Chapter 7, Problem 7.23P

(a) Imagine that a space probe could be fired as a projectile from the Earth's surface with an initial speed of 5.78 x 104 m/s relative to the Sun. What would its speed be when it is very far from the Earth (in m/s)? Ignore atmospheric friction, the effects of other planets, and the rotation of the Earth. (Consider the mass of the Sun in your calculations.) Your response is within 10% of the correct value. This may be due to roundoff error, or you could have a mistake in your calculation. Carry out all intermediate results to at least four-digit accuracy to minimize roundoff error. m/s (b) What If? The speed provided in part (a) is very difficult to achieve technologically. Often, Jupiter is used as a "gravitational slingshot" to increase the speed of a probe to the escape speed from the solar system, which is 1.85 x 104 m/s from a point on Jupiter's orbit around the Sun (if Jupiter is not nearby). If the probe is launched from the Earth's surface at a speed of 4.10 x 10 m/s relative…

As shown in the figure, a roller-coaster track includes a circular loop of radius R in a vertical plane. A car of mass m is released from rest at a height h above the bottom of the circular section and then moves freely along the track with negligible energy loss due to friction. i (a) First suppose the car barely makes it around the loop; at the top of the loop, the riders are upside down and feel weightless. Find the required height h of the release point above the bottom of the loop. (Use any variable or symbol stated above along with the following as necessary: g.) h = (b) If the car is released at some point above the minimum required height, determine the amount by which the normal force on the car at the bottom of the loop exceeds the normal force on the car at the top of the loop. (Consider the moments when the car reaches the top and when it reaches the bottom again. Use any variable or symbol stated above along with the following as necessary: g.) NB - NT = The normal force…

One of the more challenging elements in pairs figure skating competition is the "death spiral" (see the figure below), in which the female figure skater, balanced on one skate, is spun in a circle by the male skater. i The axis of rotation of the pair is vertical and through the toe of the skate on the male skater's leg that is bent backward, the toe being planted into the ice. During the one-armed maneuver first developed in the 1940s, the outstretched arm of the male skater must apply a large force to support a significant fraction of the female skater's weight and also to provide her centripetal acceleration. This force represents a danger to the structure of the wrist of the male skater. (a) Modeling the female skater, of mass 47.0 kg, as a particle, and assuming that the combined length of the two outstretched arms is 129 cm and that arms make an angle of 45.0° with the horizontal, what is the magnitude of the force (in N) exerted by the male skater's wrist if each turn is…

One popular design of a household juice machine is a conical, perforated stainless steel basket 3.30 cm high with a closed bottom of diameter 8.00 cm and open top of diameter 14.40 cm that spins at 16000 revolutions per minute about a vertical axis. Solid pieces of fruit are chopped into granules by cutters at the bottom of the spinning cone. Then the fruit granules rapidly make their way to the sloping surface where the juice is extracted to the outside of the cone through the mesh perforations. The dry pulp spirals upward along the slope to be ejected from the top of the cone. The juice is collected in an enclosure immediately surrounding the sloped surface of the cone. Pulp Motor Spinning basket Juice spout (a) What centripetal acceleration does a bit of fruit experience when it is spinning with the basket at a point midway between the top and bottom? m/s² ---Direction--- (b) Observe that the weight of the fruit is a negligible force. What is the normal force on 2.00 g of fruit at…

A satellite is in a circular orbit around the Earth at an altitude of 3.88 × 106 m. (a) Find the period of the orbit. (Hint: Modify Kepler's third law so it is suitable for objects orbiting the Earth rather than the Sun. The radius of the Earth is 6.38 × 106 m, and the mass of the Earth is 5.98 x 1024 kg.) h (b) Find the speed of the satellite. km/s (c) Find the acceleration of the satellite. m/s² toward the center of the earth

Shown below is a waterslide constructed in the late 1800's. This slide was unique for its time due to the fact that a large number of small wheels along its length made friction negligible. Riders rode a small sled down the chute which ended with a horizontal section that caused the sled and rider to skim across the water much like a flat pebble. The chute was 9.76 m high at the top and 54.3 m long. Consider a rider and sled with a combined mass of 81.0 kg. They are pushed off the top of the slide from point A with a speed of 2.90 m/s, and they skim horizontally across the water a distance of 50 m before coming to rest. 9.76 m Engraving from Scientific American, July 1888 A (a) 20.0 m/ -54.3 m- 50.0 m (b) (a) Find the speed (in m/s) of the sled and rider at point C. 14.14 m/s (b) Model the force of water friction as a constant retarding force acting on a particle. Find the magnitude (in N) of the friction force the water exerts on the sled. 162.2 N (c) Find the magnitude (in N) of the…

A small object with mass 3.60 kg moves counterclockwise with constant angular speed 1.40 rad/s in a circle of radius 2.55 m centered at the origin. It starts at the point with position vector 2.551 m. Then it undergoes an angular displacement of 9.15 rad. (a) What is its new position vector? m (b) In what quadrant is the object located and what angle does its position vector make with the positive x-axis? ---Select--- ✓ at (c) What is its velocity? m/s (d) In what direction is it moving? (Give a negative angle.) ° from the +x direction. (e) What is its acceleration? m/s² (f) What total force is exerted on the object? N

A spring with unstretched length of 14.3 cm has a spring constant of 4.63 N/m. The spring is lying on a horizontal surface, and is attached at one end to a vertical post. The spring can move freely around the post. The other end of the spring is attached to a puck of mass m. The puck is set into motion in a circle around the post with a period of 1.32 s. Assume the surface is frictionless, and the spring can be described by Hooke's law. (a) What is the extension of the spring as a function of m? (Assume x is in meters and m is in kilograms. Do not include units in your answer.) x = Your answer cannot be understood or graded. More Information x Find x (in meters) for the following masses. (If not possible, enter IMPOSSIBLE.) (b) m = 0.0700 kg x Use your result from part (a), and insert the given value for m. m (c) m 0.140 kg × Use your result from part (a), and insert the given value for m. m (d) m = 0.180 kg x Use your result from part (a), and insert the given value for m. m (e) m =…

A spacecraft in the shape of a long cylinder has a length of 100 m, and its mass with occupants is 1 860 kg. It has strayed too close to a black hole having a mass 98 times that of the Sun. The nose of the spacecraft points toward the black hole, and the distance between the nose and the center of the black hole is 10.0 km. 100 m- 10.0 km Black hole (a) Determine the total force on the spacecraft. The total force is determined by the distance from the black hole to the center of gravity of the ship which will be close to the midpoint. N (b) What is the difference in the gravitational fields acting on the occupants in the nose of the ship and on those in the rear of the ship, farthest from the black hole? (This difference in acceleration grows rapidly as the ship approaches the black hole. It puts the body of the ship under extreme tension and eventually tears it apart.) N/kg 2.56e+12