Chapter 5

Two \(25.0 \mathrm{~N}\) weights are suspended at opposite ends of a rope that passes over a light, frictionless pulley. The pulley is attached to a chain that is fastened to the ceiling. (See Figure 5.41.) Start solving this problem by making a free-body diagram of each weight. (a) What is the tension in the rope? (b) What is the tension in the chain?

Muscles are attached to bones by means of tendons. The maximum force that a muscle can exert is directly proportional to its cross-sectional area \(A\) at the widest point. We can express this relationship mathematically as \(F_{\max }=\sigma A\) where \(\sigma\) (sigma) is a proportionality constant. Surprisingly, \(\sigma\) is about the same for the muscles of all animals and has the numerical value of \(3.0 \times 10^{5}\) in SI units. The gastrocnemius muscle, in the back of the leg, has two portions, known as the medial and lateral heads. Assume that they attach to the Achilles tendon as shown in Figure \(5.45 .\) The cross sectional area of each of these two muscles is typically \(30 \mathrm{~cm}^{2}\) for many adults. What is the maximum tension they can produce in the Achilles tendon?

A man pushes on a piano of mass \(180 \mathrm{~kg}\) so that it slides at a constant velocity of \(12.0 \mathrm{~cm} / \mathrm{s}\) down a ramp that is inclined at \(11.0^{\circ}\) above the horizontal. No appreciable friction is acting on the piano. Calculate the magnitude and direction of this push (a) if the man pushes parallel to the incline, (b) if the man pushes the piano up the plane instead, also at \(12.0 \mathrm{~cm} / \mathrm{s}\) parallel to the incline, and (c) if the man pushes horizontally but still with a speed of \(12.0 \mathrm{~cm} / \mathrm{s}\)

People who do chin-ups raise their chin just over a bar (the chinning bar), supporting themselves only by their arms. Typically, the body below the arms is raised by about \(30 \mathrm{~cm}\) in a time of \(1.0 \mathrm{~s},\) starting from rest. Assume that the entire body of a \(680 \mathrm{~N}\) person who is chinning 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. Make 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.

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) Make a free-body diagram of the person during the jump. (c) In terms of this jumper's weight \(\underline{W}\). what force does the ground exert on him or her during the jump?

A large fish hangs from a spring balance supported from the roof of an elevator. (a) If the elevator has an upward acceleration of \(2.45 \mathrm{~m} / \mathrm{s}^{2}\) and the balance reads \(60.0 \mathrm{~N},\) what is the true weight of the fish? (b) Under what circumstances will the balance read \(35.0 \mathrm{~N} ?\) (c) What will the balance read if the elevator cable breaks?

A \(750.0 \mathrm{~kg}\) boulder is raised from a quarry \(125 \mathrm{~m}\) deep by a long chain having a mass of \(575 \mathrm{~kg} .\) 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 for the boulder to be lifted out at maximum acceleration if it started from rest?

At a construction site, a \(22.0 \mathrm{~kg}\) bucket of concrete is connected over a very light frictionless pulley to a \(375 \mathrm{~N}\) box on the roof of a building. (See Figure \(5.55 .)\) There is no appreciable friction on the box, since it is on roller bearings. The box starts from rest. (a) Make free- body diagrams of the bucket and the box. (b) Find the acceleration of the bucket. (c) How fast is the bucket moving after it has fallen \(1.50 \mathrm{~m}\) (assuming that the box has not yet reached the edge of the roof)?

An \(80 \mathrm{~N}\) box initially at rest is pulled by a horizontal rope on a horizontal table. The coefficients of kinetic and static friction between the box and the table are \(\frac{1}{4}\) and \(\frac{1}{2},\) respectively. What is the friction (b) \(25 \mathrm{~N}\), force on this box if the tension in the rope is (a) \(0 \mathrm{~N},]\) (c) \(39 \mathrm{~N},(\mathrm{~d}) 41 \mathrm{~N},(\mathrm{e}) 150 \mathrm{~N} ?\)

A \(2 \mathrm{~kg}\) book sits at rest on a horizontal table. The coefficient of static friction between the book and the surface is \(0.40,\) and the coefficient of kinetic friction is 0.20 . (a) What is the normal force acting on the book? (b) Is there a friction force on the book? (c) What minimum horizontal force would be required to cause the book to slide on the table? (d) If you give the book a strong horizontal push so that it begins sliding, what kind of force will cause it to come to rest? (e) What is the magnitude of this force?

A hockey puck leaves a player's stick with a speed of \(9.9 \mathrm{~m} / \mathrm{s}\) and slides \(32.0 \mathrm{~m}\) before coming to rest. Find the coefficient of friction between the puck and the ice.

(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 \(29.1 \mathrm{~m} / \mathrm{s}\) (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.)

An \(85 \mathrm{~N}\) box of oranges is being pushed across a horizontal floor. As it moves, it is slowing at a constant rate of \(0.90 \mathrm{~m} / \mathrm{s}\) each second. The push force has a horizontal component of \(20 \mathrm{~N}\) and a vertical component of \(25 \mathrm{~N}\) downward. Calculate the coefficient of kinetic friction between the box and floor.

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 coefficients of kinetic and static friction between the box and the surface are 0.200 and \(0.450,\) respectively. (a) What horizontal force must the worker apply to maintain the motion of the box? (b) If the worker stops pushing, what will be the acceleration of the box?

The coefficient of kinetic friction between a \(40 \mathrm{~kg}\) crate and the warehouse floor is \(70 \%\) of the corresponding coefficient of static friction. The crate falls off a forklift that is moving at \(3 \mathrm{~m} / \mathrm{s}\) and then slides along the warehouse floor for a distance of \(2.5 \mathrm{~m}\) before coming to rest. What is the coefficient of static friction between the crate and the floor?

One straightforward way to measure the coefficients of friction between a box and a wooden surface is illustrated in Figure \(5.59 .\) The sheet of wood can be raised by pivoting it about one edge. It is first raised to an angle \(\theta_{1}\) (which is measured) for which the box just begins to slide downward. The sheet is then immediately lowered to an angle \(\theta_{2}\) (which is also measured) for which the box slides with constant speed down the sheet. Apply Newton's second law to the box in both cases to find the coefficients of kinetic and static friction between it and the wooden sheet in terms of the measured angles \(\theta_{1}\) and \(\theta_{2}\).

With its wheels locked, a van slides down an icy, frictionless hill. What angle does the hill make with the horizontal if the acceleration of the van is \(4 \mathrm{~m} / \mathrm{s}^{2} ?\)

In emergencies involving major blood loss, the doctor will order the patient placed in the Trendelberg position, which is to raise the foot of the bed to get maximum blood flow to the brain. If the coefficient of static friction between the typical patient and the bedsheets is \(1.2,\) what is the maximum angle at which the bed can be tilted with respect to the floor before the patient begins to slide?

A toboggan approaches a snowy hill moving at \(11.0 \mathrm{~m} / \mathrm{s}\). The coefficients of static and kinetic friction between the snow and the toboggan are 0.40 and \(0.30,\) respectively, and the hill slopes upward at \(40.0^{\circ}\) above the horizontal. Find the acceleration of the toboggan (a) as it is going up the hill and (b) after it has reached its highest point and is sliding down the hill.

A \(25.0 \mathrm{~kg}\) 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 person pushes on a stationary \(125 \mathrm{~N}\) box with \(75 \mathrm{~N}\) at \(30^{\circ}\) below the horizontal, as shown in Figure 5.61 . The coefficient of static friction between the box and the horizontal floor is 0.80 . (a) Make a free-body diagram of the box. (b) What is the normal force on the box? (c) What is the friction force on the box? (d) What is the largest the friction force could be? (e) The person now replaces his push with a \(75 \mathrm{~N}\) pull at \(30^{\circ}\) above the horizontal. Find the normal force on the box in this case.

You are working for a shipping company. Your job is to stand at the bottom of an 8.0 -m-long ramp that is inclined at \(37^{\circ}\) above the horizontal. You grab packages off a conveyor belt and propel them up the ramp. The coefficient of kinetic friction between the packages and the ramp is \(\mu_{\mathrm{k}}=0.30 .\) (a) What speed do you need to give a package at the bottom of the ramp so that it has zero speed at the top of the ramp? (b) Your coworker is supposed to grab the packages as they arrive at the top of the ramp, but she misses one and it slides back down. What is its speed when it returns to you?

An atmospheric drag force with magnitude \(F_{D}=D v^{2},\) where \(v\) is speed, acts on a falling \(300 \mathrm{mg}\) raindrop that reaches a terminal velocity of \(11 \mathrm{~m} / \mathrm{s}\). (a) Show that the \(\mathrm{SI}\) units of \(D\) are \(\mathrm{kg} / \mathrm{m}\). (b) Find the value of \(D .\)

A bullet is fired horizontally from a high-powered rifle. If air drag is taken into account, is the magnitude of the bullet's acceleration after leaving the barrel greater than or less than \(g\) ? Explain.

You find that if you hang a \(1.25 \mathrm{~kg}\) weight from a vertical spring, it stretches \(3.75 \mathrm{~cm}\). (a) What is the force constant of this spring in \(\mathrm{N} / \mathrm{m} ?\) (b) How much mass should you hang from the spring so it will stretch by \(8.13 \mathrm{~cm}\) from its original, unstretched length?

An unstretched spring is \(12.00 \mathrm{~cm}\) long. When you hang an \(875 \mathrm{~g}\) weight from it, it stretches to a length of \(14.40 \mathrm{~cm}\). (a) What is the force constant (in \(\mathrm{N} / \mathrm{m}\) ) of this spring? (b) What total mass must you hang from the spring to stretch it to a total length of \(17.72 \mathrm{~cm} ?\)

A surgeon is using material from a donated heart to repair a patient's damaged aorta and needs to know the elastic characteristics of this aortal material. Tests performed on a \(16.0 \mathrm{~cm}\) strip of the donated aorta reveal that it stretches \(3.75 \mathrm{~cm}\) when a \(1.50 \mathrm{~N}\) pull is exerted on it. (a) What is the force constant of this strip of aortal material? (b) If the maximum distance it will be able to stretch when it replaces the aorta in the damaged heart is \(1.14 \mathrm{~cm},\) what is the greatest force it will be able to exert there?

A light spring having a force constant of \(125 \mathrm{~N} / \mathrm{m}\) is used to pull a \(9.50 \mathrm{~kg}\) sled on a horizontal frictionless ice rink. If the sled has an acceleration of \(2.00 \mathrm{~m} / \mathrm{s}^{2},\) by how much does the spring stretch if it pulls on the sled (a) horizontally, (b) at \(30.0^{\circ}\) above the horizontal?

You've attached a bungee cord to a wagon and are using it to pull your little sister while you take her for a jaunt. The bungee's unstretched length is \(1.3 \mathrm{~m}\), and you happen to know that your little sister weighs \(220 \mathrm{~N}\) and the wagon weighs \(75 \mathrm{~N}\). Crossing a street, you accelerate from rest to your normal walking speed of \(1.5 \mathrm{~m} / \mathrm{s}\) in \(2.0 \mathrm{~s},\) and you notice that while you're accelerating, the bungee's length increases to about \(2.0 \mathrm{~m}\). What's the force constant of the bungee cord, assuming it obeys Hooke's law?

A \(2 \mathrm{~kg}\) block is launched up a frictionless inclined plane by a spring as shown in Figure 5.68 . The plane is inclined at \(30^{\circ}\), and the spring constant is \(1000 \mathrm{~N} / \mathrm{m}\). The block is initially pushed against the spring in order to compress the spring \(0.1 \mathrm{~m},\) and then it is released. (a) Calculate the magnitude and direction of the acceleration of the block the moment after it is released. (b) Calculate the acceleration when the spring reaches the point where its compression is \(0.05 \mathrm{~m}\). (c) What are the magnitude and direction of the acceleration when the spring reaches the point where its compression is zero?

You are riding in an elevator on the way to the 18 th floor of your dormitory. The elevator is accelerating upward with \(a=1.90 \mathrm{~m} / \mathrm{s}^{2} .\) Beside you is the box containing your new computer; the box and its contents have a total mass of \(28.0 \mathrm{~kg}\). While the elevator is accelerating upward, you push horizontally on the box to slide it at constant speed toward the elevator door. If the coefficient of kinetic friction between the box and the elevator floor is \(\mu_{\mathrm{k}}=0.32,\) what magnitude of force must you apply?

A \(65.0 \mathrm{~kg}\) parachutist falling vertically at a speed of \(6.30 \mathrm{~m} / \mathrm{s}\) hits the ground, which brings him to a complete stop in a distance of \(0.92 \mathrm{~m}\) (roughly half of his height). Assuming constant acceleration after his feet first touch the ground, what is the average force exerted on the parachutist by the ground?

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 (Figure 5.73). The coefficient of static friction is \(\mu_{\mathrm{s}}\) and the coefficient of kinetic fric- tion 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?

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 in which this truck could accelerate uniformly to \(30.0 \mathrm{~m} / \mathrm{s}(\approx 60 \mathrm{mi} / \mathrm{h}) \quad\) without causing the box to slide? Include a free-body diagram of the toolbox as part of your solution. (Hint: First use Newton's second law to find the maximum acceleration that static friction can give the box, and then solve for the time required to reach \(30.0 \mathrm{~m} / \mathrm{s}\).)

An astronaut on the distant planet Xenon uses an adjustable inclined plane to measure the acceleration of gravity. The plane is frictionless, and its angle of inclination can be varied. Here is a table of the data: $$\begin{array}{lc}\hline \theta & a\left(\mathrm{~m} / \mathrm{s}^{2}\right) \\\\\hline 5.0^{\circ} & 1.20 \\ 10^{\circ} & 2.49 \\\15^{\circ} & 3.59 \\\20^{\circ} & 4.90 \\\25^{\circ} & 5.95 \\\\\hline\end{array}$$ Make a plot of the measured acceleration as a function of the sine of the angle of incline. Using a linear "best fit" to the data, determine the value of \(g\) on the planet Xenon.

Shoes for the sports of bouldering and rock climbing are designed to provide a great deal of friction between the foot and the surface of the ground. On smooth rock these shoes might have a coefficient of static friction of 1.2 and a coefficient of kinetic friction of 0.90 . For a person wearing these shoes, what's the maximum angle (with respect to the horizontal) of a smooth rock that can be walked on without slipping? A. \(42^{\circ}\) B. \(50^{\circ}\) C. \(64^{\circ}\) D. Greater than \(90^{\circ}\)

Shoes for the sports of bouldering and rock climbing are designed to provide a great deal of friction between the foot and the surface of the ground. On smooth rock these shoes might have a coefficient of static friction of 1.2 and a coefficient of kinetic friction of 0.90 . A person wearing these shoes stands on a smooth horizontal rock. She pushes against the ground to begin running. What is the maximum horizontal acceleration she can have without slipping? A. \(0.20 g\) B. \(0.75 g\) C. \(0.90 g\) D. \(1.2 g\)