Chapter 5

Two 25.0 -N weights are suspended at opposite ends of a rope that passes over a light, frictionless pulley. The pulley is attached to a chain that goes to the ceiling. (a) What is the tension in the rope? (b) What is the tension in the chain?

\(\mathrm{A} 75.0\) -kg wrecking ball hangs from a uniform heavy-duty chain having a mass of 26.0 \(\mathrm{kg}\) . (a) Find the maximum and minimum tension in the chain. (b) What is the tension at a point three-fourths of the way up from the bottom of the chain?

A picture frame hung against a wall is suspended by two wires attached to its upper corners. If the two wires make the same angle with the vertical, what must this angle be if the tension in each wire is equal to 0.75 of the weight of the frame? (Ignore any friction between the wall and the picture frame.)

A man pushes on a piano with mass 180 \(\mathrm{kg}\) so that it slides at constant velocity down a ramp that is inclined at \(11.0^{\circ}\) above the horizontal floor. Neglect any friction acting on the piano. Calculate the magnitude of the force applied by the man if he pushes (a) parallel to the incline and (b) parallel to the floor.

An astronaut is inside a \(2.25 \times 10^{6} \mathrm{kg}\) rocket that is blasting off vertically from the launch pad. You want this rocket to reach the speed of sound \((331 \mathrm{m} / \mathrm{s})\) as quickly as possible, but you also do not want the astronaut to black out. Medical tests have shown that astronauts are in danger of blacking out at an acceleration greater than 4\(g .\) (a) What is the maximum thrust the engines of the rocket can have to just barely avoid blackout? Start with a free-body diagram of the rocket. (b) What force, in terms of her weight \(w,\) does the rocket exert on the astronatt? Start with a free-body diagram of the astronaut. (c) What is the shortest time it can take the rocket to reach the speed of sound?

A 125 -kg (including all the contents) rocket has an engine that produces a constant vertical force (the thrust) of 1720 \(\mathrm{N}\) . Inside this rocket, a 15.5 -N electrical power supply rests on the floor. (a) Find the acceleration of the rocket. (b) When it has reached an altitude of \(120 \mathrm{m},\) how hard does the floor push on the power supply? (Hint: Start with a free-body diagram for the power supply.)

Genesis Crash. On September \(8,2004,\) the Genesis spacecraft crashed in the Utah desert because its parachute did not open. The 210 -kg capsule hit the ground at 311 \(\mathrm{km} / \mathrm{h}\) and penetrated the soil to a depth of 81.0 \(\mathrm{cm}\) . (a) Assuming it to be constant, what was its acceleration (in \(\mathrm{m} / \mathrm{s}^{2}\) and in \(g^{\prime} \mathrm{s} )\) during the crash? \right. (b) What force did the ground exert on the capsule during the crash? Express the force in newtons and as a multiple of the capsule's weight. (c) For how long did this force last?

A 15.0 -kg load of bricks hangs from one end of a rope that passes over a small, frictionless pulley. A 28.0 kg counterweight is suspended from the other end of the rope, as shown in Fig. E5.15. The system is released from rest. (a) Draw two free-body diagrams, one for the load of bricks and one for the counterweight. (b) What is the magnitude of the upward acceleration of the load of bricks? (c) What is the tension in the rope while the load is moving? How does the tension compare to the weight of the load of bricks? To the weight of the counterweight?

A 8.00 -kg block of ice, released from rest at the top of a \(1.50-\mathrm{m}\)-long frictionless ramp, slides downhill, reaching a speed of 2.50 \(\mathrm{m} / \mathrm{s}\) at the bottom. (a) What is the angle between the ramp and the horizontal? (b) What would be the speed of the ice at the bottom if the motion were opposed by a constant friction force of 10.0 \(\mathrm{N}\) parallel to the surface of the ramp?

A light rope is attached to a block with mass 4.00 kg that rests on a frictionless, horizontal surface. The horizontal rope passes over a frictionless, massless pulley, and a block with mass \(m\) is suspended from the other end. When the blocks are released, the tension in the rope is 10.0 \(\mathrm{N}\) (a) Draw two free-body diagrams, one for the \(4.00-\) kg block and one for the block with mass \(m\) . (b) What is the acceleration of either block? (c) Find the mass \(m\) of the hanging block. (d) How does the tension compare to the weight of the hanging block?

A 750.0 -kg boulder is raised from a quarry 125 \(\mathrm{m}\) deep by a long uniform chain having a mass of 575 kg. This chain is of uniform strength, but at any point it can support a maximum tension no greater than 2.50 times its weight without breaking. (a)What is the maximum acceleration the boulder can have and still get out of the quarry, and (b) how long does it take to be lifted out at maximum acceleration if it started from rest?

A 550 -N physics student stands on a bathroom scale in an 850 -kg (including the student) elevator that is supported by a cable. As the elevator starts moving, the scale reads 450 \(\mathrm{N}\) . (a) Find the acceleration of the elevator (magnitude and direction). (b) What is the acceleration if the scale reads 670 \(\mathrm{N}\) ? (c) If the scale reads zero, should the student worry? Explain. (d) What is the tension in the cable in parts (a) and (c)?

Force During a Jump. An average person can reach a maximum height of about 60 \(\mathrm{cm}\) when jumping straight up from a crouched position. During the jump itself, the person's body from the knees up typically rises a distance of around 50 \(\mathrm{cm}\) .To keep the calculations simple and yet get a reasonable result, assume that the entire body rises this much during the jump. (a) With what initial speed does the person leave the ground to reach a height of 60 \(\mathrm{cm} ?\) (b) Draw a free-body diagram of the person during the jump. (c) In terms of this jumper's weight \(w,\) what force does the ground exert on him or her during the jump?

A \(2540-\mathrm{kg}\) test rocket is launched vertically from the launch pad. Its fuel (of negligible mass) provides a thrust force so that its vertical velocity as a function of time is given by \(v(t)=\) \(A t+B t^{2},\) where \(A\) and \(B\) are constants and time is measured from the instant the fuel is ignited. At the instant of ignition, the rocket has an upward acceleration of 1.50 \(\mathrm{m} / \mathrm{s}^{2}\) and 1.00 s later an upward velocity of 2.00 \(\mathrm{m} / \mathrm{s}\) (a) Determine \(A\) and \(B\) , including their SI units. (b) At 4.00 s after fuel ignition, what is the acceleration of the rocket, and (c) what thrust force does the burning fuel exert on it, assuming no air resistance? Express the thrust in newtons and as a multiple of the rocket's weight. (d) What was the initial thrust due to the fuel?

A 2.00 -kg box is moving to the right with speed 9.00 \(\mathrm{m} / \mathrm{s}\) on a horizontal, frictionless surface. At \(t=0\) a horizontal force is applied to the box. The force is directed to the left and has magnitude \(F(t)=\left(6.00 \mathrm{N} / \mathrm{s}^{2}\right) t^{2}\) . (a) What distance does the box move from its position at \(t=0\) before its speed is reduced to zero? (b) If the force continues to be applied, what is the speed of the box at \(t=3.00 \mathrm{s} ?\)

In emergencies with major blood loss, the doctor will order the patient placed in the Trendelenburg position, in which the foot of the bed is raised to get maximum blood flow to the brain. If the coefficient of static friction between the typical patient and the bed sheets is \(1.20,\) what is the maximum angle at which the bed can be tilted with respect to the floor before the patient begins to slide?

A stockroom worker pushes a box with mass 11.2 \(\mathrm{kg}\) on a horizontal surface with a constant speed of 3.50 \(\mathrm{m} / \mathrm{s}\) . The coefficient of kinetic friction between the box and the surface is 0.20 (a) What horizontal force must the worker apply to maintain the motion? (b) If the force calculated in part (a) is removed, how far does the box slide before coming to rest?

A box of bananas weighing 40.0 \(\mathrm{N}\) rests on a horizontal surface. The coefficient of static friction between the box and the surface is \(0.40,\) and the coefficient of kinetic friction is 0.20 . (a) If no horizontal force is applied to the box and the box is at rest, how large is the friction force exerted on the box? (b) What is the magnitude of the friction force if a monkey applies a horizontal force of 6.0 \(\mathrm{N}\) to the box and the box is initially at rest? (c) What minimum horizontal force must the monkey apply to start the box in motion? (d) What minimum horizontal force must the monkey apply to keep the box moving at constant velocity once it has been started? If the monkey applies a horizontal force of \(18.0 \mathrm{N},\) what is the magnitude of the friction force and what is the box's acceleration?

A 45.0 -kg crate of tools rests on a horizontal floor. You exert a gradually increasing horizontal push on it and observe that the crate just begins to move when your force exceeds 313 \(\mathrm{N}\) . After that you must reduce your push to 208 \(\mathrm{N}\) to keep it moving at a steady 25.0 \(\mathrm{cm} / \mathrm{s}\) (a) What are the coefficients of static and kinetic friction between the crate and the floor? (b) What push must you exert to give it an acceleration of 1.10 \(\mathrm{m} / \mathrm{s}^{2} ?\) (c) Suppose you were performing the same experiment on this crate but were doing it on the moon instead, where the acceleration due to gravity is 1.62 \(\mathrm{m} / \mathrm{s}^{2}\) . (i) What magnitude push would cause it to move? (ii) What would its acceleration be if you maintained the push in part (b)?

Some sliding rocks approach the base of a hill with a speed of 12 \(\mathrm{m} / \mathrm{s} .\) The hill rises at \(36^{\circ}\) above the horizontal and has coefficients of kinetic and static friction of 0.45 and \(0.65,\) respectively, with these rocks. (a) Find the acceleration of the rocks as they slide up the hill. (b) Once a rock reaches its highest point, will it stay there or slide down the hill? If it stays there, show why. If it slides down, find its acceleration on the way down.

A pickup truck is carrying a toolbox, but the rear gate of the truck is missing, so the box will slide out if it is set moving. The coefficients of kinetic and static friction between the box and the bed of the truck are 0.355 and 0.650 , respectively. Starting from rest, what is the shortest time this truck could accelerate uniformly to 30.0 \(\mathrm{m} / \mathrm{s}\) without causing the box to slide? Include a free-body diagram of the toolbox as part of your solution.

(a) If the coefficient of kinetic friction between tires and dry pavement is \(0.80,\) what is the shortest distance in which you can stop an automobile by locking the brakes when traveling at 28.7 \(\mathrm{m} / \mathrm{s}\) (about 65 \(\mathrm{mi} / \mathrm{h} ) ?\) (b) On wet pavement the coefficient of kinetic friction may be only \(0.25 .\) How fast should you drive on wet pavement in order to be able to stop in the same distance as in part (a)? (Note: Locking the brakes is not the safest way to stop.)

A 25.0 -kg box of textbooks rests on a loading ramp that makes an angle \(\alpha\) with the horizontal. The coefficient of kinetic friction is \(0.25,\) and the coefficient of static friction is \(0.35 .\) (a) As the angle \(\alpha\) is increased, find the minimum angle at which the box starts to slip. (b) At this angle, find the acceleration once the box has begun to move. (c) At this angle, how fast will the box be moving after it has slid 5.0 \(\mathrm{m}\) along the loading ramp?

A box with mass \(m\) is dragged across a level floor having a coefficient of kinetic friction \(\mu_{k}\) by a rope that is pulled upward at an angle \(\theta\) above the horizontal with a force of magnitude \(F .\) (a) In terms of \(m, \mu_{k}, \theta,\) and \(g,\) obtain an expression for the magnitude of the force required to move the box with constant speed. (b)Knowing that you are studying physics, a CPR instructor asks you how much force it would take to slide a 90 -kg patient across a floor at constant speed by pulling on him at an angle of \(25^{\circ}\) above the horizontal. By dragging some weights wrapped in an old pair of pants down the hall with a spring balance, you find that \(\mu_{\mathrm{k}}=0.35 .\) Use the result of part (a) to answer the instructor's question.

You throw a baseball straight up. The drag force is proportional to \(v^{2} .\) In terms of \(g,\) what is the \(y\) -component of the ball's acceleration when its speed is half its terminal speed and (a) it is moving up? It is moving back down?

A small car with mass 0.800 kg travels at constant speed on the inside of a track that is a vertical circle with radius 5.00 m (Fig. E5.42). If the normal force exerted by the track on the car when it is at the top of the track (point \(B\) is 6.00 \(\mathrm{N},\) what is the normal force on the car when it is at the bottom of the track (point \(A ) ?\)

A machine part consists of a thin 40.0 -cm-long bar with small 1.15 -kg masses fastened by screws to its ends. The screws can support a maximum force of 75.0 \(\mathrm{N}\) without pulling out. This bar rotates about an axis perpendicular to it at its center. (a) As the bar is turning at a constant rate on a horizontal, frictionless surface, what is the maximum speed the masses can have without pulling out the screws? (b) Suppose the machine is redesigned so that the bar turns at a constant rate in a vertical circle. Will one of the screws be more likely to pull out when the mass is at the top of the circle or at the bottom? Use a free-body diagram to see why. (c) Using the result of part (b), what is the greatest speed the masses can have without pulling a screw?

A flat (unbanked) curve on a highway has a radius of 220.0 \(\mathrm{m} .\) A car rounds the curve at a speed of 25.0 \(\mathrm{m} / \mathrm{s}\) . (a) What is the minimum coefficient of friction that will prevent sliding? (b) Suppose the highway is icy and the coefficient of friction between the tires and pavement is only one-third what you found in part (a). What should be the maximum speed of the car so it can round the curve safely?

A 1125 -kg car and a 2250 -kg pickup truck approach a curve on the expressway that has a radius of 225 \(\mathrm{m}\) . (a) At what angle should the highway engineer bank this curve so that vehicles traveling at 65.0 \(\mathrm{mi} / \mathrm{h}\) 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 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 (Fig. E5.46). Each arm supports a seat suspended from a cable 5.00 m long, the upper end of the cable being fastened to the arm at a point 3.00 \(\mathrm{m}\) from the central shaft. (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?

A small button placed on a horizontal rotating platform with diameter 0.320 \(\mathrm{m}\) will revolve with the platform when it is brought up to a speed of 40.0 rev/min, provided the button is no more than 0.150 \(\mathrm{m}\) from the axis. (a) What is the coefficient of static friction between the button and the platform? (b) How far from the axis can the button be placed, without slipping, if the platform rotates at 60.0 \(\mathrm{rev} / \mathrm{min}\) ?

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 \(800 \mathrm{m},\) how many revolutions per minute are needed for the "artificial gravity" acceleration to be 9.80 \(\mathrm{m} / \mathrm{s}^{2}\) (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 \(\left(3.70 \mathrm{m} / \mathrm{s}^{2}\right) .\) How many revolutions per minute are needed in this case?

The Cosmoclock 21 Ferris wheel in Yokohama City, Japan, has a diameter of 100 \(\mathrm{m}\) . 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 \(\mathrm{N}\) 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 thepassenger's apparent weight at the highest point were zero? (d) What then would be the passenger's apparent weight at the lowest point?

An airplane flies in a loop (a circular path in a vertical plane) of radius 150 \(\mathrm{m} .\) The pilot's head always points toward the center of the loop. The speed of the airplane is not constant; the airplane goes slowest at the top of the loop and fastest at the bottom. (a) At the top of the loop, the pilot feels weightless. What is the speed of the airplane at this point? (b) At the bottom of the loop, the speed of the airplane is 280 \(\mathrm{km} / \mathrm{h}\) . What is the apparent weight of the pilot at this point? His true weight is 700 \(\mathrm{N}\) .

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 \(95.0 \mathrm{m} / \mathrm{s},\) what is the minimum radius of the circle for the acceleration at this point not to exceed 4.00\(g ?\) (b) What is the apparent weight of the pilot at the lowest point of the pullout?

You tie a cord to a pail of water, and you swing the pail in a vertical circle of radius 0.600 \(\mathrm{m} .\) What minimum speed must you give the pail at the highest point of the circle if no water is to spill from it?

A bowling ball weighing 71.2 \(\mathrm{N}(16.0 \mathrm{lb})\) 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 \(\mathrm{m} / \mathrm{s} .\) (a) What is the acceleration of the bowling ball, in magnitude and direction, at this instant? (b) What is the tension in the rope at this instant?

While a person is walking, his arms swing through approximately a \(45^{\circ}\) angle in \(\frac{1}{2}\) s. As a reasonable approximation, we can assume that the arm moves with constant speed during each swing. A typical arm is 70.0 \(\mathrm{cm}\) long, measured from the shoulder joint. (a) What is the acceleration of a \(1.0-\mathrm{g}\) drop of blood in the fingertips at the bottom of the swing? (b) Draw a free-body diagram of the drop of blood in part (a). ( c)Find the force that the blood vessel must exert on the drop of blood in part (a). Which way does this force point? (d) What force would the blood vessel exert if the arm were not swinging?

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. P5.60). (a) Draw a free-body diagram for the ball. (b) How hard does the surface of the ramp push on the ball? (c) What is the tension in the wire?

People who do chinups raise their chin just over a bar (the chinning bar), supporting themselves with only their arms. Typically, the body below the arms is raised by about 30 \(\mathrm{cm}\) in a time of 1.0s, starting from rest. Assume that the entire body of a \(680-\mathrm{N}\) person doing chin-ups is raised this distance and that half the 1.0 \(\mathrm{s}\) 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 then apply it to find the force his arms must exert on him during the accelerating part of the chin-up.

People (especially the elderly) who are prone to falling can wear hip pads to cushion the impact on their hip from a fall. Experiments have shown that if the speed at impact can be reduced to 1.3 \(\mathrm{m} / \mathrm{s}\) or less, the hip will usually not fracture. Let us investigate the worst-case scenario in which a 55 -kg person completely loses her footing (such as on icy pavement) and falls a distance of \(1.0 \mathrm{m},\) the distance from her hip to the ground. We shall assume that the person's entire body has the same acceleration, which, in reality, would not quite be true. (a) With what speed does her hip reach the ground? (b) A typical hip pad can reduce the person's speed to 1.3 \(\mathrm{m} / \mathrm{s}\) over a distance of 2.0 \(\mathrm{cm} .\) Find the acceleration (assumed to be constant) of this person's hip while she is slowing down and the force the pad exerts on it. (c) The force in part (b) is very large. To see whether it is likely to cause injury, calculate how long it lasts.

A 3.00 -kg box that is several hundred meters above the surface of the earth 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 this results in a tension in the rope of \(T(t)=(36.0 \mathrm{N} / \mathrm{s}) t .\) 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 \(t=1.00 \mathrm{s}\) and (ii) \(t=3.00 \mathrm{s} ?\) (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?

A 5.00 -kg box sits at rest at the bottom of a ramp that is 8.00 \(\mathrm{m}\) long and that is inclined at \(30.0^{\circ}\) above the horizontal. The coefficient of kinetic friction is \(\mu_{\mathrm{k}}=0.40\) , and the coefficient of static friction is \(\mu_{\mathrm{s}}=0.50 .\) 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 4.00 \(\mathrm{s} ?\)

Two boxes connected by a light horizontal rope are on a horizontal surface, as shown in Fig. P5.35. The coefficient of kinetic friction between each box and the surface is \(\mu_{\mathrm{k}}=0.30\) . One box (box \(B\) ) has mass \(5.00 \mathrm{kg},\) and the other box (box \(A )\) has mass \(m .\) A force \(F\) with magnitude 40.0 \(\mathrm{N}\) and direction \(53.1^{\circ}\) above the horizontal is applied to the \(5.00-\mathrm{kg}\) box, and both boxes move to the right with \(a=1.50 \mathrm{m} / \mathrm{s}^{2}\) . (a) What is the tension \(T\) in the rope that connects the boxes? (b) What is the mass \(m\) of the second box?

A 6.00 -kg box sits on a ramp that is inclined at \(37.0^{\circ}\) above the horizontal. The coefficient of kinetic friction between the box and the ramp is \(\mu_{k}=0.30 .\) What horizontal force is required to move the box up the incline with a constant acceleration of 4.20 \(\mathrm{m} / \mathrm{s}^{2}\) ?

Two bicycle tires are set rolling with the same initial speed of 3.50 \(\mathrm{m} / \mathrm{s}\) on a long, straight road, and the distance each travels before its speed is reduced by half is measured. One tire is inflated to a pressure of 40 psi and goes 18.1 m; the other is at 105 psi and goes 92.9 \(\mathrm{m} .\) What is the coefficient of rolling friction \(\mu_{\mathrm{r}}\) for each? Assume that the net horizontal force is due to rolling friction only.

A block with mass \(M\) is attached to the lower end of a vertical, uniform rope with mass \(m\) and length \(L\) A constant upward force \(\vec{\boldsymbol{F}}\) is applied to the top of the rope, causing the rope and block to accelerate upward. Find the tension in the rope at a distance \(x\) from the top end of the rope, where \(x\) can have any value from 0 to \(L .\)

A block with mass \(m_{1}\) is placed on an inclined plane with slope angle \(\alpha\) and is connected to a second hanging block with mass \(m_{2}\) by a cord passing over a small, frictionless pulley (Fig. P5.68). The coefficient of static friction is \(\mu_{\mathrm{s}}\) and the coefficient of kinetic friction is \(\mu_{\mathrm{k}}\) (a) Find the mass \(m_{2}\) for which block \(m_{1}\) moves up the plane at constant speed once it is set in motion. (b) Find the mass \(m_{2}\) for which block \(m_{1}\) 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?

Block \(A\) in Fig. P5.72 weighs 60.0 \(\mathrm{N} .\) The coefficient of static friction between the block and the surface on which it rests is \(0.25 .\) The weight \(w\) is 12.0 \(\mathrm{N}\) 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.

You are standing on a bathroom scale in an elevator in a tall building. Your mass is 64 \(\mathrm{kg}\) . The elevator starts from rest and travels upward with a speed that varies with time according to \(v(t)=\left(3.0 \mathrm{m} / \mathrm{s}^{2}\right) t+\left(0.20 \mathrm{m} / \mathrm{s}^{3}\right) t^{2} .\) When \(t=4.0 \mathrm{s},\) what is the reading of the bathroom scale?