Chapter 5

(I) If the coefficient of kinetic friction between a \(22-\mathrm{kg}\) crate and the floor is 0.30 , what horizontal force is required to move the crate at a steady speed across the floor? What horizontal force is required if \(\mu_{\mathrm{k}}\) is zero?

The Problems in this Section are ranked \(1,\) II, or III according to estimated difficulty, with \((1)\) Problems being easiest. Level (III) Problems are meant mainly as a challenge for the best students, for "extra credit." The Problems are arranged by Sections, meaning that the reader should have read up to and inciuding that Section, but this Chapter also has a group of General Problems that are not arranged by Section and not ranked. \(\begin{array}{l}{\text { (1) If the coefficient of kinetic friction between a } 22 \text { -kg crate }} \\ {\text { and the floor is } 0.30 \text { , what horizontal force is required to }} \\ {\text { move the crate at a steady speed across the floor? What }} \\ {\text { horizontal force is required if } \mu_{k} \text { is zero? }}\end{array}\)

(I) A force of \(35.0 \mathrm{~N}\) is required to start a 6.0 -kg box moving across a horizontal concrete floor. ( \(a\) ) What is the coefficient of static friction between the box and the floor? \((b)\) If the 35.0-N force continues, the box accelerates at \(0.60 \mathrm{~m} / \mathrm{s}^{2}\). What is the coefficient of kinetic friction?

(I) Suppose you are standing on a train accelerating at \(0.20 \mathrm{~g}\). What minimum coefficient of static friction must exist between your feet and the floor if you are not to slide?

(I) The coefficient of static friction between hard rubber and normal street pavement is about \(0.90 .\) On how steep a hill (maximum angle) can you leave a car parked?

(I) What is the maximum acceleration a car can undergo if the coefficient of static friction between the tires and the ground is \(0.90 ?\)

(II) \((a)\) A box sits at rest on a rough \(33^{\circ}\) inclined plane. Draw the free-body diagram, showing all the forces acting on the box. (b) How would the diagram change if the box were sliding down the plane. ( \(c\) ) How would it change if the box were sliding up the plane after an initial shove?

(II) A \(25.0-\mathrm{kg}\) box is released on a \(27^{\circ}\) incline and accelerates down the incline at \(0.30 \mathrm{~m} / \mathrm{s}^{2}\). Find the friction force impeding its motion. What is the coefficient of kinetic friction?

(II) A skier moves down a \(27^{\circ}\) slope at constant speed. What can you say about the coefficient of friction, \(\mu_{\mathrm{k}}\) ? Assume the speed is low enough that air resistance can be ignored.

(II) A wet bar of soap slides freely down a ramp \(9.0 \mathrm{~m}\) long inclined at \(8.0^{\circ} .\) How long does it take to reach the bottom? Assume \(\mu_{\mathrm{k}}=0.060 .\)

(II) A box is given a push so that it slides across the floor. How far will it go, given that the coefficient of kinetic friction is 0.15 and the push imparts an initial speed of \(3.5 \mathrm{~m} / \mathrm{s} ?\)

(II) \((a)\) Show that the minimum stopping distance for an automobile traveling at speed \(v\) is equal to \(v^{2} / 2 \mu_{\mathrm{s}} g,\) where \(\mu_{\mathrm{s}}\) is the coefficient of static friction between the tires and the road, and \(g\) is the acceleration of gravity. (b) What is this distance for a \(1200-\mathrm{kg}\) car traveling \(95 \mathrm{~km} / \mathrm{h}\) if \(\mu_{\mathrm{s}}=0.65 ?\) (c) What would it be if the car were on the Moon (the acceleration of gravity on the Moon is about \(g / 6\) ) but all else stayed the same?

(II) A 1280 -kg car pulls a 350 -kg trailer. The car exerts a horizontal force of \(3.6 \times 10^{3} \mathrm{~N}\) against the ground in order to accelerate. What force does the car exert on the trailer? Assume an effective friction coefficient of 0.15 for the trailer.

(II) Police investigators, examining the scene of an accident involving two cars, measure 72 -m-long skid marks of one of the cars, which nearly came to a stop before colliding. The coefficient of kinetic friction between rubber and the pavement is about \(0.80 .\) Estimate the initial speed of that car assuming a level road.

(1I) Police investigators, examining the scene of an accident involving two cars, measure 72 -m-long skid marks of one of the cars, which nearly came to a stop before colliding. The coefficient of kinetic friction between rubber and the pave- ment is about \(0.80 .\) Estimate the initial speed of that car assuming a level road.

(II) Piles of snow on slippery roofs can become dangerous projectiles as they melt. Consider a chunk of snow at the ridge of a roof with a slope of \(34^{\circ} .(a)\) What is the minimum value of the coefficient of static friction that will keep the snow from sliding down? \((b)\) As the snow begins to melt the coefficient of static friction decreases and the snow finally slips. Assuming that the distance from the chunk to the edge of the roof is \(6.0 \mathrm{~m}\) and the coefficient of kinetic friction is \(0.20,\) calculate the speed of the snow chunk when it slides off the roof. \((c)\) If the edge of the roof is \(10.0 \mathrm{~m}\) above ground, estimate the speed of the snow when it hits the ground.

(II) A small box is held in place against a rough vertical wall by someone pushing on it with a force directed upward at \(28^{\circ}\) above the horizontal. The coefficients of static and kinetic friction between the box and wall are 0.40 and 0.30 , respectively. The box slides down unless the applied force has magnitude \(23 \mathrm{~N}\). What is the mass of the box?

(II) Two crates, of mass 65 \(\mathrm{kg}\) and 125 \(\mathrm{kg}\) , are in contact and at rest on a horizontal surface (Fig, \(32 ) . \mathrm{A} 650\) -N force is exerted on the 65 -kg crate. If the coefficient of kinetic friction is \(0.18,\) calculate \((a)\) the acceleration of the system, and \((b)\) the force that each crate exerts on the other. (c) Repeat with the crates reversed.

(II) The crate shown in Fig. 33 lies on a plane tilted at an angle \(\theta=25.0^{\circ}\) to the horizontal, with \(\mu_{\mathrm{k}}=0.19\) . (a) Determine the acceleration of the crate as it slides down the plane. (b) If the crate starts from rest 8.15 \(\mathrm{m}\) up the plane from its base, what will be the crate's speed when it reaches the bottom of the incline?

(II) A flatbed truck is carrying a heavy crate. The coefficient of static friction between the crate and the bed of the truck is \(0.75 .\) What is the maximum rate at which the driver can decelerate and still avoid having the crate slide against the cab of the truck?

(II) A small block of mass \(m\) is given an initial speed \(v_{0}\) up a ramp inclined at angle \(\theta\) to the horizontal. It travels a distance \(d\) up the ramp and comes to rest. ( \(a\) ) Determine a formula for the coefficient of kinetic friction between block and ramp. (b) What can you say about the value of the coefficient of static friction?

(II) \(\mathrm{A}\) 75-kg snowboarder has an initial velocity of 5.0 \(\mathrm{m} / \mathrm{s}\) at the top of a \(28^{\circ}\) incline (Fig. \(36 ) .\) After sliding down the 110 -m long incline (on which the coefficient of kinetic friction is \(\mu_{k}=0.18\) ), the snowboarder has attained a velocity \(v .\) The snowboarder then slides along a flat surface (on which \(\mu_{k}=0.15\) and comes to rest after a distance \(x .\) Use Newton's second law to find the snowboarder's acceleration while on the incline and while on the flat surface. Then use these accelerations to determine \(x .\)

(II) A child slides down a slide with a \(34^{\circ}\) incline, and at the bottom her speed is precisely half what it would have been if the slide had been frictionless. Calculate the coefficient of kinetic friction between the slide and the child.

(III) A 3.0 -kg block sits on top of a \(5.0-\mathrm{kg}\) block which is on a horizontal surface. The 5.0 -kg block is pulled to the right with a force \(\vec{\mathbf{F}}\) as shown in Fig. \(39 .\) The coefficient of static friction between all surfaces is 0.60 and the kinetic coeffi- cient is \(0.40 .\) (a) What is the minimum value of \(F\) needed to move the two blocks? (b) If the force is 10\(\%\) greater than your answer for \((a),\) what is the acceleration of each block?

(III) A 4.0-kg block is stacked on top of a \(12.0-\mathrm{kg}\) block, which is accelerating along a horizontal table at \(a=5.2 \mathrm{~m} / \mathrm{s}^{2}\) (Fig. 5-40). Let \(\mu_{\mathrm{k}}=\mu_{\mathrm{s}}=\mu .\)( \(a\) ) What minimum coefficient of friction \(\mu\) between the two blocks will prevent the \(4.0-\mathrm{kg}\) block from sliding off? \((b)\) If \(\mu\) is only half this minimum value, what is the acceleration of the \(4.0-\mathrm{kg}\) block with respect to the table, and \((c)\) with respect to the \(12.0-\mathrm{kg}\) block? (d) What is the force that must be applied to the \(12.0-\mathrm{kg}\) block in \((a)\) and in \((b)\), assuming that the table is frictionless?

(III) A 4.0 -kg block is stacked on top of a 12.0 -kg block, which is accelerating along a horizontal table at \(a=5.2 \mathrm{m} / \mathrm{s}^{2}\) (Fig. \(40 ) .\) Let \(\mu_{\mathrm{k}}=\mu_{\mathrm{s}}=\mu .\) (a) What minimum coefficient of friction \(\mu\) between the two blocks will prevent the 4.0 -kg block from sliding off? (b) If \(\mu\) is only half this minimum value, what is the acceleration of the 4.0 -kg block with respect to the table, and \((c)\) with respect to the 12.0 -kg block? (d) What is the force that must be applied to the 12.0 -kg block in \((a)\) and in \((b),\) assuming that the table is frictionless?

(III) A small block of mass \(m\) rests on the rough, sloping side of a triangular block of mass \(M\) which itself rests on a horizontal frictionless table as shown in Fig. 5-41. If the coefficient of static friction is \(\mu,\) determine the minimum horizontal force \(F\) applied to \(M\) that will cause the small block \(m\) to start moving up the incline.

(I) What is the maximum speed with which a \(1200-\mathrm{kg}\) car can round a turn of radius \(80.0 \mathrm{~m}\) on a flat road if the coefficient of friction between tires and road is \(0.65 ?\) Is this result independent of the mass of the car?

(I) What is the maximum speed with which a 1200 -kg car can round a turn of radius 80.0 \(\mathrm{m}\) on a flat road if the coeffi- cient of friction between tires and road is 0.65\(?\) Is this result independent of the mass of the car?

(I) A child sitting \(1.20 \mathrm{~m}\) from the center of a merry-goaround moves with a speed of \(1.30 \mathrm{~m} / \mathrm{s} .\) Calculate \((a)\) the centripetal acceleration of the child and \((b)\) the net horizontal force exerted on the child \((\) mass \(=22.5 \mathrm{~kg})\)

(I) A jet plane traveling \(1890 \mathrm{~km} / \mathrm{h}(525 \mathrm{~m} / \mathrm{s})\) pulls out of a dive by moving in an arc of radius \(4.80 \mathrm{~km} .\) What is the plane's acceleration in \(g\) 's?

(II) Is it possible to whirl a bucket of water fast enough in a vertical circle so that the water won't fall out? If so, what is the minimum speed? Define all quantities needed.

(II) How fast (in rpm) must a centrifuge rotate if a particle \(8.00 \mathrm{~cm}\) from the axis of rotation is to experience an acceleration of \(125,000 g\) 's?

(II) At what minimum speed must a roller coaster be traveling when upside down at the top of a circle (Fig. \(5-42\) ) so that the passengers do not fall out? Assume a radius of curvature of \(7.6 \mathrm{~m}\)

(II) A sports car crosses the bottom of a valley with a radius of curvature equal to \(95 \mathrm{~m}\). At the very bottom, the normal force on the driver is twice his weight. At what speed was the car traveling?

(II) How large must the coefficient of static friction be between the tires and the road if a car is to round a level curve of radius \(85 \mathrm{~m}\) at a speed of \(95 \mathrm{~km} / \mathrm{h} ?\)

(II) Suppose the space shuttle is in orbit \(400 \mathrm{~km}\) from the Earth's surface, and circles the Earth about once every 90 min. Find the centripetal acceleration of the space shuttle in its orbit. Express your answer in terms of \(g,\) the gravitational acceleration at the Earth's surface.

(II) A bucket of mass \(2.00 \mathrm{~kg}\) is whirled in a vertical circle of radius \(1.10 \mathrm{~m} .\) At the lowest point of its motion the tension in the rope supporting the bucket is \(25.0 \mathrm{~N}\). \((a)\) Find the speed of the bucket. \((b)\) How fast must the bucket move at the top of the circle so that the rope does not go slack?

(II) How many revolutions per minute would a \(22-\mathrm{m}-\) diameter Ferris wheel need to make for the passengers to feel "weightless" at the topmost point?

(II) Use dimensional analysis (Section \(1-7\) ) to obtain the form for the centripetal acceleration, \(a_{\mathrm{R}}=v^{2} / r\)

(II) Use dimensional analysis to obtain the form for the centripetal acceleration, \(a_{\mathrm{R}}=v^{2} / r\) .

(II) A jet pilot takes his aircraft in a vertical loop (Fig. \(5-43)\). ( \(a\) ) If the jet is moving at a speed of \(1200 \mathrm{~km} / \mathrm{h}\) at the lowest point of the loop, determine the minimum radius of the circle so that the centripetal acceleration at the lowest point does not exceed \(6.0 \mathrm{~g}\) 's. \((b)\) Calculate the \(78-\mathrm{kg}\) pilot's effective weight (the force with which the seat pushes up on him) at the bottom of the circle, and \((c)\) at the top of the circle (assume the same speed).

(II) A proposed space station consists of a circular tube that will rotate about its center (like a tubular bicycle tire), Fig. \(5-44 .\) The circle formed by the tube has a diameter of about \(1.1 \mathrm{~km} .\) What must be the rotation speed (revolutions per day) if an effect equal to gravity at the surface of the Earth \((1.0 g)\) is to be felt?

(II) On an ice rink two skaters of equal mass grab hands and spin in a mutual circle once every \(2.5 \mathrm{~s}\). If we assume their arms are each \(0.80 \mathrm{~m}\) long and their individual masses are \(60.0 \mathrm{~kg}\), how hard are they pulling on one another?

(II) A coin is placed \(12.0 \mathrm{~cm}\) from the axis of a rotating turntable of variable speed. When the speed of the turntable is slowly increased, the coin remains fixed on the turntable until a rate of 35.0 rpm (revolutions per minute) is reached, at which point the coin slides off. What is the coefficient of static friction between the coin and the turntable?

(II) The design of a new road includes a straight stretch that is horizontal and flat but that suddenly dips down a steep hill at \(22^{\circ} .\) The transition should be rounded with what minimum radius so that cars traveling \(95 \mathrm{~km} / \mathrm{h}\) will not leave the road (Fig. \(5-45) ?\)

(II) A 975-kg sports car (including driver) crosses the rounded top of a hill (radius \(=88.0 \mathrm{~m})\) at \(12.0 \mathrm{~m} / \mathrm{s}\) s. Determine \((a)\) the normal force exerted by the road on the car, (b) the normal force exerted by the car on the 72.0 -kg driver, and (c) the car speed at which the normal force on the driver equals zero.

(II) A \(975-\mathrm{kg}\) sports car (including driver) crosses the rounded top of a hill (radius = 88.0 \(\mathrm{m} )\) at 12.0 \(\mathrm{m} / \mathrm{s}\) . Determine \((a)\) the normal force exerted by the road on the car, \((b)\) the normal force exerted by the car on the \(72.0-\mathrm{kg}\) driver, and \((c)\) the car speed at which the normal force on the driver equals zero.

(II) Two blocks, with masses \(m_{\mathrm{A}}\) and \(m_{\mathrm{B}},\) are connected to each other and to a central post by cords as shown in Fig. \(5-46 .\) They rotate about the post at frequency \(f\) (revolutions per second) on a frictionless horizontal surface at distances \(r_{\mathrm{A}}\) and \(r_{\mathrm{B}}\) from the post. Derive an algebraic expression for the tension in each segment of the cord (assumed massless).

(II) Tarzan plans to cross a gorge by swinging in an arc from a hanging vine (Fig. \(47 ) .\) If his arms are capable of exerting a force of 1350 \(\mathrm{N}\) on the rope, what is the maximum speed he can tolerate at the lowest point of his swing? His mass is 78 \(\mathrm{kg}\) and the vine is 5.2 \(\mathrm{m}\) long.