can objects in a system have momentum while the momentum of the system is zero? Explain your answer.A sprinter accelerates out of the starting blocks. Can you consider him as a closed system? Explain.A rocket in deep space (zero gravity) accelerates by firing hot gas out of its thrusters. Does the rocket constitute a closed system? Explain.Two objects of equal mass are moving with equal and opposite velocities when they collide. Can all the kinetic energy be lost in the collision?Describe a system for which momentum is conserved but mechanical energy is not. Now the reverse: Describe a system for which kinetic energy is conserved but momentum is not.Momentum for a system can be conserved in one direction while not being conserved in another. What is the angle between the directions? Give an example.Suppose a fireworks shell explodes, breaking into three large pieces for which air resistance is negligible. How does the explosion affect the motion of the center of mass? How would it be affected if the pieces experienced significantly more air resistance than the intact shell?It is possible for the velocity of a rocket to be greater than the exhaust velocity of the gases it ejects. When that is the case, the gas velocity and gas momentum are in the same direction as that of the rocket. How is the rocket still able to obtain thrust by ejecting the gases?An elephant and a hunter are having a confrontation. a. Calculate the momentum of the 2000.0-kg elephant charging the hunter at a speed of 7.50 m/s. b. Calculate the ratio of the elephant’s momentum to the momentum of a 0.0400-kg tranquilizer dart fired at a speed of 600 m/s. c. What is the momentum of the 90.0-kg hunter running at 7.40 m/s after missing the elephant?A skater of mass 40 kg is carrying a box of mass 5 kg. The skater has a speed of 5 m/s with respect to the floor and is gliding without any friction on a smooth surface. a. Find the momentum of the box with respect to the floor. b. Find the momentum of the box with respect to the floor after she puts the box down on the frictionless skating surface.A car of mass 2000 kg is moving with a constant velocity of 10 m/s due east. What is the momentum of the car?The mass of Earth is 5.971024kg and its orbital radius is an average of 1.501011m . Calculate the magnitude of its linear momentum at the location in the diagram.If a rainstorm drops 1 cm of rain over an area of 10km2 in the period of 1 hour, what is the momentum of the rain that falls in one second? Assume the terminal velocity of a raindrop is 10 m/s.What is the average momentum of an avalanche that moves a 40-cm-thick layer of snow over an area of 100 m by 500 m over a distance of 1 km down a hill in 5.5 s? Assume a density of 350kg/m3 for the snow.What is the average momentum of a 70.0-kg sprinter who runs the 100-m dash in 9.65 s?A 75.0-kg person is riding in a car moving at 20.0 m/s when the car runs into a bridge abutment (see the following figure). a. Calculate the average force on the person if he is stopped by a padded dashboard that compresses an average of 1.00 cm. b. Calculate the average force on the person if he is stopped by an air bag that compresses an average of 15.0 cm.One hazard of space travel is debris left by previous missions. There are several thousand objects orbiting Earth that are large enough to be detected by radar, but there are far greater numbers of very small objects, such as flakes of paint. Calculate the force exerted by a 0.100-mg chip of paint that strikes a spacecraft window at a relative speed of 4.00103m/s , given the collision lasts 6.00108s .A cruise ship with a mass of 1.00107kg strikes a pier at a speed of 0.750 m/s. It comes to rest after traveling 6.00 m, damaging the ship, the pier, and the tugboat captain’s finances. Calculate the average force exerted on the pier using the concept of impulse. (Hint: First calculate the time it took to bring the ship to rest, assuming a constant force.)Calculate the final speed of a 110-kg rugby player who is initially running at 8.00 m/s butt collides head-on with a padded goalpost and experiences a backward force of l.76104N for 5.50102s .Water from a fire hose is directed horizontally against a wall at a rate of 50.0 kg/s and a speed of 42.0 m/s. Calculate the force exerted on the wall, assuming the water’s horizontal momentum is reduced to zero.A 0.450-kg hammer is moving horizontally at 7.00 m/s when it strikes a nail and comes to rest after driving the nail 1.00 cm into a board. Assume constant acceleration of the hammer-nail pair. a. Calculate the duration of the impact. b. What was the average force exerted on the nail?31P The x-component of a force on a 46-g golf ball by a 7-iron versus time is plotted in the following figure: Find the x-component of the impulse during the intervals i. [0,50ms] , and ii. [50ms,100ms] Find the change in the x-component of the momentum during the intervals iii. [0,50ms] , and iv. [50ms,100ms]A hockey puck of mass 150 g is sliding due east on a frictionless table with a speed of 10 m/s. Suddenly, a constant force of magnitude 5 N and direction due north is applied to the puck for 1.5 s. Find the north and east components of the momentum at the end of the 1.3-s interval.A ball of mass 250 g is thrown with an initial velocity of 25 m/s at an angle of 30 with the horizontal direction. Ignore air resistance. What is the momentum of the ball after 0.2 s? (Do this problem by finding the components of the momentum first, and then constructing the magnitude and direction of the momentum vector from the components.)35P Two identical pucks collide elastically on an air hockey table. Puck 1 was originally at rest; puck 2 has an incoming speed of 6.00 m/s and scatters at an angle of 30 with respect to its incoming direction. What is the velocity (magnitude and direction) of puck 1 after the collision?The figure below shows a bullet of mass 200 g traveling horizontally towards the east with speed 400 m/s, which strikes a block of mass 1.5 kg that is initially at rest on a frictionless table. After striking the block, the bullet is embedded in the block and the block and the bullet move together as one unit. a. What is the magnitude and direction of the velocity of the block/bullet combination immediately after the impact? b. What is the magnitude and direction of the impulse by the block on the bullet? c. What is the magnitude and direction of the impulse from the bullet on the block? d. If it took 3 ms for the bullet to change the speed from 400 m/s to the final speed after impact, what is the average force between the block and the bullet during this time?A 20-kg child is coasting at 3.3 m/s over flat ground in a 4.0-kg wagon. The child drops a 1.0-kg ball out the back of the wagon. What is the final speed of the child and wagon?A 4.5 kg puffer fish expands to 40 of its mass by taking in water. When the puffer fish is threatened, it releases the water toward the threat to move quickly forward. What is the ratio of the speed of the puffer fish forward to the speed of the expelled water backwards?Explain why a cannon recoils when it fires a shell.Two figure skaters are coasting in the same direction, with the leading skater moving at 5.5 m/s and the trailing skating moving at 6.2 m/s. When the trailing skater catches up with the leading skater, he picks her up without applying any horizontal forces on his skates. If the trailing skater is 50 heavier than the 50-kg leading skater, what is their speed after he picks her up?A 2000-kg railway freight car coasts at 4.4 m/s underneath a grain terminal, which dumps grain directly down into the freight car. If the speed of the loaded freight car must not go below 3.0 m/s, what is the maximum mass of grain that it can accept?A 5.50-kg bowling ball moving at 9.00 m/s collides with a 0.850-kg bowling pin, which is scattered at an angle to the initial direction of the bowling ball and with a speed of 15.0 m/s. a. Calculate the final velocity (magnitude and direction) of the bowling ball. b. Is the collision elastic?Ernest Rutherford (the first New Zealander to be awarded the Nobel Prize in Chemistry) demonstrated that nuclei were very small and dense by scattering helium-4 nuclei from gold-197 nuclei. The energy of the incoming helium nucleus was 8.001013J , and the masses of the helium and gold nuclei were 6.681027kg and 3.291025kg , respectively (note that their mass ratio is 4 to 197). a. If a helium nucleus scatters to an angle of 120 during an elastic collision with a gold nucleus, calculate the helium nucleus’s final speed and the final velocity (magnitude and direction) of the gold nucleus. b. What is the final kinetic energy of the helium nucleus?A 90.0-kg ice hockey player hits a 0.150-kg puck, giving the puck a velocity of 45.0 m/s. If both are initially at rest and if the ice is frictionless, how far does the player recoil in the time it takes the puck to reach the goal 15.0 m away?A 100-g firecracker is launched vertically into the air and explodes into two pieces at the peak of its trajectory. If a 72-g piece is projected horizontally to the left at 20 m/s, what is the speed and direction of the other piece?In an elastic collision, a 400-kg bumper car collides directly from behind with a second, identical bumper car that is traveling in the same direction. The initial speed of the leading bumper car is 5.60 m/s and that of the trailing car is 6.00 m/s. Assuming that the mass of the drivers is much, much less than that of the bumper cars, what are their final speeds?Repeat the preceding problem if the mass of the leading bumper car is 30.0 greater than that of the trailing bumper car.An alpha particle (4He) undergoes an elastic collision with a stationary uranium nucleus (235U) . What percent of the kinetic energy of the alpha particle is transferred to the uranium nucleus? Assume the collision is one- dimensional.You are standing on a very slippery icy surface and throw a 1-kg football horizontally at a speed of 6.7 m/s. What is your velocity when you release the football? Assume your mass is 65 kg.A 35-kg child rides a relatively massless sled down a hill and then coasts along the flat section at the bottom, where a second 35-kg child jumps on the sled as it passes by her. If the speed of the sled is 3.5 m/s before the second child jumps on, what is its speed after she jumps on?A boy sleds down a hill and onto a frictionless ice- covered lake at 10.0 m/s. In the middle of the lake is a 1000-kg boulder. When the sled crashes into the boulder; he is propelled backwards from the boulder. The collision is an elastic collision. If the boy’s mass is 40.0 kg and the sled’s mass is 2.50 kg, what is the speed of the sled and the boulder after the collision?53P A billiard ball, labeled 1, moving horizontally strikes another billiard ball, labeled 2, at rest. Before impact, ball 1 was moving at a speed of 3.00 m/s, and after impact it is moving at 0.50 m/s at 50 from the original direction. If the two balls have equal masses of 300 g, what is the velocity of the ball 2 after the impact?A projectile of mass 2.0 kg is fired in the air at an angle of 40.0 to the horizon at a speed of 50.0 m/s. At the highest point in its flight, the projectile breaks into three parts of mass 1.0 kg, 0.7 kg, and 0.3 kg. The 1.0-kg part falls straight down after breakup with an initial speed of 10.0 m/s, the 0.7-kg part moves in the original forward direction, and the 0.3-kg part goes straight up. Launch a. Find the speeds of the 0.3-kg and 0.7-kg pieces immediately after the break-up. b. How high from the break-up point does the 0.3-kg piece go before coming to rest? c. Where does the 0.7-kg piece land relative to where it was fired from?Two asteroids collide and stick together. The first asteroid has mass of 15103kg and is initially moving at 770 m/s. The second asteroid has mass of 20103kg and is moving at 1020 m/s. Their initial velocities made an angle of 20 with respect to each other. What is the final speed and direction with respect to the velocity of the first asteroid?57P A proton traveling at 3.0106m/s scatters elastically from an initially stationary alpha particle and is deflected at an angle of 85 with respect to its initial velocity. Given that the alpha particle has four times the mass of the proton, what percent of its initial kinetic energy does the proton retain after the collision?59P A family is skating. The father (75 kg) skates at 8.2 m/s and collides and sticks to the mother (50 kg), who was initially moving at 3.3 m/s and at 45 with respect to the father’s velocity. The pair then collides with their daughter (30 kg), who was stationary, and the three slide off together. What is their final velocity?61P Two cars of the same mass approach an extremely icy four-way perpendicular intersection. Car A travels northward at 30 m/s and car B is travelling eastward. They collide and stick together, traveling at 28 north of east. What was the initial velocity of car B?Three point masses are placed at the corners of a triangle as shown in the figure below. Find the center of mass of the three-mass system.Two particles of masses m1 and m2 separated by a horizontal distance D are released from the same height h at the same time. Find the vertical position of the center of mass of these two particles at a time before the two particles strike the ground. Assume no air resistance.Two particles of masses m1 and m2 separated by a horizontal distance D are let go from the same height h at different times. Particle 1 starts at t=0 , and particle 2 is let go at t=T . Find the vertical position of the center of mass at a time before the first particle strikes the ground. Assume no air resistance.Two particles of masses m1 and m2 , move uniformly in different circles of radii R1 and R2 R2 about origin in the x, y-plane. The x- and y-coordinates of the center of mass and that of particle 1 are given as follows (where length is in meters and tin seconds): x1(t)=4cos(2t) , y1(t)=4sin(2t) and: xCM(t)=4cos(2t) , yCM(t)=3sin(2t) . a. Find the radius of the circle in which particle 1 moves. b. Find the x- and y-coordinates of particle 2 and the radius of the circle this particle moves.Two particles of masses m1 and m2 move uniformly in different circles of radii R1 and R1 about the origin in the x, y-plane. The coordinates of the two particles in meters are given as follows ( z=0 for both). Here t is in seconds: x1(t)=4cos(2t) y1(t)=4sin(2t) x2(t)=2cos(3t2) y2(t)=2sin(3t2) a. Find the radii of the circles of motion of both particles. b. Find the x- and y-coordinates of the center of mass. c. Decide if the center of mass moves in a circle by plotting its trajectory.Find the center of mass of a one-meter long rod, made of 50 cm of iron (density 8gcm3 ) and 50 cm of aluminum (density 2.7gcm3 ).Find the center of mass of a rod of length L whose mass density changes from one end to the other quadratically. That is, if the rod is laid out along the x-axis with one end at the origin and the other end at x=L , the density is given by (x)=0+(10)(xL)2 , where 0 and 1 are constant values.Find the center of mass of a rectangular block of length a and width b that has a nonuniform density such that when the rectangle is placed in the x, y-plane with one corner at the origin and the block placed in the first quadrant with the two edges along the x- and y-axes, the density is given by (x,y)=0x , where 0 is a constant.Find the center of mass of a rectangular material of length a and width b made up of a material of nonuniform density. The density is such that when the rectangle is placed in the xy-plane, the density is given by (x,y)=0xy .A cube of side a is cut out of another cube of side b as shown in the figure below. Find the location of the center of mass of the structure. (Hint: Think of the missing part as a negative mass overlapping a positive mass.)Find the center of mass of a cone of uniform density that has a radius R at the base, height h, and mass M. Let the origin be at the center of the base of the cone and have +z going through the cone vertex.Find the center of mass of a thin wire of mass in and length L bent in a semicircular shape. Let the origin be at the center of the semicircle and have the wire arc from the +x axis, cross the +y axis, and terminate at the -x axis.Find the center of mass of a uniform thin semicircular plate of radius R. Let the origin be at the center of the semicircle, the plate arc from the +x axis to the -x axis, and the z axis be perpendicular to the plate.Find the center of mass of a sphere of mass M and radius R and a cylinder of mass in, radius r, and height h arranged as shown below. Express your answers in a coordinate system that has the origin at the center of the cylinder.(a) A 5.00-kg squid initially at rest ejects 0.250 kg of fluid with a velocity of 10.0 m/s. What is the recoil velocity of the squid if the ejection is done in 0.100 s and there is a 5.00-N frictional force opposing the squid’s movement? (b) How much energy is lost to work done against friction?A rocket takes off from Earth and reaches a speed of 100 m/s in 10.0 s. If the exhaust speed is 1500 m/s and the mass of fuel burned is 100 kg, what was the initial mass of the rocket?Repeat the preceding problem but for a rocket that takes off from a space station, where there is no gravity other than the negligible gravity due to the space station.How much fuel would be needed for a 1000-kg rocket (this is its mass with no fuel) to take off from Earth and reach 1000 m/s in 30 s? The exhaust speed is 1000 m/s.What exhaust speed is required to accelerate a rocket in deep space from 800 m/s to 1000 m/s in 5.0 s if the total rocket mass is 1200 kg and the rocket only has 50 kg of fuel left?Unreasonable Results Squids have been reported to jump from the ocean and travel 30.0 m (measured horizontally) before re-entering the water. (a) Calculate the initial speed of the squid if it leaves the water at an angle of 20.0 , assuming negligible lift from the air and negligible air resistance. (b) The squid propels itself by squirting water. What fraction of its mass would it have to eject in order to achieve the speed found in the previous part? The water is ejected at 12.0 m/s; gravitational force and friction are neglected. (c) What is unreasonable about the results? (d) Which premise is unreasonable, or which premises are inconsistent?Two 70-kg canoers paddle in a single, 50-kg canoe. Their paddling moves the canoe at 1.2 m/s with respect to the water, and the river they’re in flows at 4 m/s with respect to the land. What is their momentum with respect to the land?Which has a larger magnitude of momentum: a 3000-kg elephant moving at 40 km/h or a 60-kg cheetah moving at 112 km/h?A driver applies the brakes and reduces the speed of her car by 20 , without changing the direction in which the car is moving. By how much does the car’s momentum change?You friend claims that momentum is mass multiplied by velocity, so things with more mass have more momentum. Do you agree? Explain.Dropping a glass on a cement floor is more likely to break the glass than if it is dropped from the same height on a grass lawn. Explain in terms of the impulse.Your 1500-kg sports car accelerates from 0 to 30 m/s in 10 s. What average force is exerted on it during this acceleration?A ball of mass m is dropped. What is the formula for the impulse exerted on the ball from the instant it is dropped to an arbitrary time later? Ignore air resistance.Repeat the preceding problem, but including a drag force due to air of fdrag=bv .A 5.0-g egg falls from a 90-cm-high counter onto the floor and breaks. What impulse is exerted by the floor on the egg?A car crashes into a large tree that does not move. The car goes from 30 m/s to 0 in 1.3 m. (a) What impulse is applied to the driver by the seatbelt, assuming he follows the same motion as the car? (b) What is the average force applied to the driver by the seatbelt?Two hockey players approach each other head on, each traveling at the same speed Vi . They collide and get tangled together, falling down and moving off at a speed Vi/5 . What is the ratio of their masses?You are coasting on your 10-kg bicycle at 15 m/s and a 5.0-g bug splatters on your helmet. The bug was initially moving at 2.0 m/s in the same direction as you. If your mass is 60 kg, (a) what is the initial momentum of you plus your bicycle? (b) What is the initial momentum of the bug? (c) What is your change in velocity due to the collision with the bug? (d) What would the change in velocity have been if the bug were traveling in the opposite direction?A load of gravel is dumped straight down into a 30000-kg freight car coasting at 2.2 m/s on a straight section of a railroad. If the freight car’s speed after receiving the gravel is 1.5 m/s, what mass of gravel did it receive?Two carts on a straight track collide head on. The first cart was moving at 3.6 m/s in the positive x direction and the second was moving at 2.4 m/s in the opposite direction. After the collision, the second car continues moving in its initial direction of motion at 0.24 m/s. If the mass of the second car is 5.0 times that of the first, what is the mass and final velocity of the first car?A 100-kg astronaut finds himself separated from his spaceship by 10 m and moving away from the spaceship at 0.1 m/s. To get back to the spaceship, he throws a 10-kg tool bag away from the spaceship at 5.0 m/s. How long will he take to return to the spaceship?Derive the equations giving the final speeds for two objects that collide elastically, with the mass of the objects being m1 and m2 , and the initial speeds being v1,i and v2,i=0 (i.e., second object is initially stationary).Repeat the preceding problem for the case when the initial speed of the second object is nonzero.A child sleds down a hill and collides at 5.6 m/s into a stationary sled that is identical to his. The child is launched forward at the same speed, leaving behind the two sleds that lock together and slide forward more slowly. What is the speed of the two sleds after this collision?For the preceding problem, find the final speed of the case of each sled for the case of an elastic collision.A 90-kg football player jumps vertically into the air to catch a 0.50-kg football that is thrown essentially horizontally at him at 17 m/s. What is his horizontal speed after catching the ball?Three skydivers are plummeting earthward. They are initially holding onto each other, but then push apart. Two skydivers of mass 70 and 80 kg gain horizontal velocities of 1.2 m/s north and 1.4 m/s southeast, respectively. What is the horizontal velocity of the third skydiver, whose mass is 55 kg?Two billiard balls are at rest and touching each other on a pool table. The cue ball travels at 3.8 m/s along the line of symmetry between these balls and strikes them simultaneously. If the collision is elastic, what is the velocity of the three balls after the collision?105AP 106AP Repeat the preceding problem if the balls collide when the center of ball 1 is at the origin and the center of ball 2 is at the point (0.2R) .Repeat the preceding problem if the balls collide when the center of ball 1 is at the origin and the center of ball 2 is at the point (3R/2,R/2) .Where is the center of mass of a semicircular wire of radius R that is centered on the origin, begins and ends on the x axis, and lies in the x, y plane?Where is the center of mass of a slice of pizza that was cut into eight equal slices? Assume the origin is at the apex of the slice and measure angles with respect to an edge of the slice. The radius of the pizza is R.If 1 of the Earth’s mass were transferred to the Moon, how far would the center of mass of the Earth-Moon-population system move? The mass of the Earth is 5.971024kg and that of the Moon is 7.341022kg . The radius of the Moon’s orbit is about 3.84105m .You friend wonders how a rocket continues to climb into the sky once it is sufficiently high above the surface of Earth so that its expelled gasses no longer push on the surface. How do you respond?To increase the acceleration of a rocket, should you throw rocks out of the front window of the rocket or out of the back window?A 65-kg person jumps from the first floor window of a burning building and lands almost vertically on the ground with a horizontal velocity of 3 m/s and vertical velocity of -9 m/s. Upon impact with the ground he is brought to rest in a short time. The force experienced by his feet depends on whether he keeps his knees stiff or bends them. Find the force on his feet in each case. a. First find the impulse on the person from the impact on the ground. Calculate both its magnitude and direction. b. Find the average force on the feet if the person keeps his leg stiff and straight and his center of mass drops by only 1 cm vertically and 1 cm horizontally during the impact. c. Find the average force on the feet if the person bends his legs throughout the impact so that his center of mass drops by 50 cm vertically and 5 cm horizontally during the impact. d. Compare the results of part (b) and (c), and draw conclusions about which way is better. You will need to find the time the impact lasts by making reasonable assumptions about the deceleration. Although the force is not constant during the impact, working with constant average force for this problem is acceptable.Two projectiles of mass m1 and m2 , are fired at the same speed but in opposite directions from two launch sites separated by a distance D. They both reach the same spot in their highest point and strike there. As a result of the impact they stick together and move as a single body afterwards. Find the place they will land.Two identical objects (such as billiard balls) have a one-dimensional collision in which one is initially motionless. After the collision, the moving object is stationary and the other moves with the same speed as the other originally had. Show that both momentum and kinetic energy are conserved.A ramp of mass M is at rest on a horizontal surface. A small cart of mass m is placed at the top of the ramp and released. What are the velocities of the ramp and the cart relative to the ground at the instant the cart leaves the ramp?Find the center of mass of the structure given in the figure below. Assume a uniform thickness of 20 cm, and a uniform density of 1g/cm3 Check Your Understanding The fan blades on a turbofan jet engine (shown below) accelerate from rest up to a rotation rate of 40.0 rev/s in 20 s. The increase in angular velocity of the fan is constant in time. (The GE90-11OB1 turbofan engine mounted on a Boeing 777, as shown, is currently the largest turbofan engine in the world, capable of thrusts of 330-510 kN.) (a) What is the average angular acceleration? (b) What is the instantaneous angular acceleration at any time during the first 20 s?Check Your Understanding A centrifuge used in DNA extraction spins at a maximum rate of 7000 rpm, producing a “g-force on the sample that is 6000 times the force of gravity. If the centrifuge takes 10 seconds to come to rest from the maximum spin rate: (a) What Is the angular acceleration of the centrifuge? (b) What is the angular displacement of the centrifuge during this time?Check Your Understanding A boy jumps on a merry-go-round with a radius of 5 m that is at rest. It starts accelerating at a constant rate up to an angular velocity of 5 rad/s in 20 seconds. What is the distance travelled by the boy?Check Your Understanding A nuclear submarine propeller has a moment of inertia of 800.0kgm2 . If the submerged propeller has a rotation rate of 4.0 rev/s when the engine is cut, what is the rotation rate of the propeller after 5.0 s when water resistance has taken 50,000 J out of the system?Check Your Understanding What is the moment of inertia of a cylinder of radius R and mass m about an axis through a point on the surface, as shown below? A uniform thin disk about an axis through the center Integrating to find the moment of inertia of a two-dimensional object is a little bit trickier, but one shape is commonly done at this level of study—a uniform thin disk about an axis through its center (Figure 10.27). Figure 10.27 Calculating the moment of inertia for a thin disk about an axis through its center.Check Your Understanding A large ocean-going ship runs aground near the coastline, similar to the fate of the Costa Concordia, and lies at an angle as shown below. Salvage crews must apply a torque to right the ship in order to float the vessel for transport. A force of 5.0105Nacting at point A must be applied to right the ship. What is the torque about the point of contact of the ship with the ground (Figure 10.36)?Check Your Understanding The fan blades on a jet engine have a moment of inertia 30.0kgm2 . In 10s, they rotate counterclockwise from rest up to a rotation rate of 20 rev/s. (a) What torque must be applied to the blades to achieve this angular acceleration? (b) What is the torque required to bring the fan blades rotating at 20 rev/s to a rest in 20s?Check Your Understanding A constant torque of 500kNm is applied to a wind turbine to keep it rotating at 6 rad/s. What is the power required to keep the turbine rotating?A clock is mounted on the wall. As you look at it, what is the direction of the angular velocity vector of the second hand?What is the value of the angular acceleration of the second hand of the clock on the wall?A baseball bat is swung. Do all points on the bat have the same angular velocity? The same tangential speed?The blades of a blender on a counter are rotating clockwise as you look into it from the top. If the blender is put to a greater speed what direction Is the angular acceleration of the blades?If a rigid body has a constant angular acceleration, what is the functional form of the angular velocity In terms of the time variable?If a rigid body has a constant angular acceleration, what is the functional form of the angular position?If the angular acceleration of a rigid body is zero, what is the functional form of the angular velocity?A massless tether with a masses tied to both ends rotates about a fixed axis through the center. Can the total acceleration of the tether mass combination be zero If the angular velocity Is constant?Explain why centripetal acceleration changes the direction of velocity in circular motion but not its magnitude.In circular motion, a tangential acceleration can change the magnitude of the velocity but not its direction. Explain your answer.Suppose a piece of food is on the edge of a rotating microwave oven plate. Does it experience nonzero tangential acceleration, centripetal acceleration, or both when: (a) the plate starts to spin faster? (b) The plate rotates at constant angular velocity? (c) The plate slows to a halt?What if another planet the same size as Earth were put into orbit around the Sun along with Earth. Would the moment of inertia of the system increase, decrease, or stay the same?A solid sphere is rotating about an axis through its center at a constant rotation rate. Another hollow sphere of the same mass and radius is rotating about its axis through the center at the same rotation rate. Which sphere has a greater rotational kinetic energy?If a child walks toward the center of a merry-go-round, does the moment of inertia increase or decrease?A discus thrower rotates with a discus in his hand before letting it go. (a) How does his moment of inertia change after releasing the discus? (b) What would be a good approximation to use in calculating the moment of inertia of the discus thrower and discus?Does increasing the number of blades on a propeller increase or decrease its moment of inertia, and why?The moment of inertia of a long rod spun around an axis through one end perpendicular to its length is mL2/3 . Why is this moment of inertia greater than it would be if you spun a point mass m at the location of the center of mass of the rod (at L/2 ) (that would be mL2/4 )?Why is the moment of inertia of a hoop that has a mass M and a radius R greater than the moment of inertia of a disk that has the same mass and radius?What three factors affect the torque created by a force relative to a specific pivot point?Give an example in which a small force exerts a large torque. Give another example in which a large force exerts a small torque.When reducing the mass of a racing bike, the greatest benefit is realized from reducing the mass of the tires and wheel rims. Why does this allow a racer to achieve greater accelerations than would an identical reduction in the mass of the bicycle’s frame?Can a single force produce a zero torque?Can a set of forces have a net torque that is zero and a net force that is not zero?Can a set of forces have a net force that is zero and a net torque that is not zero?25CQ If you were to stop a spinning wheel with a constant force, where on the wheel would you apply the force to produce the maximum negative acceleration?27CQ Calculate the angular velocity of Earth.A track star runs a 400-m race on a 400-m circular track in 45 s. What is his angular velocity assuming a constant speed?A wheel rotates at a constant rate of 2.0103rev/min . (a) What is its angular velocity in radians per second? (b) Through what angle does it turn in 10 s? Express the solution in radians and degrees.A particle moves 3.0 m along a circle of radius 1.5 m. (a) Through what angle does it rotate? (b) If the particle makes this trip in 1.0 s at a constant speed, what is its angular velocity? (c) What is its acceleration?A compact disc rotates at 500 rev/min. If the diameter of the disc is 120 mm, (a) what is the tangential speed of a point at the edge of the disc? (b) At a point halfway to the center of the disc?Unreasonable results. The propeller of an aircraft is spinning at 10 rev/s when the pilot shuts off the engine. The propeller reduces its angular velocity at a constant 2.0rad/s2 for a time period of 40 s. What is the rotation rate of the propeller in 40 s? Is this a reasonable situation?A gyroscope slows from an initial rate of 32.0 rad/s at a rate of 0.700rad/s2 . How long does it take to come to rest?On takeoff, the propellers on a UAV (unmanned aerial vehicle) increase their angular velocity form rest at a rate of =(25.0t)rad/s for 3.0 s. (a) What is the instantaneous angular velocity of the propellers at t=2.0s ? (b) What is the angular acceleration?The angular position of a rod varies as 20.0t2radians from time t=0 . The rod has two beads on it as shown in the following figure, one at 10 cm from the ratation axis and the other at 20 cm from the rotation axis. (a) What is the instantaneous angular velocity of the rod at t=5s ? (b) What is the angular acceleration of the rod? (c) What are the tangential speeds of the beads at t=5s ? (d) What are the tangential acceleration of the beads at t=5s ? (e) What are the centripetal accelerations of the beads at t=5s ?A wheel has a constant angular acceleration of 5.0rad/s2 . Starting from rest, it turns through 300 rad. (a) What is its final angular velocity? (b) How much time elapses while it turns through the 300 radians?During a 6.0-s time interval, a fly-wheel with a constant angular acceleration turns through 500 radians that acquire an angular velocity of 100 rad/s. (a) What is the angular velocity at the beginning of the 6.0 s? (b) What is the angular acceleration of the fly-wheel?The angular velocity of a rotating rigid body increases from 500 to 1500 rev/min in 120 s. (a) What is the angular acceleration of the body? (b) Through what angle does it turn in this 120 s?A flywheel slows from 600 to 400 rev/min while rotating through 40 revolutions. (a) What is the angular acceleration of the flywheel? (b) How much time elapses during the 40 revolutions?A wheel 1.0 m in diameter rotates with an angular acceleration of 4.0rad/s2 . (a) If the wheel’s initial angular velocity is 2.0 rad/s, what is its angular velocity after 10 s? (b) Through what angle does it rotate in the 10-s interval? (c) What are the tangential speed and acceleration of a point on the rim of the wheel at the end of the 10-s interval?A vertical wheel with a diameter of 50 cm starts from rest and rotates with a constant angular acceleration of 5.0rad/s2 around a fixed axis through its center counterclockwise. (a) Where is the point that is initially at the bottom of the wheel at t=10s ? (b) What is the point’s linear acceleration at this instant?A circular disk of radius 10 cm has a contant angular acceleration of 1.0rad/s2 ; at t=0 its angular velocity is 2.0 rad/s. (a) Determine the disk’s angular velocity at t=5.0s (b) What is the angle it has rotated through during this time? (c) What is the tangential acceleration of a point on the disk at t=5.0s ?The angular velocity vs. time for a fan on a hovercraft is shown below. (a) What is the angle through which the fan blades rotate in the first 8 seconds? (b) Verify your result using the kinematic equations.A rod of length 20 cm has two beads attached to its ends. The rod with beads starts rotating from rest. If the beads are to have a tangential speed of 20 m/s in 7 s, what is the angular acceleration of the rod to achieve this?At its peak, a tornado is 60.0 m in diameter and carries 500 km/h winds. What is its angular velocity in revolutions per second?A man stands on a merry-go-round that is rotating at 2.5 rad/s. If the coefficient of static friction between the man’s shoes and the merry-go-round is s=0.5 , how far from the axis of rotation can he stand without sliding?An ultracentrifuge accelerates from to 100,000 rpm in 2.00 min. (a) What is the average angular acceleration in rad/s2 ? (b) What is the tangential acceleration of a point 9.50 cm from the axis of rotation? (c) What is the centripetal acceleration in m/s2 and multiples of g of this point at full rpm? (d) What is the total distance travelled by a point 9.5 cm from the axis of totation of the ultracentrifuge?A wind turbine is rotating conterclockwise at 0.5 rev/s and slows to a stop in 10 s. Its blades are 20 m in length. (a) What is the angular acceleration of the turbine? (b) What is the centripetal acceleration of the tip fo the blades at t=0s ? (c) What is the magnitude and direction of the total linear acceleration of the tip of the blades at t=0s ?What is (a) the angular speed and (b) the linear speed of a point on Earth’s surface at latitude 30N . Take the radius of the Earth to be 6309 km. (c) At what latitude would your linear speed be 10 m/s?A child with mass 40 kg sits on the edge of a merry-go-round at a distance of 3.0 m from its axis of rotation. The merry-go-round accelerates from rest up to 0.4 rev/s in 10 s. If the coefficient of static friction between the child and the surface of the merry-go-round is 0.6, does the child fall off before 5 s?A bicycle wheel with radius 0.3 m rotates from rest to 3 rev/s in 5 s. What is the magnitude and direction of the total acceleration vector at the edge of the wheel at 1.0 s?The angular velocity of a flywheel with radius 1.0 m varies according to (t)=2.0t . Plot ac(t) and at(t) from t=0 to 3.0 s for r=1.0m . Analyze these results to explain when acat and when acat for a point on the flywheel at a radius of 1.0 m.A system of point particles is shown in the following figure. Each particle has mass 0.3 kg and they all lie in the same place. (a) What is the moment of inertia of the system about the given axis? (b) If the system rotates at 5 rev/s, what is its rotational kinetic energy?(a) Calculate the rotational kinetic energy of Earth on its axis. (b) What is the rotational kinetic energy of Earth in its orbit around the Sun?Calculate the rotational kinetic energy of a 12-kg motorcycle wheel if its angular velocity is 120 rad/s and its inner radius is 0.280 m and outer radius 0.330 m.A baseball pitcher throws the ball in a motion where there is rotation of the forearm about the elbow joint as well as other movements. If the linear velocity of the ball relative to the elbow joint is 20.0 m/s at a distance of 0.480 m from the joint and the moment of inertia of the forearm is 0.500kg-m2 , what is the rotational kinetic energy of the forearm?A diver goes into a somersault during a dive by tucking her limbs. If her rotational kindetic energy is 100 J and her moment of inertia in the tuck is 9.0kgm2 , what is her rotational rate during the somersault?An aircraft is coming in for a landing at 300 meters height when the propeller falls off. The aircraft is flying at 40.0 m/s horizontally. The propeller has rotation rate of 20 rev/s, a moment of inertia fo 70.0kgm2 , and mass of 200 kg. Neglect air resistance. (a) With what translational velocity does the propeller hit the ground? (b) What is the rotation rate of the propeller at impact?If air resistance is present in the preceding problem and reduces the propeller’s rotational kinetic energy at impact by 30 , what is the propeller’s rotation rate at impact?A neutron star of mass 21030kg and radius 10 km rotates with a period of 0.02 seconds. What is its rotational kinetic energy?An electric sander consisting of a rotating disk of mass 0.7 kg and radius 10 cm rotates at 15 rev/sec. When applied to a rough wooden wall the rotation rate decreases by 20 . (a) What is the final rotational kinetic energy of the rotating disk? (a) How much has its rotational kinetic energy decreased?A system consists of a disk of mass 2.0 kg and radius 50 cm upon which is mounted an annular cylinder of mass 1.0 kg with inner radius 20 cm and outer radius 30 cm (see below). The system rotates about an axis through the center of the disk and annular cylinder at 10 rev/s. (a) What is the moment of inertia of the system? (b) What is its rotational kinetic energy?While punting a football, a kicker rotates his leg about the hip joint. The moment of inertia of the leg is 3.75kgm2 and its rotational kinetic energy is 175 J. (a) What is the angular velocity of the leg? (b) What is the velocity of tip of the punter’s shoe if it is 1.05 m from the hip joint?Using the parallel axis theorem, what is the moment of inertia of the rod of mass m about the axis shown below?Find the moment of inertia of the rod in the previous problem by direct integration.A uniform rod of mass 1.0 kg and length 2.0 m is free to rotate about one end (see the following figure). If the rod is released from rest at an angle of 60 with respect to the horizontal, what is the speed of the tip of the rod as it passes the horizontal position?A pendulum consists of a rod of mass 2 kg and length 1 m with a solid sphere at one end with mass 0.3 kg and radius 20 cm (see the following figure). If the pendulum is released from rest at an angle of 30 , what is the angular velocity at the lowest point?A solid sphere of radius 10 cm is allowed to rotate freely about an axis. The sphere is given a sharp blow so that its center of mass starts from the position shown in the following figure with speed 15 cm/s. What is the maximum angle that the diameter makes with the vertical?Calculate the moment of inertia by direct integration of a thin rod of mass M and length L about an axis through the rod L/3 , as shown below. Check your answer with the parallel-axis theorem.Two flywheel fo negligible mass and different radii are bounded together and rotate about common axis (see below). The smaller flywheel of radius 30 cm has a cord that has a pulling force of 50 N on it. What pulling force needs to be applied to the cord connecting the larger flywheel of radius 50 cm such that the combination does not rotate?The cylindrical head bolts on a car are to be tightened with a torque of 62.0Nm . If a mechanic uses a wrench of length 20 cm, what perpendicular force must he exert on the end of the wrench to tighten a bolt correctly?(a) When opening a door, 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? (b) Does it matter if you push at the same height as the hinges? There is only one pair of hinges.When tightening a bolt, you push perpendicularly on a wrench with a force of 165 N at a distance of 0.140 m from the center of the bolt. How much torque are you exerting in newton-meters (relative to the center of the bolt)?What hanging mass must be placed on the cord to keep the pulley from rotating (see the following figure)? The mass on the frictionless plane is 5.0 kg. The inner radius of the pulley is 20 cm and the outer radius is 30 cm.A simple pendulum consists of a massless tether 50 cm in length connected to a pivot and a small mass of 1.0 kg attached at the other end. What is the torque about the pivot when the pendulum makes an angle of 40 with respect to the vertical?Calculate the torque about the z-axis that is out of the page at the origin in the following figure, given that F1=3N , F2=2N , F3=3N , F4=1.8N .A seesaw has length 10.0 m and uniform mass 10.0 kg and is resting at an angle of 30 with respect to the ground (see the following figure). The pivot is located at 6.0 m. What magnitude of force needs to be applied perpendicular to the seesaw at the raised end so as to allow the seesaw to barely start to rotate?A pendulum consists ofa rod of mass 1 kg and length 1 m connected to a pivot with a solid sphere attached at the other end with mass 0.5 kg and radius 30 cm. What is the torque about the pivot when the pendulum makes an angle of 30 with respect to the vertical?A torque of 5.00103Nm is required to raise a drawbridge (see the following figure). What is the tension necessary to produce this torque? Would it be easier to raise the drawbridge it the angle were larger or smaller?A horizontal beam of length 3 m and mass 2.0 kg has a mass of 1.0 kg and width 0.2 m sitting at the end of the beam (see the following figure). What is the torque of the system about the support at the wall?What force must be applied to end of a rod along the x-axis of length 2.0 m in order to produce a torque on the rod about the origin of 8.0k Nm ?What is the torque abot the origin of the force if it is applied at the point whose position is:You have a grindstone (a disk) that is 90.0 kg, has a 0.340-m radius, and is turning at 90.0 rpm, and you press a steel axe against it with a radial force of 20.0 N. (a) Assuming the kinetic coefficient of friction between steel and stone is 0.20, calculate the angular acceleration of the grindstine. (b) How many turns will the stone make before coming to rest?Suppose you exert a force of 180 N tangential to a 0.280-m-radius, 75.0-kg grindstone (a solid disk). (a) What torque is exerted? (b) What is the angular acceleration assuming negligible opposing friction? (c) What is the angular acceleration if there is an opposing frictional force of 20.0 N exerted 1.50 cm from the axis?A flywheel (l=50kgm2) starting from rest acquires an angular velocity of 200.0 rad/s while subject to a constant torque from a motor for 5 s. (a) What is the angular acceleration of the flywhell? (b) What is the magnitude of the torque?A constant torque is applied to a rigid body whose moment of inertia is 4.0kgm2 around the axis of rotation. If the wheel starts from rest and attains an angular velocity of 20.0 rad/s in 10.0 s, what is the applied torque?A torque of 50.0Nm is applied to a grinding wheel (l=20.0kgm2) for 20 s. (a) If it starts from rest, what is the angular velocity of the grinding wheel after the troque is removed? (b) Through what angle does the wheel move while the torque is applied?A flywheel (I=100.0kgm2) rotating at 500.0 rev/min is brought to rest by friction in 2.0 min. What is the frictional torque on the flywheel?A uniform cylindrical grinding wheel of mass 50.0 kg and diameter 1.0 m is turned on by an electric motor. The friction in the bearings is negligible. (a) What torque must be applied to the wheel to bring it from rest to 120 rev/min in 20 revolutions? (b) A tool whose coefficent of kinetic friction with the wheel with a force of 40.0 N. What torque must be supplied by the motor to keep the wheel rotating at a constant angular velocity?Suppose when Earth was created, it was not rotating. However, after the application of a uniform torque after 6 days, it was rotating at 1 rev/day. (a) What was the angular acceleration during the 6 days? (b) What torque was applied to Earth during this period? (c) What force tangent to Earth at its equator would produce this torque?A pulley of moment of inertia 2.0kgm2 is mounted on a wall as shown in the following figure. Light strings are wrapped around two circumferences of the pulley and weights are attached. What are (a) the angular acceleration of the pulley and (b) the linear acceleration of the weights? Assume the following data: r1=50cm , r2=20cm , m1=1.0kg , m2=2.0kg A block of mass 3 kg slides down an inclined plane at an angle of 45 with a massless tether attached to a pulley with mass 1 kg and radius 0.5 m at the top of the incline (see the following figure). The pulley can be approximated as a disk. The coefficient of kinetic friction on the plane is 0.4. What is the acceleration of the block?The cart shown below moves across the table top as the block falls. What is the acceleration of the cart? Neglect friction and assume the following data:A uniform rod of mass and length is held vertically by two strings of negligible mass, as shown below. (a) Immediately after the string is cut, what is the linear acceleration of the free end of the stick? (b) Of the middle of the stick?A thin stick of mass 0.2 kg and length L=0.5m is attached to the rim of a metal disk of mass M=2.0kg and radius R=0.3m . The stick is free to rotate around a horizontal axis through its other end (see the following figure). (a) If the combinatin is related with the stick horizontal, what is the speed of the center of the disk when the stick is vertical? (b) What is the acceleration fo the center of the disk at the stick is released? (c) At the instant the stick passes through the vertical?A wind turbine rotates at 20 rev/min. If its power output is 2.0 MW, what is the torque produced on the turbine from the wind?A clay cylinder of radius 20 cm on a potter’s wheel spins at a constant rate of 10 rev/s. The potter applies a force of 10 N to the clay with his hands where the coefficient of friction is 0.1 between his hands and the clay. What is the power that the potter has to deliver to the wheel to keep it rotating at this constant rate?A uniform cylindrical grindstion has a mass of 10 kg and a radius of 12 cm. (a) What is the rotational kinetic energy of the grindstone when it is rotating at 1.5103rev/min ? (b) After the grindstone’s motor is turned off, a knife blade is pressed against the outer edge coefficient of kinetic friction between the grindstone and the blade is 0.80. Use the work energy theorem to determine how many turns the grindstone makes before it stops.A uniform disk of mass 500 kg and radius 0.25 m is mounted on frictionless bearings so it can rotate freely around a vertical axis through its center (see the following figure). A cord is wrapped around the rim fo the disk and pulled with a force of 10 N. (a) How much work has the force done at the instant the disk has completed three revolutions, starting from rest? (b) Determine the torque due to the force, then calculate the work done by this torque at the instant the disk has completed three revolutions? (c) What is the angular velocity at the instant? (d) What is the power output of the force at that instant?A propeller is accelerated from rest to an angular velocity of 1000 rev/min over a period of 6.0 seconds bya constant torque of 2.0103Nm . (a) What is the moment of inertia of the propeller? (b) What power is being provided to the propeller 3.0 s after it starts rotating?A sphere of mass 1.0 kg and radius 0.5 m is attached to the end of a massless rod of length 3.0 m. The rod rotates about an axis that is at the opposite end of the sphere (see below). The system rotates horizontally about the axis at a constant 400 rev/min. After rotating at this angular speed in a vacuum, air resistance is introduced and provides a force 0.15 N on the sphere opposite to the direction of motion. What is the power provided by air resistance to the system 100.0 s after air resistance is introduced?A uniform rod of length L and mass M is held vertically with one end resting on the floor as shown below. When the rod is released, it rotates around its lower end until it hits the floor. Assuming the lower end of the rod does not slip, what is the linear velocity of the upper end when is hits the floor?An athlete in a gym applies a constant force of 50 N to the pedals of a bicycle to keep the rotation rate of the wheel at 10 rev/s. The length of the pedal arms is 30 cm. What is the power delivered to the bicycle by the athlete?A 2-kg block on a frictionless inclined plane at 45 has a cord attached to a pulley of mass 1 kg and radius 20 cm (see the following figure). (a) What is the acceleration of the block down the plane? (b) What is the work done by the gravitational force to move the block 50 cm?Small bodies of mass m1 and m2 are attached to opposite ends of a thin rigid rod of length L and M. The rod is mounted so that it is free to rotate in a horizontal plane around a vertical axis (see below). What distance d from m1 should the rotational axis be so that minimum amount of work is required to set the rod rotating at an angular velocity ?A cyclist is riding such that the wheels of the bicycle have a rotation rate of 3.0 rev/s. If the cyclist brakes such that the rotation rate of the wheels decrease at a rate of 0.3rev/s2 , how long does it take for the cyclist to come to a complete stop?Calculate the angular velocity of the orbital motion of Earth around the Sun.A phonograph turntable rotating at 331/3rev/min slows down and stops in 1.0 min. (a) What is the turntable’s angular acceleration assuming it is constant? (b) How many revolutions does the turntable make while stopping?With the aid of a string, a gyroscope is accelerated from rest to 32 rad/s in 0.40 s under a constant angular acceleration. (a) What is its angular acceleration in rad/s2 ? (b) How many revolutions does it go through in the process?Suppose a piece of dust has fallen on a CD. If the spin rate of the CD is 500 rpm, and the piece of dust is 4.3 cm from the center, what is the total distance traveled by the dust in 3 minutes? (Ignore accelerations due to getting the CD rotating.)A system of point particles is rotating about a fixed axis at 4 rev/s. The particles are fixed with respect to each other. The masses and distances to the axis of the point particles are m1=0.1kg , r1=0.2m , m2=0.05kg , r2=0.4m , m3=0.5kg , r3=0.01m . (a) What is the moment of inertia of the system? (b) What is the rotational kinetic energy of the system?Calculate the moment of inertia of a skater given the following information. (a) The 60.0-kg skater is approximated as a cylinder that has a 0.110-m radius. b) The skater with arms extended is approximated by a cylinder that is 52.5 kg, has a 0.110-m radius, and has two 0.900-m-long arms which are 3.75 kg each and extend straight out from the cylinder like rods rotated about their ends.A stick of length 1.0 m and mass 6.0 kg is free to rotate about a horizontal axis through the center. Small bodies of masses 4.0 and 2.0 kg are attached to its two ends (see the following figure). The stick is released from the horizontal position. What is the angular velocity of the stick when it swings through the vertical?A pendulum consists of a rod of length 2 m and mass 3 kg with a solid sphere of mass 1 kg and radius 0.3 m attached at one end. The axis of rotation is as shown below. What is the angular velocity of the pendulum at its lowest point if it is released from rest at an angle of 30 ?Calculate the torque of the 40-N force around the axis through O and perpendicular to the plane of the page as shown below.Two children push on opposite sides of a door during play. 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.118AP An automobile engine can produce 200Nm of torque. Calculate the angular acceleration produced if 95.0 of this torque is applied to the drive shaft, axle, and rear wheels of a car, given the following information. The car is suspended so that the wheels can turn freely. Each wheel acts like a 15.0-kg disk that has a 0.180-m radius. The walls of each tire act like a 2.00-kg annular ring that has inside radius of 0.180 m and outside radius of 0.320 m. The tread of each tire acts like a 10.0-kg hoop of radius 0.330 m. The 14.0-kg axle acts like a rod that has a 2.00-cm radius. The 30.0-kg drive shaft acts like a rod that has a 3.20-cm radius.A grindstone with a mass of 50 kg and radius 0.8 m maintains a constant rotation rate of 4.0 rev/s by a motor while a knife is pressed against the edge with a force of 5.0 N. The coefficient of kinetic friction between the grindstone and the blade is 0.8. What is the power provided by the motor to keep the grindstone at the constant rotation rate?The angular acceleration of a rotating rigid body is given by =(2.0-3.0t)rad/s2 . If the body starts rotating from rest at t=0 , (a) what is the angular velocity? (b) Angular position? (c) What angle does it rotate through in 10 s? (d) Where does the vector perpendicular to the axis of rotation indicating 0 at t=0 lie at t=10s ?Earth’s day has increased by 0.002 s in the last century. If this increase in Earth’s period is constant, how long will it take for Earth to come to rest?A disk of mass m , radius R , and area A has a surface mass density =mrAR (see the following figure). What is the moment of inertia of the disk about an axis through the center?Zorch, an archenemy of Rotation Man, decides to slow Earth’s rotation to once per 28.0 h by exerting an opposing force at and parallel to the equator. Rotation Man is not immediately concerned, because he knows Zorch can only exert a force of 4.00107N (a little greater than a Saturn V rocket’s thrust). How long must Zorch push with this force to accomplish his goal? (This period gives Rotation Man time to devote to other villains.)A cord is wrapped around the rim of a solid cylinder of radius 0.25 m, and a constant force of 40 N is exerted on the cord shown, as shown in the following figure. The cylinder is mounted on frictionless bearings, and its moment of inertia is 6.0kgm2 . (a) Use the work energy theorem to calculate the angular velocity of the cylinder after 5.0 m of cord have been removed. (b) If the 40-N force is replaced by a 40-N weight, what is the angular velocity of the cylinder after 5.0 m of cord have unwound?Check Your Understanding A hollow cylinder is on an incline at an angle of 60 . The coefficient of static friction on the surface is s=0.6 . (a) Does the cylinder roll without slipping? (b) Will a solid cylinder roll without slipping?Check Your Understanding A proton spiraling around a magnetic field executes circular motion in the plane of the paper, as shown below. The circular path has a radius of 0.4 m and the proton has velocity 4.0106m/s . What is the angular momentum of the proton about the origin?Check Your Understanding Which has greater angular momentum: a solid sphere of mass m rotating at a constant angular frequency 0 about the z-axis, or a solid cylinder of same mass and rotation rate about the z-axis?Check Your Understanding A merry-go-round at a playground is rotating at 4.0 rev/min. Three children jump on and increase the moment of inertia of the merry-go-round/children rotating system by sw 25 . What is the new rotation rate?Check Your Understanding A top has a precession frequency of 5.0 rad/s on Earth. What is its precession frequency on the Moon?Can a round object released from rest at the top of a frictionless incline undergo rolling motion?A cylindrical can of radius R is rolling across a horizontal surface without slipping. (a) After one complete revolution of the can, what is the distance that its center of mass has moved? (b) Would this distance be greater or smaller if slipping occurred?A wheel is released from the top on an incline. Is the wheel most likely to slip if the incline is steep or gently sloped?Which rolls down an inclined plane faster, a hollow cylinder or a solid sphere? Both have the same mass and radius.A hollow sphere and a hollow cylinder of the same radius and mass roll up an incline without slipping and have the same initial center of mass velocity. Which object reaches a greater height before stopping?Can you assign an angular momentum to a particle without first defining a reference point?For a particle traveling in a straight line, are there any points about which the angular momentum is zero? Assume the line intersects the origin.Under what conditions does a rigid body have angular momentum but not linear momentum?If a particle is moving with respect to a chosen origin it has linear momentum. What conditions must exist for this particle’s angular momentum to be zero about the chosen origin?If you know the velocity of a particle, can you say anything about the particle’s angular momentum?What is the purpose of the small propeller at the back of a helicopter that rotates in the plane perpendicular to the large propeller?Suppose a child walks from the outer edge of a rotating merry-go-round to the inside. Does the angular velocity of the merry-go-round increase, decrease, or remain the same? Explain your answer. Assume the merry-go-round is spinning without friction.As the rope of a tethered ball winds around a pole, what happens to the angular velocity of the ball?Suppose the polar ice sheets broke free and floated toward Earth’s equator without melting. What would happen to Earth’s angular velocity?Explain why stars spin faster when they collapse.

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