Chapter 10

A hand exerciser utilizes a coiled spring. A force of \(89.0 \mathrm{~N}\) is required to compress the spring by \(0.0191 \mathrm{~m}\). Determine the force needed to compress the spring by \(0.0508 \mathrm{~m}\).

An archer, about to shoot an arrow, is applying a force of \(+240 \mathrm{~N}\) to a drawn bowstring. The bow behaves like an ideal spring whose spring constant is \(480 \mathrm{~N} / \mathrm{m}\). What is the displacement of the bowstring?

A \(0.70\) -kg block is hung from and stretches a spring that is attached to the ceiling. A second block is attached to the first one, and the amount that the spring stretches from its unstrained length triples. What is the mass of the second block?

A \(0.70-\mathrm{kg}\) block is hung from and stretches a spring that is attached to the ceiling. A second block is attached to the first one, and the amount that the spring stretches from its unstrained length triples. What is the mass of the second block?

A person who weighs \(670 \mathrm{~N}\) steps onto a spring scale in the bathroom, and the spring compresses by \(0.79 \mathrm{~cm}\). (a) What is the spring constant? (b) What is the weight of another person who compresses the spring by \(0.34 \mathrm{~cm}\) ?

A person who weighs 670 N steps onto a spring scale in the bathroom, and the spring compresses by \(0.79 \mathrm{~cm}\). (a) What is the spring constant? (b) What is the weight of another person who compresses the spring by \(0.34 \mathrm{~cm} ?\)

A car is hauling a \(92-\mathrm{kg}\) trailer, to which it is connected by a spring. The spring constant is \(2300 \mathrm{~N} / \mathrm{m}\). The car accelerates with an acceleration of \(0.30 \mathrm{~m} / \mathrm{s}^{2} .\) By how much does the spring stretch?

A car is hauling a 92-kg trailer, to which it is connected by a spring. The spring constant is \(2300 \mathrm{~N} / \mathrm{m}\). The car accelerates with an acceleration of \(0.30 \mathrm{~m} / \mathrm{s}^{2} .\) By how much does the spring stretch?

In a room that is \(2.44 \mathrm{~m}\) high, a spring (unstrained length \(=0.30 \mathrm{~m}\) ) hangs from the ceiling. A board whose length is \(1.98 \mathrm{~m}\) is attached to the free end of the spring. The board hangs straight down, so that its \(1.98-\mathrm{m}\) length is perpendicular to the floor. The weight of the board ( 104 N) stretches the spring so that the lower end of the board just extends to, but does not touch, the floor. What is the spring constant of the spring?

In \(0.750 \mathrm{~s},\) a \(7.00-\mathrm{kg}\) block is pulled through a distance of \(4.00 \mathrm{~m}\) on a frictionless horizontal surface, starting from rest. The block has a constant acceleration and is pulled by means of a horizontal spring that is attached to the block. The spring constant of the spring is \(415 \mathrm{~N} / \mathrm{m}\). By how much does the spring stretch?

Interactive Solution \(10.9\) at discusses a method used to solve this problem. To measure the static friction coefficient between a \(1.6-\mathrm{kg}\) block and a vertical wall, the setup shown in the drawing is used. A spring (spring constant \(=510 \mathrm{~N} / \mathrm{m}\) ) is attached to the block. Someone pushes on the end of the spring in a direction perpendicular to the wall until the block does not slip downward. If the spring in such a setup is compressed by \(0.039 \mathrm{~m}\), what is the coefficient of static friction?

A small ball is attached to one end of a spring that has an unstrained length of 0.200 \(\mathrm{m}\). The spring is held by the other end, and the ball is whirled around in a horizontal circle at a speed of \(3.00 \mathrm{~m} / \mathrm{s}\). The spring remains nearly parallel to the ground during the motion and is observed to stretch by \(0.010 \mathrm{~m}\). By how much would the spring stretch if it were attached to the ceiling and the ball allowed to hang straight down, motionless?

A \(30.0\) -kg block is resting on a flat horizontal table. On top of this block is resting a \(15.0\) kg block, to which a horizontal spring is attached, as the drawing illustrates. The spring constant of the spring is \(325 \mathrm{~N} / \mathrm{m}\). The coefficient of kinetic friction between the lower block and the table is \(0.600\), and the coefficient of static friction between the two blocks is \(0.900\). A horizontal force \(\overrightarrow{\mathbf{F}}\) is applied to the lower block as shown. This force is increasing in such a way as to keep the blocks moving at a constant speed. At the point where the upper block begins to slip on the lower block, determine (a) the amount by which the spring is compressed and (b) the magnitude of the force \(\overrightarrow{\mathbf{F}}\).

A \(30.0-\mathrm{kg}\) block is resting on a flat horizontal table. On top of this block is resting a \(15.0-\) kg block, to which a horizontal spring is attached, as the drawing illustrates. The spring constant of the spring is \(325 \mathrm{~N} / \mathrm{m}\). The coefficient of kinetic friction between the lower block and the table is 0.600 , and the coefficient of static friction between the two blocks is 0.900 . A horizontal force \(\overrightarrow{\mathbf{F}}\) is applied to the lower block as shown. This force is increasing in such a way as to keep the blocks moving at a constant speed. At the point where the upper block begins to slip on the lower block, determine (a) the amount by which the spring is compressed and (b) the magnitude of the force \(\overrightarrow{\mathbf{F}}\).

A \(15.0\) -kg block rests on a horizontal table and is attached to one end of a massless, horizontal spring. By pulling horizontally on the other end of the spring, someone causes the block to accelerate uniformly and reach a speed of \(5.00 \mathrm{~m} / \mathrm{s}\) in \(0.500 \mathrm{~s}\). In the process, the spring is stretched by \(0.200 \mathrm{~m}\). The block is then pulled at a constant speed of \(5.00 \mathrm{~m} / \mathrm{s}\), during which time the spring is stretched by only \(0.0500 \mathrm{~m}\). Find (a) the spring constant of the spring and (b) the coefficient of kinetic friction between the block and the table.

A \(15.0-\mathrm{kg}\) block rests on a horizontal table and is attached to one end of a massless, horizontal spring. By pulling horizontally on the other end of the spring, someone causes the block to accelerate uniformly and reach a speed of \(5.00 \mathrm{~m} / \mathrm{s}\) in \(0.500 \mathrm{~s}\). In the process, the spring is stretched by \(0.200 \mathrm{~m}\). The block is then pulled at a constant speed of \(5.00 \mathrm{~m} / \mathrm{s}\), during which time the spring is stretched by only \(0.0500 \mathrm{~m}\). Find (a) the spring constant of the spring and (b) the coefficient of kinetic friction between the block and the table

A loudspeaker diaphragm is producing a sound for 2.5 s by moving back and forth in simple harmonic motion. The angular frequency of the motion is \(7.54 \times 10^{4} \mathrm{rad} / \mathrm{s} .\) How many times does the diaphragm move back and forth?

Atoms in a solid are not stationary, but vibrate about their equilibrium positions. Typically, the frequency of vibration is about \(f=2.0 \times 10^{12} \mathrm{~Hz},\) and the amplitude is about \(1.1 \times 10^{-11} \mathrm{~m}\). For a typical atom, what is its (a) maximum speed and (b) maximum acceleration?

In Concept Simulation 10.3 at you can explore the concepts that are important in this problem. A block of mass \(m=0.750 \mathrm{~kg}\) is fastened to an unstrained horizontal spring whose spring constant is \(k=82.0 \mathrm{~N} / \mathrm{m} .\) The block is given a displacement of \(+0.120 \mathrm{~m}\) where the \(+\) sign indicates that the displacement is along the \(+x\) axis, and then released from rest. (a) What is the force (magnitude and direction) that the spring exerts on the block just before the block is released? (b) Find the angular frequency \(\omega\) of the resulting oscillatory motion. (c) What is the maximum speed of the block? (d) Determine the magnitude of the maximum acceleration of the block.

A person bounces up and down on a trampoline, while always staying in contact with it. The motion is simple harmonic motion, and it takes 1.90 s to complete one cycle. The height of each bounce above the equilibrium position is \(45.0 \mathrm{~cm} .\) Determine (a) the amplitude and (b) the angular frequency of the motion. (c) What is the maximum speed attained by the person?

Objects of equal mass are oscillating up and down in simple harmonic motion on two different vertical springs. The spring constant of spring 1 is \(174 \mathrm{~N} / \mathrm{m}\). The motion of the object on spring 1 has twice the amplitude as the motion of the object on spring \(2 .\) The magnitude of the maximum velocity is the same in each case. Find the spring constant of spring 2

Multiple-Concept Example 6 reviews the principles that play a role in this problem. A bungee jumper, whose mass is \(82 \mathrm{~kg}\), jumps from a tall platform. After reaching his lowest point, he continues to oscillate up and down, reaching the low point two more times in \(9.6 \mathrm{~s}\). Ignoring air resistance and assuming that the bungee cord is an ideal spring, determine its spring constant.

Interactive Solution \(\underline{10.21}\) at presents a model for solving this problem. A spring (spring constant \(=112 \mathrm{~N} / \mathrm{m}\) ) is mounted on the floor and is oriented vertically. A 0.400 kg block is placed on top of the spring and pushed down to start it oscillating in simple harmonic motion. The block is not attached to the spring. (a) Obtain the frequency (in \(\mathrm{Hz}\) ) of the motion. (b) Determine the amplitude at which the block will lose contact with the spring.

The drawing shows a top view of a frictionless horizontal surface, where there are two springs with particles of mass \(m_{1}\) and \(m_{2}\) attached to them. Each spring has a spring constant of \(120 \mathrm{~N} / \mathrm{m}\). The particles are pulled to the right and then released from the positions shown in the drawing. How much time passes before the particles are side by side for the first time at \(x=0 \mathrm{~m}\) if \((\mathrm{a}) \mathrm{m}_{1}=m_{2}=3.0 \mathrm{~kg}\) and \((\mathrm{b}) m_{1}=3.0 \mathrm{~kg}\) and \(m_{2}=27 \mathrm{~kg} ?\)

An archer pulls the bowstring back for a distance of \(0.470 \mathrm{~m}\) before releasing the arrow. The bow and string act like a spring whose spring constant is \(425 \mathrm{~N} / \mathrm{m}\). (a) What is the elastic potential energy of the drawn bow? (b) The arrow has a mass of \(0.0300 \mathrm{~kg}\). How fast is it traveling when it leaves the bow?

A spring is hung from the ceiling. A \(0.450\) -kg block is then attached to the free end of the spring. When released from rest, the block drops \(0.150 \mathrm{~m}\) before momentarily coming to rest. (a) What is the spring constant of the spring? (b) Find the angular frequency of the block's vibrations.

A spring is hung from the ceiling. A \(0.450-\mathrm{kg}\) block is then attached to the free end of the spring. When released from rest, the block drops \(0.150 \mathrm{~m}\) before momentarily coming to rest. (a) What is the spring constant of the spring? (b) Find the angular frequency of the block's vibrations.

A rifle fires a \(2.10 \times 10^{-2} \mathrm{~kg}\) pellet straight upward, because the pellet rests on a compressed spring that is released when the trigger is pulled. The spring has a negligible mass and is compressed by \(9.10 \times 10^{-2} \mathrm{~m}\) from its unstrained length. The pellet rises to a maximum height of \(6.10 \mathrm{~m}\) above its position on the compressed spring. Ignoring air resistance, determine the spring constant.

A rifle fires a \(2.10 \times 10^{-2}\) kg pellet straight upward, because the pellet rests on a compressed spring that is released when the trigger is pulled. The spring has a negligible mass and is compressed by \(9.10 \times 10^{-2} \mathrm{~m}\) from its unstrained length. The pellet rises to a maximum height of \(6.10 \mathrm{~m}\) above its position on the compressed spring. Ignoring air resistance, determine the spring constant.

A vertical spring with a spring constant of \(450 \mathrm{~N} / \mathrm{m}\) is mounted on the floor. From directly above the spring, which is unstrained, a \(0.30-\mathrm{kg}\) block is dropped from rest. It collides with and sticks to the spring, which is compressed by \(2.5 \mathrm{~cm}\) in bringing the block to a momentary halt. Assuming air resistance is negligible, from what height (in \(\mathrm{cm}\) ) above the compressed spring was the block dropped?

Refer to Interactive Solution \(\underline{10.29}\) at for help in solving this problem. A heavy-duty stapling gun uses a \(0.140-\mathrm{kg}\) metal rod that rams against the staple to eject it. The rod is pushed by a stiff spring called a "ram spring" \((k=32000 \mathrm{~N} / \mathrm{m})\). The mass of this spring may be ignored. Squeezing the handle of the gun first compresses the ram spring by \(3.0 \times 10^{-2} \mathrm{~m}\) from its unstrained length and then releases it. Assuming that the ram spring is oriented vertically and is still compressed by \(0.8 \times 10^{-2} \mathrm{~m}\) when the downwardmoving ram hits the staple, find the speed of the ram at the instant of contact.

A \(1.00 \times 10^{-2} \mathrm{~kg}\) block is resting on a horizontal frictionless surface and is attached to a horizontal spring whose spring constant is \(124 \mathrm{~N} / \mathrm{m}\). The block is shoved parallel to the spring axis and is given an initial speed of \(8.00 \mathrm{~m} / \mathrm{s}\), while the spring is initially unstrained. What is the amplitude of the resulting simple harmonic motion?

A horizontal spring is lying on a frictionless surface. One end of the spring is attached to a wall while the other end is connected to a movable object. The spring and object are compressed by \(0.065 \mathrm{~m}\), released from rest, and subsequently oscillate back and forth with an angular frequency of \(11.3 \mathrm{rad} / \mathrm{s}\). What is the speed of the object at the instant when the spring is stretched by \(0.048 \mathrm{~m}\) relative to its unstrained length?

A 1.1-kg object is suspended from a vertical spring whose spring constant is \(120 \mathrm{~N} /\) \(\mathrm{m}\). (a) Find the amount by which the spring is stretched from its unstrained length. (b) The object is pulled straight down by an additional distance of \(0.20 \mathrm{~m}\) and released from rest. Find the speed with which the object passes through its original position on the way up.

A \(1.1-\mathrm{kg}\) object is suspended from a vertical spring whose spring constant is \(120 \mathrm{~N} /\) \(\mathrm{m}\) (a) Find the amount by which the spring is stretched from its unstrained length. (b) The object is pulled straight down by an additional distance of \(0.20 \mathrm{~m}\) and released from rest. Find the speed with which the object passes through its original position on the way up.

A \(1.00 \times 10^{-2} \mathrm{~kg}\) bullet is fired horizontally into a \(2.50-\mathrm{kg}\) wooden block attached to one end of a massless, horizontal spring \((k=845 \mathrm{~N} / \mathrm{m})\). The other end of the spring is fixed in place, and the spring is unstrained initially. The block rests on a horizontal, frictionless surface. The bullet strikes the block perpendicularly and quickly comes to a halt within it. As a result of this completely inelastic collision, the spring is compressed along its axis and causes the block/bullet to oscillate with an amplitude of \(0.200 \mathrm{~m}\). What is the speed of the bullet?

A \(1.00 \times 10^{-2}\) kg bullet is fired horizontally into a \(2.50-\mathrm{kg}\) wooden block attached to one end of a massless, horizontal spring \((k=845 \mathrm{~N} / \mathrm{m})\). The other end of the spring is fixed in place, and the spring is unstrained initially. The block rests on a horizontal, frictionless surface. The bullet strikes the block perpendicularly and quickly comes to a halt within it. As a result of this completely inelastic collision, the spring is compressed along its axis and causes the block/bullet to oscillate with an amplitude of \(0.200 \mathrm{~m}\). What is the speed of the bullet?

A 70.0 -kg circus performer is fired from a cannon that is elevated at an angle of \(40.0^{\circ}\) above the horizontal. The cannon uses strong elastic bands to propel the performer, much in the same way that a slingshot fires a stone. Setting up for this stunt involves stretching the bands by \(3.00 \mathrm{~m}\) from their unstrained length. At the point where the performer flies free of the bands, his height above the floor is the same as that of the net into which he is shot. He takes 2.14 s to travel the horizontal distance of \(26.8 \mathrm{~m}\) between this point and the net. Ignore friction and air resistance and determine the effective spring constant of the firing mechanism.

If the period of a simple pendulum is to be \(2.0 \mathrm{~s},\) what should be its length?

A simple pendulum is made from a \(0.65\) -m-long string and a small ball attached to its free end. The ball is pulled to one side through a small angle and then released from rest. After the ball is released, how much time elapses before it attains its greatest speed?

A simple pendulum is made from a \(0.65-\mathrm{m}\) -long string and a small ball attached to its free end. The ball is pulled to one side through a small angle and then released from rest. After the ball is released, how much time elapses before it attains its greatest speed?

A spiral staircase winds up to the top of a tower in an old castle. To measure the height of the tower, a rope is attached to the top of the tower and hung down the left of the staircase. However, nothing is available with which to measure the length of the rope. Therefore, at the bottom of the rope a small object is attached so as to form a simple pendulum that just clears the floor. The period of the pendulum is measured to be \(9.2 \mathrm{~s}\). What is the height of the tower?

The length of a simple pendulum is \(0.79 \mathrm{~m}\) and the mass of the particle (the "bob") at the end of the cable is \(0.24 \mathrm{~kg}\). The pendulum is pulled away from its equilibrium position by an angle of \(8.50^{\circ}\) and released from rest. Assume that friction can be neglected and that the resulting oscillatory motion is simple harmonic motion. (a) What is the angular frequency of the motion? (b) Using the position of the bob at its lowest point as the reference level, determine the total mechanical energy of the pendulum as it swings back and forth. (c) What is the bob's speed as it passes through the lowest point of the swing?

Multiple-Concept Example 11 explores the concepts that are important in this problem. Pendulum A is a physical pendulum made from a thin, rigid, and uniform rod whose length is \(d\). One end of this rod is attached to the ceiling by a frictionless hinge, so the rod is free to swing back and forth. Pendulum B is a simple pendulum whose length is also \(d\). Obtain the ratio \(T_{\mathrm{A}} / T_{\mathrm{B}}\) of their periods for small-angle oscillations.

A point on the surface of a solid sphere (radius \(=R)\) is attached directly to a pivot on the ceiling. The sphere swings back and forth as a physical pendulum with a small amplitude. What is the length of a simple pendulum that has the same period as this physical pendulum? Give your answer in terms of \(R\).

A student's CD player is mounted on four cylindrical rubber blocks. Each cylinder has a height of \(0.030 \mathrm{~m}\) and a cross-sectional area of \(1.2 \times 10^{-3} \mathrm{~m}^{2}\), and the shear modulus for rubber is \(2.6 \times 10^{6} \mathrm{~N} / \mathrm{m}^{2}\). If a horizontal force of magnitude \(32 \mathrm{~N}\) is applied to the CD player, how far will the unit move sideways? Assume that each block is subjected to one-fourth of the force.

Between each pair of vertebrae in the spinal column is a cylindrical disc of cartilage. Typically, this disc has a radius of about \(3.0 \times 10^{-2} \mathrm{~m}\) and a thickness of about \(7.0 \times 10^{-3} \mathrm{~m}\). The shear modulus of cartilage is \(1.2 \times 10^{7} \mathrm{~N} / \mathrm{m}^{2}\). Suppose a shearing force of magnitude \(11 \mathrm{~N}\) is applied parallel to the top surface of the disc while the bottom surface remains fixed in place. How far does the top surface move relative to the bottom surface?

A piece of mohair taken from an Angora goat has a radius of \(31 \times 10^{-6} \mathrm{~m}\). What is the least number of identical pieces of mohair that should be used to suspend a \(75-\mathrm{kg}\) person, so the strain \(\Delta L / L_{0}\) experienced by each piece is less than \(0.010 ?\) Assume that the tension is the same in all the pieces.

A die is designed to punch holes with a radius of \(1.00 \times 10^{-2} \mathrm{~m}\) in a metal sheet that is \(3.0 \times 10^{-3} \mathrm{~m}\) thick, as the drawing illustrates. To punch through the sheet, the die must exert a shearing stress of \(3.5 \times 10^{8} \mathrm{~Pa}\). What force \(\overrightarrow{\mathbf{F}}\) must be applied to the die?

A \(1.0 \times 10^{-3}-\mathrm{kg}\) house spider is hanging vertically by a thread that has a Young's modulus of \(4.5 \times 10^{9} \mathrm{~N} / \mathrm{m}^{2}\) and a radius of \(13 \times 10^{-6} \mathrm{~m}\). Suppose that a \(95-\mathrm{kg}\) person is hanging vertically on an aluminum wire. What is the radius of the wire that would exhibit the same strain as the spider's thread, when the thread is stressed by the full weight of the spider?