When a crate with mass 25.0 kg is placed on a ramp that is inclined at an angle a below the horizontal, it slides down the ramp with an acceleration of \(4.9 \ \mathrm{m/s}^2\). The ramp is not frictionless. To increase the acceleration of the crate, a downward vertical force \(\vec{F}\) is applied to the top of the crate. What must F be in order to increase the acceleration of the crate so that it is \(9.8 \ \mathrm{m/s}^2\)? How does the value of F that you calculate compare to the weight of the crate?

A man sits in a seat that is hanging from a rope. The rope passes over a pulley suspended from the ceiling, and the man holds the other end of the rope in his hands. What is the tension in the rope, and what force does the seat exert on him? Draw a free-body force diagram for the man.

“In general, the normal force is not equal to the weight.” Give an example in which these two forces are equal in magnitude, and at least two examples in which they are not.

A clothesline hangs between two poles. No matter how tightly the line is stretched, it sags a little at the center. Explain why.

You drive a car up a steep hill at constant speed. Discuss all of the forces that act on the car. What pushes it up the hill?

For medical reasons, astronauts in outer space must determine their body mass at regular intervals. Devise a scheme for measuring body mass in an apparently weightless environment.

To push a box up a ramp, which requires less force: pushing horizontally or pushing parallel to the ramp? Why?

A woman in an elevator lets go of her briefcase, but it does not fall to the floor. How is the elevator moving?

A block rests on an inclined plane with enough friction to prevent it from sliding down. To start the block moving, is it easier to push it up the plane or down the plane? Why?

A crate slides up an inclined ramp and then slides down the ramp after momentarily stopping near the top. There is kinetic friction between the surface of the ramp and the crate. Which is greater? (i) The crate’s acceleration going up the ramp; (ii) the crate’s acceleration going down the ramp; (iii) both are the same. Explain.

A crate of books rests on a level floor. To move it along the floor at a constant velocity, why do you exert less force if you pull it at an angle \(\theta\) above the horizontal than if you push it at the same angle below the horizontal?

In a world without friction, which of the following activities could you do (or not do)? Explain your reasoning. (a) Drive around an unbanked highway curve; (b) jump into the air; (c) start walking on a horizontal sidewalk; (d) climb a vertical ladder; (e) change lanes while you drive.

When you stand with bare feet in a wet bathtub, the grip feels fairly secure, and yet a catastrophic slip is quite possible. Explain this in terms of the two coefficients of friction.

You are pushing a large crate from the back of a freight elevator to the front as the elevator is moving to the next floor. In which situation is the force you must apply to move the crate the least, and in which is it the greatest: when the elevator is accelerating upward, when it is accelerating downward, or when it is traveling at constant speed? Explain.

It is often said that “friction always opposes motion.” Give at least one example in which (a) static friction causes motion, and (b) kinetic friction causes motion.

If there is a net force on a particle in uniform circular motion, why doesn’t the particle’s speed change?

A curve in a road has a bank angle calculated and posted for 80 km/h. However, the road is covered with ice, so you cautiously plan to drive slower than this limit. What might happen to your car? Why?

You swing a ball on the end of a lightweight string in a horizontal circle at constant speed. Can the string ever be truly horizontal? If not, would it slope above the horizontal or below the horizontal? Why?

The centrifugal force is not included in the free-body diagrams of Figs. 5.34b and 5.35. Explain why not.

A professor swings a rubber stopper in a horizontal circle on the end of a string in front of his class. He tells Caroline, in the front row, that he is going to let the string go when the stopper is directly in front of her face. Should Caroline worry?

To keep the forces on the riders within allowable limits, many loop-the-loop roller coaster rides are designed so that the loop is not a perfect circle but instead has a larger radius of curvature at the bottom than at the top. Explain.

A tennis ball drops from rest at the top of a tall glass cylinder—first with the air pumped out of the cylinder so that there is no air resistance, and again after the air has been readmitted to the cylinder. You examine multiflash photographs of the two drops. Can you tell which photo belongs to which drop? If so, how?

You throw a baseball straight upward with speed \(v_0\). When the ball returns to the point from where you threw it, how does its speed compare to \(v_0\) (a) in the absence of air resistance and (b) in the presence of air resistance? Explain.

You throw a baseball straight upward. If you do not ignore air resistance, how does the time required for the ball to reach its maximum height compare to the time required for it to fall from its maximum height back down to the height from which you threw it? Explain.

You have two identical tennis balls and fill one with water. You release both balls simultaneously from the top of a tall building. If air resistance is negligible, which ball will strike the ground first? Explain. What if air resistance is not negligible?

A ball is dropped from rest and feels air resistance as it falls. Which of the graphs in Fig. Q5.25 best represents its acceleration as a function of time?

A ball is dropped from rest and feels air resistance as it falls. Which of the graphs in Fig. Q5.26 best represents its vertical velocity component as a function of time, if the +y-direction is taken to be downward?

When a batted baseball moves with air drag, when does the ball travel a greater horizontal distance? (i) While climbing to its maximum height; (ii) while descending from its maximum height back to the ground; (iii) the same for both. Explain in terms of the forces acting on the ball.

“A ball is thrown from the edge of a high cliff. Regardless of the angle at which it is thrown, due to air resistance, the ball will eventually end up moving vertically downward.” Justify this statement.

A 1125 kg car and a 2250 kg pickup truck approach a curve on a highway that has a radius of 225 m. (a) At what angle should the highway engineer bank this curve so that vehicles traveling at 65.0 mi/h can safely round it regardless of the condition of their tires? Should the heavy truck go slower than the lighter car? (b) As the car and truck round the curve at 65.0 mi/h, find the normal force on each one due to the highway surface.

The “Giant Swing” at a county fair consists of a vertical central shaft with a number of horizontal arms attached at its upper end. Each arm supports a seat suspended from a cable 5.00 m long, and the upper end of the cable is fastened to the arm at a point 3.00 m from the central shaft (Fig. E5.50). (a) Find the time of one revolution of the swing if the cable supporting a seat makes an angle of \(30.0^{\circ}\) with the vertical. (b) Does the angle depend on the weight of the passenger for a given rate of revolution?

In another version of the “Giant Swing” (see Exercise 5.50), the seat is connected to two cables, one of which is horizontal (Fig. E5.51). The seat swings in a horizontal circle at a rate of 28.0 rpm (rev/min). If the seat weighs 255 N and an 825 N person is sitting in it, find the tension in each cable.

A steel ball with mass m is suspended from the ceiling at the bottom end of a light, 15.0-m-long rope. The ball swings back and forth like a pendulum. When the ball is at its lowest point and the rope is vertical, the tension in the rope is three times the weight of the ball, so T = 3mg. (a) What is the speed of the ball as it swings through this point? (b) What is the speed of the ball if T = mg at this point, where the rope is vertical?

Rotating Space Stations. One problem for humans living in outer space is that they are apparently weightless. One way around this problem is to design a space station that spins about its center at a constant rate. This creates “artificial gravity” at the outside rim of the station. (a) If the diameter of the space station is 800 m, how many revolutions per minute are needed for the “artificial gravity” acceleration to be \(9.80 \ \mathrm{m/s}^2\)? (b) If the space station is a waiting area for travelers going to Mars, it might be desirable to simulate the acceleration due to gravity on the Martian surface \((3.70 \ \mathrm{m/s}^2)\). How many revolutions per minute are needed in this case?

The Cosmo Clock 21 Ferris wheel in Yokohama, Japan, has a diameter of 100 m. Its name comes from its 60 arms, each of which can function as a second hand (so that it makes one revolution every 60.0 s). (a) Find the speed of the passengers when the Ferris wheel is rotating at this rate. (b) A passenger weighs 882 N at the weight-guessing booth on the ground. What is his apparent weight at the highest and at the lowest point on the Ferris wheel? (c) What would be the time for one revolution if the passenger’s apparent weight at the highest point were zero? (d) What then would be the passenger’s apparent weight at the lowest point?

A small rock with mass m is attached to a light string of length L and whirled in a vertical circle of radius R. (a) What is the minimum speed v at the rock’s highest point for which it stays in a circular path? (b) If the speed at the rock’s lowest point in its circular path is twice the value found in part (a), what is the tension in the string when the rock is at this point?

A 50.0 kg stunt pilot who has been diving her airplane vertically pulls out of the dive by changing her course to a circle in a vertical plane. (a) If the plane’s speed at the lowest point of the circle is 95.0 m>s, what is the minimum radius of the circle so that the acceleration at this point will not exceed 4.00g? (b) What is the apparent weight of the pilot at the lowest point of the pullout?

Stay Dry! You tie a cord to a pail of water and swing the pail in a vertical circle of radius 0.600 m. What minimum speed must you give the pail at the highest point of the circle to avoid spilling water?

A bowling ball weighing 71.2 N (16.0 lb) is attached to the ceiling by a 3.80 m rope. The ball is pulled to one side and released; it then swings back and forth as a pendulum. As the rope swings through the vertical, the speed of the bowling ball is 4.20 m/s. At this instant, what are (a) the acceleration of the bowling ball, in magnitude and direction, and (b) the tension in the rope?

Two ropes are connected to a steel cable that supports a hanging weight (Fig. P5.59). (a) Draw a freebody diagram showing all of the forces acting at the knot that connects the two ropes to the steel cable. Based on your diagram, which of the two ropes will have the greater tension? (b) If the maximum tension either rope can sustain without breaking is 5000 N, determine the maximum value of the hanging weight that these ropes can safely support. Ignore the weight of the ropes and of the steel cable.

An adventurous archaeologist crosses between two rock cliffs by slowly going hand over hand along a rope stretched between the cliffs. He stops to rest at the middle of the rope (Fig. P5.60). The rope will break if the tension in it exceeds \(2.50 \times 10^4 \ \mathrm{N}\), and our hero’s mass is 90.0 kg. (a) If the angle \(\theta\) is \(10.0^{\circ}\), what is the tension in the rope? (b) What is the smallest value \(\theta\) can have if the rope is not to break?

In a repair shop a truck engine that has mass 409 kg is held in place by four light cables (Fig. P5.61). Cable A is horizontal, cables B and D are vertical, and cable C makes an angle of \(37.1^{\circ}\) with a vertical wall. If the tension in cable A is 722 N, what are the tensions in cables B and C?

In Fig. P5.62 a worker lifts a weight w by pulling down on a rope with a force \(\vec{F}\) . The upper pulley is attached to the ceiling by a chain, and the lower pulley is attached to the weight by another chain. Draw one or more free-body diagrams to find the tension in each chain and the magnitude of \(\vec{F}\), in terms of w, if the weight is lifted at constant speed. Assume that the rope, pulleys, and chains have negligible weights.

CP A small block sits at one end of a flat board that is 3.00 m long. The coefficients of friction between the block and the board are \(\mu_s = 0.600\) and \(\mu_k = 0.400\). The end of the board where the block sits is slowly raised until the angle the board makes with the horizontal is \(\alpha_0\), and then the block starts to slide down the board. If the angle is kept equal to \(\alpha_0\) as the block slides, what is the speed of the block when it reaches the bottom of the board?

A horizontal wire holds a solid uniform ball of mass m in place on a tilted ramp that rises \(35.0^{\circ}\) above the horizontal. The surface of this ramp is perfectly smooth, and the wire is directed away from the center of the ball (Fig. P5.64). (a) Draw a free-body diagram of the ball. (b) How hard does the surface of the ramp push on the ball? (c) What is the tension in the wire?

CP A box of mass 12.0 kg sits at rest on a horizontal surface. The coefficient of kinetic friction between the surface and the box is 0.300. The box is initially at rest, and then a constant force of magnitude F and direction \(37.0^{\circ}\) below the horizontal is applied to the box; the box slides along the surface. (a) What is F if the box has a speed of 6.00 m/s after traveling a distance of 8.00 m? (b) What is F if the surface is frictionless and all the other quantities are the same? (c) What is F if all the quantities are the same as in part (a) but the force applied to the box is horizontal?

CP A box is sliding with a constant speed of 4.00 m/s in the +x-direction on a horizontal, frictionless surface. At x = 0 the box encounters a rough patch of the surface, and then the surface becomes even rougher. Between x = 0 and x = 2.00 m, the coefficient of kinetic friction between the box and the surface is 0.200; between x = 2.00 m and x = 4.00 m, it is 0.400. (a) What is the x-coordinate of the point where the box comes to rest? (b) How much time does it take the box to come to rest after it first encounters the rough patch at x = 0?

CP BIO Forces During Chin-ups. When you do a chin-up, you raise your chin just over a bar (the chinning bar), supporting yourself with only your arms. Typically, the body below the arms is raised by about 30 cm in a time of 1.0 s, starting from rest. Assume that the entire body of a 680 N person doing chin-ups is raised by 30 cm, and that half the 1.0 s is spent accelerating upward and the other half accelerating downward, uniformly in both cases. Draw a free-body diagram of the person’s body, and use it to find the force his arms must exert on him during the accelerating part of the chin-up.

CP CALC A 2.00 kg box is suspended from the end of a light vertical rope. A time-dependent force is applied to the upper end of the rope, and the box moves upward with a velocity magnitude that varies in time according to \(v(t) = (2.00 \ \mathrm{m/s}^2)t +(0.600 \ \mathrm{m/s}^3)t^2\). What is the tension in the rope when the velocity of the box is 9.00 m/s?

CALC A 3.00 kg box that is several hundred meters above the earth’s surface is suspended from the end of a short vertical rope of negligible mass. A time-dependent upward force is applied to the upper end of the rope and results in a tension in the rope of T(t) = (36.0 N/s)t. The box is at rest at t = 0. The only forces on the box are the tension in the rope and gravity. (a) What is the velocity of the box at (i) t = 1.00 s and (ii) t = 3.00 s? (b) What is the maximum distance that the box descends below its initial position? (c) At what value of t does the box return to its initial position?

CP A 5.00 kg box sits at rest at the bottom of a ramp that is 8.00 m long and is inclined at \(30.0^{\circ}\) above the horizontal. The coefficient of kinetic friction is \(\mu_k = 0.40\), and the coefficient of static friction is \(\mu_s = 0.43\). What constant force F, applied parallel to the surface of the ramp, is required to push the box to the top of the ramp in a time of 6.00 s?

When a crate with mass 25.0 kg is placed on a ramp that is inclined at an angle a below the horizontal, it slides down the ramp with an acceleration of \(4.9 \ \mathrm{m/s}^2\). The ramp is not frictionless. To increase the acceleration of the crate, a downward vertical force \(\vec{F}\) is applied to the top of the crate. What must F be in order to increase the acceleration of the crate so that it is \(9.8 \ \mathrm{m/s}^2\)? How does the value of F that you calculate compare to the weight of the crate?

A large crate is at rest on a horizontal floor. The coefficient of static friction between the crate and the floor is 0.400. A force \(\vec{F}\) is applied to the crate in a direction \(30.0^{\circ}\) above the horizontal. The minimum value of F required to get the crate to start sliding is 380 N. What is the mass of the crate?

CP An 8.00 kg box sits on a ramp that is inclined at \(33.0^{\circ}\) above the horizontal. The coefficient of kinetic friction between the box and the surface of the ramp is \(\mu_k = 0.300\). A constant horizontal force F = 26.0 N is applied to the box (Fig. P5.73), and the box moves down the ramp. If the box is initially at rest, what is its speed 2.00 s after the force is applied?

CP In Fig. P5.74, \(m_1 = 20.0 \ \mathrm{kg}\) and \(\alpha = 53.1^{\circ}\). The coefficient of kinetic friction between the block of mass \(m_1\) and the incline is \(\mu_k = 0.40\). What must be the mass \(m_2\) of the hanging block if it is to descend 12.0 m in the first 3.00 s after the system is released from rest?

CP You place a book of mass 5.00 kg against a vertical wall. You apply a constant force \(\vec{F}\) to the book, where F = 96.0 N and the force is at an angle of \(60.0^{\circ}\) above the horizontal (Fig. P5.75). The coefficient of kinetic friction between the book and the wall is 0.300. If the book is initially at rest, what is its speed after it has traveled 0.400 m up the wall?

Block A in Fig. P5.76 weighs 60.0 N. The coefficient of static friction between the block and the surface on which it rests is 0.25. The weight w is 12.0 N, and the system is in equilibrium. (a) Find the friction force exerted on block A. (b) Find the maximum weight w for which the system will remain in equilibrium.

A block with mass \(m_1\) is placed on an inclined plane with slope angle \(\alpha\) and is connected to a hanging block with mass \(m_2\) by a cord passing over a small, frictionless pulley (Fig. P5.74). The coefficient of static friction is \(\mu_s\), and the coefficient of kinetic friction is \(\mu_k\). (a) Find the value of \(m_2\) for which the block of mass \(m_1\) moves up the plane at constant speed once it is set in motion. (b) Find the value of \(m_2\) for which the block of mass \(m_1\) moves down the plane at constant speed once it is set in motion. (c) For what range of values of \(m_2\) will the blocks remain at rest if they are released from rest?

DATA BIO The Flying Leap of a Flea. High-speed motion pictures (3500 frames/second) of a jumping \(210 \ \mu g\) flea yielded the data to plot the flea’s acceleration as a function of time, as shown in Fig. P5.78. (See “The Flying Leap of the Flea,” by M. Rothschild et al., Scientific American, November 1973.) This flea was about 2 mm long and jumped at a nearly vertical takeoff angle. Using the graph, (a) find the initial net external force on the flea. How does it compare to the flea’s weight? (b) Find the maximum net external force on this jumping flea. When does this maximum force occur? (c) Use the graph to find the flea’s maximum speed.

Block A in Fig. P5.79 weighs 1.20 N, and block B weighs 3.60 N. The coefficient of kinetic friction between all surfaces is 0.300. Find the magnitude of the horizontal force \(\vec{F}\) necessary to drag block B to the left at constant speed (a) if A rests on B and moves with it (Fig. P5.79a), (b) if A is held at rest (Fig. P5.79b).