Physics for Scientists and Engineers with Modern Physics

Raymond A. Serway, John W. Jewett, Jr.

Chapter 15

Oscillatory Motion - all with Video Answers

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Chapter Questions

Problem 1

A 0.60 -kg block attached to a spring with force constant 130 $\mathrm{N} / \mathrm{m}$ is free to move on a frictionless, horizontal surface as in Active Figure $15.1 .$ The block is released from rest when the spring is stretched 0.13 $\mathrm{m}$ . At the instant the block is released, find (a) the force on the block and (b) its acceleration.

Matt Braby

Matt Braby

Numerade Educator

Problem 2

When a 4.25-kg object is placed on top of a vertical spring, the spring compresses a distance of 2.62 cm. What is the force constant of the spring?

Keshav Singh

Numerade Educator

Problem 3

A vertical spring stretches 3.9 cm when a 10-g object is hung from it. The object is replaced with a block of mass 25 g that oscillates up and down in simple harmonic motion. Calculate the period of motion.

Michael Sorola

Numerade Educator

Problem 4

In an engine, a piston oscillates with simple harmonic motion so that its position varies according to the
expression $$ x=5.00 \cos \left(2 t+\frac{\pi}{6}\right) $$
where $x$ is in centimeters and $t$ is in seconds. At $t=0$ , find (a) the position of the particle, $(\mathrm{b})$ its velocity, and $(\mathrm{c})$ its acceleration. Find $(\mathrm{d})$ the period and $(\mathrm{e})$ the amplitude of the motion.

Dading Chen

Numerade Educator

Problem 5

The position of a particle is given by the expression $x=$ $4.00 \cos (3.00 \pi t+\pi),$ where $x$ is in meters and $t$ is in seconds. Determine (a) the frequency and (b) period of the motion, (c) the amplitude of the motion, (d) the phase constant, and (e) the position of the particle at $t=0.250 \mathrm{s}$ .

Matt Braby

Numerade Educator

Problem 6

A piston in a gasoline engine is in simple harmonic motion. The engine is running at the rate of 3600 rev/min. Taking the extremes of its position relative to its center point as $\pm 5.00 \mathrm{cm},$ find the magnitudes of the (a) maximum velocity and (b) maximum acceleration of the piston.

Keshav Singh

Numerade Educator

Problem 7

A 1.00 -kg object is attached to a horizontal spring. The spring is initially stretched by 0.100 $\mathrm{m}$ , and the object is released from rest there. It proceeds to move without friction. The next time the speed of the object is zero is 0.500 $\mathrm{s}$ later. What is the maximum speed of the object?

Matt Braby

Numerade Educator

Problem 8

A simple harmonic oscillator takes 12.0 s to undergo five complete vibrations. Find (a) the period of its motion, (b) the frequency in hertz, and (c) the angular frequency in radians per second.

Keshav Singh

Numerade Educator

Problem 9

A 7.00 -kg object is hung from the bottom end of a vertical spring fastened to an overhead beam. The object is set into vertical oscillations having a period of 2.60 $\mathrm{s}$ . Find the force constant of the spring.

Matt Braby

Numerade Educator

Problem 10

(a) A hanging spring stretches by 35.0 cm when an object of mass 450 g is hung on it at rest. In this situation, we define its position as $x=0$. The object is pulled down an additional 18.0 cm and released from rest to oscillate without friction. What is its position x at a moment 84.4 s later? (b) Find the distance traveled by the vibrating object in part (a). (c) What If? Another hanging spring stretches
by 35.5 cm when an object of mass 440 g is hung on it at rest. We define this new position as $x=0$. This object is also pulled down an additional 18.0 cm and released from rest to oscillate without friction. Find its position 84.4 s later. (d) Find the distance traveled by the object in part (c). (e) Why are the answers to parts (a) and (c) so different when the initial data in parts (a) and (c) are so similar and
the answers to parts (b) and (d) are relatively close? Does this circumstance reveal a fundamental difficulty in calculating the future?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 11

A particle moves along the $x$ axis. It is initially at the position $0.270 \mathrm{m},$ moving with velocity 0.140 $\mathrm{m} / \mathrm{s}$ and acceleration $-0.320 \mathrm{m} / \mathrm{s}^{2}$ . Suppose it moves as a particle under constant acceleration for 4.50 s. Find (a) its position and (b) its velocity at the end of this time interval. Next, assume it moves as a particle in simple harmonic motion for 4.50 s and $x=0$ is its equilibrium position. Find $(\mathrm{c})$ its position and $(\mathrm{d})$ its velocity at the end of this time interval.

Matt Braby

Numerade Educator

Problem 12

A ball dropped from a height of 4.00 m makes an elastic collision with the ground. Assuming no mechanical energy is lost due to air resistance, (a) show that the ensuing motion is periodic and (b) determine the period of the motion. (c) Is the motion simple harmonic? Explain.

Keshav Singh

Numerade Educator

Problem 13

A particle moving along the $x$ axis in simple harmonic motion starts from its equilibrium position, the origin, at $t=0$ and moves to the right. The amplitude of its motion is $2.00 \mathrm{cm},$ and the frequency is 1.50 $\mathrm{Hz}$ . (a) Find an expression for the position of the particle as a function of time. Determine (b) the maximum speed of the particle and (c) the earliest time $(t>0)$ at which the particle has this speed. Find (d) the maximum positive acceleration of the particle and (e) the earliest time $(t>0)$ at which the particle has this acceleration. (f) Find the total distance traveled by the particle between $t=0$ and $t=1.00$ s.

Matt Braby

Numerade Educator

Problem 14

A 1.00 -kg glider attached to a spring with a force constant of 25.0 $\mathrm{N} / \mathrm{m}$ oscillates on a frictionless, horizontal air track. At $t=0,$ the glider is released from rest at $x=-3.00 \mathrm{cm}$
(that is, the spring is compressed by 3.00 $\mathrm{cm}$ ). Find (a) the period of the glider's motion, (b) the maximum values of its speed and acceleration, and $(c)$ the position, velocity, and acceleration as functions of time.

Penny Riley

Numerade Educator

Problem 15

A $0.500-\mathrm{kg}$ object attached to a spring with a force constant of 8.00 $\mathrm{N} / \mathrm{m}$ vibrates in simple harmonic motion with an amplitude of $10.0 \mathrm{cm} .$ Calculate the maximum value of its (a) speed and (b) acceleration, (c) the speed and (d) the acceleration when the object is 6.00 $\mathrm{cm}$ from the equilibrium position, and (e) the time interval required for the
object to move from $x=0$ to $x=8.00 \mathrm{cm} .$

Matt Braby

Numerade Educator

Problem 16

You attach an object to the bottom end of a hanging vertical spring. It hangs at rest after extending the spring 18.3 cm. You then set the object vibrating. (a) Do you have enough information to find its period? (b) Explain your answer and state whatever you can about its period.

Keshav Singh

Numerade Educator

Problem 17

To test the resiliency of its bumper during low-speed collisions, a 1000 -kg automobile is driven into a brick wall. The car's bumper behaves like a spring with a force constant $5.00 \times 10^{6} \mathrm{N} / \mathrm{m}$ and compresses 3.16 $\mathrm{cm}$ as the car is brought to rest. What was the speed of the car before impact, assuming no mechanical energy is transformed or transferred away during impact with the wall?

Matt Braby

Numerade Educator

Problem 18

A 200 -g block is attached to a horizontal spring and executes simple harmonic motion with a period of 0.250 s. The total energy of the system is 2.00 $\mathrm{J}$ . Find (a) the force constant of the spring and (b) the amplitude of the motion.

Keshav Singh

Numerade Educator

Problem 19

A 50.0 -g object connected to a spring with a force constant of 35.0 $\mathrm{N} / \mathrm{m}$ oscillates with an amplitude of 4.00 $\mathrm{cm}$ on a frictionless, horizontal surface. Find (a) the total energy of the system and (b) the speed of the object when its position is $1.00 \mathrm{cm} .$ Find $(\mathrm{c})$ the kinetic energy and (d) the potential energy when its position is $3.00 \mathrm{cm} .$

Matt Braby

Numerade Educator

Problem 20

A 2.00-kg object is attached to a spring and placed on a frictionless, horizontal surface. A horizontal force of 20.0 N is required to hold the object at rest when it is pulled 0.200 m from its equilibrium position (the origin of the x axis). The object is now released from rest from this stretched position, and it subsequently undergoes simple harmonic oscillations. Find (a) the force constant of the spring, (b) the frequency of the oscillations, and (c) the maximum speed of the object. (d) Where does this maximum speed occur? (e) Find the maximum acceleration of the object. (f) Where does the maximum acceleration occur? (g) Find the total energy of the oscillating system. Find (h) the speed and (i) the acceleration of the object when its position is equal to one-third the maximum value.

Deepak Kohli

Numerade Educator

Problem 21

A simple harmonic oscillator of amplitude A has a total energy E. Determine (a) the kinetic energy and
(b) the potential energy when the position is one-third the amplitude. (c) For what values of the position does the kinetic energy equal one-half the potential energy? (d) Are there any values of the position where the kinetic energy is greater than the maximum potential energy? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 22

A 65.0-kg bungee jumper steps off a bridge with a light bungee cord tied to her body and to the bridge.
The unstretched length of the cord is 11.0 m. The jumper reaches the bottom of her motion 36.0 m below the bridge before bouncing back. We wish to find the time interval between her leaving the bridge and her arriving at the bottom of her motion. Her overall motion can be separated into an 11.0-m free fall and a 25.0-m section of simple harmonic oscillation. (a) For the free-fall part, what is the
appropriate analysis model to describe her motion? (b) For what time interval is she in free fall? (c) For the simple harmonic oscillation part of the plunge, is the system of the bungee jumper, the spring, and the Earth isolated or non-isolated? (d) From your response in part (c) find the spring constant of the bungee cord. (e) What is the location of the equilibrium point where the spring force balances the
gravitational force exerted on the jumper? (f) What is the angular frequency of the oscillation? (g) What time interval is required for the cord to stretch by 25.0 m? (h) What is the total time interval for the entire 36.0-m drop?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 23

A 0.250-kg block resting on a frictionless, horizontal surface is attached to a spring whose force constant is 83.8 N/m as in Figure P15.23. A horizontal force F S causes the spring to stretch a distance of 5.46 cm from its equilibrium position. (a) Find the magnitude of F S. (b) What is the total energy stored in the system when the spring is stretched? (c) Find the magnitude of the acceleration of the block just after the applied force is removed. (d) Find the speed of the block when it first reaches the equilibrium position. (e) If the surface is not frictionless but the block still reaches the equilibrium position, would your answer to part (d) be larger or smaller? (f) What other information would you need to know to find the actual answer to part (d) in this case? (g) What is the largest value of the coefficient of friction that would allow the block to reach the equilibrium position?

William Affel

Numerade Educator

Problem 24

A $326-$ g object is attached to a spring and executes simple harmonic motion with a period of 0.250 s. If the total energy of the system is 5.83 $\mathrm{J}$ , find (a) the maximum speed of the object, (b) the force constant of the spring, and (c) the amplitude of the motion.

Averell Hause

Carnegie Mellon University

Problem 25

A 326-g object is attached to a spring and executes simple harmonic motion with a period of 0.250 s. If the total energy of the system is 5.83 J, find (a) the maximum speed of the object, (b) the force constant of the spring, and (c) the amplitude of the motion.

Matt Braby

Numerade Educator

Problem 26

While driving behind a car traveling at 3.00 m/s, you notice that one of the car’s tires has a small hemispherical bump on its rim as shown in Figure P15.25. (a) Explain why the bump, from your viewpoint behind the car, executes simple harmonic motion. (b) If the radii of the car’s tires are 0.300 m, what
is the bump’s period of oscillation?

Keshav Singh

Numerade Educator

Problem 27

A simple pendulum makes 120 complete oscillations in 3.00 $\mathrm{min}$ at a location where $g=9.80 \mathrm{m} / \mathrm{s}^{2} .$ Find (a) the period of the pendulum and $(\mathrm{b})$ its length.

Matt Braby

Numerade Educator

Problem 28

A particle of mass m slides without friction inside a hemispherical bowl of radius R. Show that if the particle starts from rest with a small displacement from equilibrium, it moves in simple harmonic motion with an angular frequency equal to that of a simple pendulum of length $R .$
That is, $\omega=\sqrt{g / R} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 29

A physical pendulum in the form of a planar object moves in simple harmonic motion with a frequency of 0.450 $\mathrm{Hz}$ . The pendulum has a mass of 2.20 $\mathrm{kg}$ , and the pivot is. located 0.350 $\mathrm{m}$ from the center of mass. Determine the moment of inertia of the pendulum about the pivot point.

Matt Braby

Numerade Educator

Problem 30

If A physical pendulum in the form of a planar object moves in simple harmonic motion with a frequency $f$ . The pendulum has a mass $m$ , and the pivot is located a distance $d$ from the center of mass. Determine the moment of inertia of the pendulum about the pivot point.

Keshav Singh

Numerade Educator

Problem 31

A simple pendulum has a mass of 0.250 kg and a length of 1.00 m. It is displaced through an angle of 15.08 and then released. Using the analysis model of a particle in simple harmonic motion, what are (a) the maximum speed of the bob, (b) its maximum angular acceleration, and (c) the maximum restoring force on the bob? (d) What If? Solve parts (a) through (c) again by using analysis models introduced in earlier chapters. (e) Compare the answers.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 32

Consider the physical pendulum of Figure 15.17. (a) Represent its moment of inertia about an axis passing through its center of mass and parallel to the axis passing through its pivot point as $I_{\mathrm{CM}}$. Show that its period is $$ T=2 \pi \sqrt{\frac{I_{\mathrm{CM}}+m d^{2}}{m g d}} $$
where $d$ is the distance between the pivot point and the center of mass. (b) Show that the period has a minimum value when $d$ satisfies $m d^{2}=I_{\mathrm{CM}} .$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 33

A simple pendulum is 5.00 $\mathrm{m}$ long. What is the period of small oscillations for this pendulum if it is located in an elevator (a) accelerating upward at 5.00 $\mathrm{m} / \mathrm{s}^{2}$ (b) Accelerating downward at 5.00 $\mathrm{m} / \mathrm{s}^{2}$ (c) What is the period of this pendulum if it is placed in a truck that is accelerating horizontally at 5.00 $\mathrm{m} / \mathrm{s}^{2} ?$

Matt Braby

Numerade Educator

Problem 34

A very light rigid rod of length 0.500 m extends straight out from one end of a meterstick. The combination is suspended from a pivot at the upper end of the rod as shown in Figure P15.34.
The combination is then pulled out by a small angle and released. (a) Determine the period of oscillation of the system. (b) By what percentage does the period differ from the period of a simple pendulum 1.00 m long?

Sheh Lit Chang

University of Washington

Problem 35

A watch balance wheel (Fig. P15.35) has a period of oscillation of 0.250 s. The wheel is constructed so that its mass of 20.0 g is concentrated around a rim of radius 0.500 cm. What are (a) the wheel’s moment of inertia and (b) the torsion constant of the attached spring?

Matt Braby

Numerade Educator

Problem 36

A small object is attached to the end of a string to form a simple pendulum. The period of its harmonic motion is measured for small angular displacements and three lengths. For lengths of 1.000 m, 0.750 m, and 0.500 m, total time intervals for 50 oscillations of 99.8 s, 86.6 s, and 71.1 s are measured with a stopwatch. (a) Determine the period of motion for each length. (b) Determine the mean value of g obtained from these three independent measurements and compare it with the accepted value. (c) Plot $T^{2}$ versus L and obtain a value for g from the slope of your best-fit straight-line graph. (d) Compare the value found in part (c) with that obtained in part (b).

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 37

A pendulum with a length of 1.00 $\mathrm{m}$ is released from an initial angle of $15.0^{\circ} .$ After 1000 $\mathrm{s}$ , its amplitude has been reduced by friction to $5.50^{\circ} .$ What is the value of $b / 2 \mathrm{m}$ ?

Matt Braby

Numerade Educator

Problem 38

Show that the time rate of change of mechanical energy for a damped, undriven oscillator is given by
$d E / d t=-b v^{2}$ and hence is always negative. To do so, differentiate the expression for the mechanical energy of an oscillator, $E=\frac{1}{2} m v^{2}+\frac{1}{2} k x^{2},$ and use Equation $15.31 .$

Keshav Singh

Numerade Educator

Problem 39

A 10.6 -kg object oscillates at the end of a vertical spring that has a spring constant of $2.05 \times 10^{4} \mathrm{N} / \mathrm{m}$ . The effect of air resistance is represented by the damping coefficient $b=$ $3.00 \mathrm{N} \cdot \mathrm{s} / \mathrm{m} .$ (a) Calculate the frequency of the damped
oscillation. (b) By what percentage does the amplitude of the oscillation decrease in each cycle? (c) Find the time interval that elapses while the energy of the system drops to 5.00$\%$ of its initial value.

Matt Braby

Numerade Educator

Problem 40

Show that Equation 15.32 is a solution of Equation 15.31 provided that $b^{2}<4 m k$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 41

As you enter a fine restaurant, you realize that you have accidentally brought a small electronic timer from home instead of your cell phone. In frustration, you drop the timer into a side pocket of your suit coat, not realizing that the timer is operating. The arm of your chair presses the light cloth of your coat against your body at one spot. Fabric with a length L hangs freely below that spot, with the timer at the bottom. At one point during your dinner, the timer goes off and a buzzer and a vibrator turn on and off with a frequency of 1.50 Hz. It makes the hanging part of your coat swing back and forth with remarkably large amplitude,
drawing everyone’s attention. Find the value of L.

Matt Braby

Numerade Educator

Problem 42

A baby bounces up and down in her crib. Her mass is 12.5 kg, and the crib mattress can be modeled as a light spring with force constant 700 N/m. (a) The baby soon learns to bounce with maximum amplitude and minimum effort by bending her knees at what frequency? (b) If she were to use the mattress as a trampoline—losing contact with it for part of each cycle—what minimum amplitude of oscillation does she require?

Keshav Singh

Numerade Educator

Problem 43

A. $00-\mathrm{kg}$ object attached to a spring moves without friction $(b=0)$ and is driven by an external force given by the expression $F=3.00 \sin (2 \pi t),$ where $F$ is in newtons and $t$ is in seconds. The force constant of the spring is 20.0 $\mathrm{N} / \mathrm{m}$ . Find $(\mathrm{a})$ the resonance angular frequency of the system, (b) the angular frequency of the driven system, and (c) the
amplitude of the motion.

Matt Braby

Numerade Educator

Problem 44

A block weighing 40.0 $\mathrm{N}$ is suspended from a spring that has a force constant of 200 $\mathrm{N} / \mathrm{m}$ . The system is undamped $(b=0)$ and is subjected to a harmonic driving force of frequency 10.0 $\mathrm{Hz}$ , resulting in a forced-motion amplitude of $2.00 \mathrm{cm} .$ Determine the maximum value of the driving force.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 45

Damping is negligible for a 0.150 -kg object hanging from a light, $6.30-\mathrm{N} / \mathrm{m}$ spring. A sinusoidal force with an amplitude of 1.70 $\mathrm{N}$ drives the system. At what frequency
will the force make the object vibrate with an amplitude of 0.440 $\mathrm{m}$ ?

Matt Braby

Numerade Educator

Problem 46

Considering an undamped, forced oscillator $(b=0)$ show that Equation 15.35 is a solution of Equation $15.34,$ with an amplitude given by Equation $15.36 .$

Luis Rios

Numerade Educator

Problem 47

The mass of the deuterium molecule $\left(\mathrm{D}_{2}\right)$ is twice that of the hydrogen molecule $\left(\mathrm{H}_{2}\right) .$ If the vibrational frequency of $\mathrm{H}_{2}$ is $1.30 \times 10^{14} \mathrm{Hz}$ , what is the vibrational frequency of $\mathrm{D}_{2} ?$ Assume the "spring constant" of attracting forces is the same for the two molecules.

Matt Braby

Numerade Educator

Problem 48

This problem extends the reasoning of Problem 59 in Chapter $9 .$ Two gliders are set in motion on an
air track. Glider 1 has mass $m_{1}=0.240 \mathrm{kg}$ and moves to the right with speed 0.740 $\mathrm{m} / \mathrm{s}$ . It will have a rear-end collision with glider $2,$ of mass $m_{2}=0.360 \mathrm{kg}$ , which initially moves to the right with speed 0.120 $\mathrm{m} / \mathrm{s}$ . A light spring of force constant 45.0 $\mathrm{N} / \mathrm{m}$ is attached to the back end of glider 2 as shown in Figure $\mathrm{P} 9.59 .$ When glider 1 touches the spring, superglue instantly and permanently makes it stick to its end of the spring. (a) Find the common speed the
two gliders have when the spring is at maximum compres-sion. (b) Find the maximum spring compression distance. The motion after the gliders become attachession of the cencombination of $(1)$ the constant-velocity motion of the center of mass of the two-glider system found in part (a) and
(2) simple harmonic motion of the gliders relative to the center of mass. (c) Find the energy of the center-of-mass motion. (d) Find the energy of the oscillation.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 49

An object of mass $m$ moves in simple harmonic motion with amplitude 12.0 $\mathrm{cm}$ on a light spring. Its maximum acceleration is 108 $\mathrm{cm} / \mathrm{s}^{2}$ . Regard $m$ as a variable. (a) Find the period $T$ of the object. (b) Find its frequency $f .$ (c) Find the maximum speed $v_{\text { max }}$ of the object. (d) Find the total energy $E$ of the object-spring system. (e) Find the force constant $k$ of the spring. (f) Describe the pattern of dependence of each of the quantities $T, f, v_{\max }, E,$ and $k$ on $m .$

Matt Braby

Numerade Educator

Problem 50

A rock rests on a concrete sidewalk. An earthquake strikes, making the ground move vertically
in simple harmonic motion with a constant frequency of 2.40 Hz and with gradually increasing amplitude. (a) With what amplitude does the ground vibrate when the rock begins to lose contact with the sidewalk? Another rock is sitting on the concrete bottom of a swimming pool full of water. The earthquake produces only vertical motion, so the water does not slosh from side to side. (b) Present a convincing argument that when the ground vibrates with the amplitude found in part (a), the submerged rock also barely loses contact with the floor of the swimming pool.

Luis Rios

Numerade Educator

Problem 51

A small ball of mass M is attached to the end of a uniform rod of equal mass M and length L that is pivoted at the top (Fig. P15.51). Determine the tensions in the rod (a) at the pivot and (b) at the point P when the system is stationary. (c) Calculate the period of oscillation for small displacements from equilibrium and (d) determine this period for L 5 2.00 m.

Matt Braby

Numerade Educator

Problem 52

An object attached to a spring vibrates with simple harmonic motion as described by Figure P15.52. For this motion, find (a) the amplitude, (b) the period, (c) the angular frequency, (d) the maximum speed, (e) the maximum acceleration, and (f) an equation for its position x as a function of time.

Keshav Singh

Numerade Educator

Problem 53

A large block P attached to a light spring executes horizontal, simple harmonic motion as it
slides across a frictionless surface with a frequency $f=1.50 \mathrm{Hz}$. Block B rests on it as shown in Figure P15.53, and the coefficient of static friction between the two is $\mu_{s}=0.600$. What maximum amplitude of oscillation can the system have if block B is not to slip?

Matt Braby

Numerade Educator

Problem 54

A large block P attached to a light spring executes horizontal, simple harmonic motion as it slides across a frictionless surface with a frequency f. Block B rests on it as shown in Figure P15.53, and the coefficient of static friction between the two is $\mu_{s}$. What maximum amplitude of oscillation can the system have if block B is not to slip?

Keshav Singh

Numerade Educator

Problem 55

A pendulum of length $L$ and mass $M$ has a spring of force constant $k$ connected to it at a distance $h$ below its point of suspension (Fig. Pl5.55). Find the frequency of vibration of the system for small values of the amplitude (small $\theta )$ Assume the vertical suspension rod of length $L$ is rigid, but
ignore its mass.

Matt Braby

Numerade Educator

Problem 56

A particle with a mass of 0.500 $\mathrm{kg}$ is attached to a horizontal spring with a force constant of $50.0 \mathrm{N} / \mathrm{m} .$ At the moment $t=0,$ the particle has its maximum speed of 20.0 $\mathrm{m} / \mathrm{s}$ and is moving to the left. (a) Determine the particle's equation of motion, specifying its position as a function of time. (b) Where in the motion is the potential energy three
val required for the particle to move from $x=0$ to $x=$ $1.00 \mathrm{m} .$ (d) Find the length of a simple pendulum with the same period.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 57

A horizontal plank of mass 5.00 $\mathrm{kg}$ and length 2.00 $\mathrm{m}$ is pivoted at one
end. The plank's other end. is supported by a spring of force constant 100 $\mathrm{N} / \mathrm{m}$ is displaced by a small angle $\theta$ from its horizontal equilibrium position and released. Find the angular frequency with which the plank moves with simple harmonic motion.

Matt Braby

Numerade Educator

Problem 58

A horizontal plank of mass $m$ and length $L$ is pivoted at one end. The plank's other end is supported by a spring of force constant $k$ (Fig. PIS. $P 15.57 )$ . The plank is displaced by a small angle $\theta$ from its horizontal equilibrium position and released. Find the angular frequency with which the plank
moves with simple harmonic motion.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 59

A particle of mass 4.00 kg is attached to a spring with a force constant of 100 N/m. It is oscillating
on a frictionless, horizontal surface with an amplitude of 2.00 m. A 6.00-kg object is dropped vertically on top of the 4.00-kg object as it passes through its equilibrium point. The two objects stick together. (a) What is the new amplitude of the vibrating system after the collision? (b) By what factor has the period of the system changed? (c) By how much does the energy of the system change as a result of the collision? (d) Account for the change in energy.

Matt Braby

Numerade Educator

Problem 60

A simple pendulum with a length of 2.23 m and a mass of 6.74 kg is given an initial speed of 2.06 m/s at its equilibrium position. Assume it undergoes simple harmonic motion. Determine (a) its period, (b) its total energy, and (c) its maximum angular displacement.

Ren Jie Tuieng

Numerade Educator

Problem 61

Review. One end of a light spring with force constant $k=100 \mathrm{N} / \mathrm{m}$ is attached to
a vertical wall. A light string is tied to the other end of the horizontal spring. As shown in Figure Pl5.61, thestring changes from horizontal to vertical as it passes over a pulley of mass $M$ in the shape of a solid disk of radius $R=2.00 \mathrm{cm} .$ The pulley is free to turn on a fixed, smooth axle. The vertical section of the string supports an object of mass $m=200 \mathrm{g}$ . The string does not slip at its contact with the pulley. The object is pulled downward a small distance and released. (a) What is the angular frequency $\omega$ of oscillation of the object in terms of the mass $M ?$ (b) What is the highest possible value of the angular frequency of oscillation of the object? (C) What is the highest possible value of the angular frequency of oscillation of the object if the pulley radius
is doubled to $R=4.00 \mathrm{cm}$ ?

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 62

People who ride motorcycles and bicycles learn to look out for bumps in the road and especially for wash-boarding, a condition in which many equally spaced ridges are worn into the road. What is so bad about wash boarding? A motorcycle has several springs and shock absorbers in its suspension, but you can model it as a single spring supporting a block. You can estimate the force constant by thinking about how far the spring compresses when a heavy rider sits on the seat. A motorcyclist traveling at highway speed must be particularly careful of washboard bumps that are a certain distance apart. What is the order
of magnitude of their separation distance?

Deepak Kohli

Numerade Educator

Problem 63

A ball of mass $m$ is connected to two rubber bands of length $L,$ each under tension $T$ as shown in Figure $\mathrm{P} 15.63$ . The ball is displaced by a small distance $y$ perpendicular to the length of the rubber bands. Astaning the tension does not change, show that (a) the restoring force is $-(2 T / L) y$
and (b) the system exhibits simple harmonic motion with an angular frequency $\omega=\sqrt{2 T / m L}$ .

Matt Braby

Numerade Educator

Problem 64

When a block of mass $M,$ connected to the end of a spring of mass $m_{s}=7.40 \mathrm{g}$ and force constant $k,$ is set into simple harmonic motion, the period of its motion is
$$T=2 \pi \sqrt{\frac{M+\left(m_{s} / 3\right)}{k}}$$ A two-part experiment is conducted with the use of blocks of various masses suspended vertically from the spring as shown in Figure P15.64. (a) Static extensions of 17.0, 29.3, 35.3, 41.3, 47.1, and 49.3 cm are measured for M values of
20.0, 40.0, 50.0, 60.0, 70.0, and 80.0 g, respectively. Construct a graph of Mg versus x and perform a linear least-squares fit to the data. (b) From the slope of your graph, determine a value for k for this spring. (c) The system is now set into simple harmonic motion, and periods are measured with a stopwatch. With M 5 80.0 g, the total time interval required for ten oscillations is measured to
be 13.41 s. The experiment is repeated with M values of 70.0, 60.0, 50.0, 40.0, and 20.0 g, with corresponding time intervals for ten oscillations of 12.52, 11.67, 10.67, 9.62, and 7.03 s. (d) Compute the experimental value for T from each of these measurements. (e) Plot a graph of T 2 versus
M and (f) determine a value for k from the slope of the linear least-squares fit through the data points. (g) Compare this value of k with that obtained in part (b). (h) Obtain a value for $m_{s}$ from your graph and compare it with the given value of 7.40 g.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 65

A light balloon filled with helium of density 0.179 $\mathrm{kg} / \mathrm{m}^{3}$ is tied to a light string of length L 5 3.00 m. The string is tied to the ground forming an “inverted” simple pendulum (Fig. 15.65a). If the balloon is displaced slightly from equilibrium as in Figure P15.65b and released, (a) show that the motion is simple harmonic and (b) determine the period of the motion. Take the density of air to be 1.20 $\mathrm{kg} / \mathrm{m}^{3}$. Hint: Use an analogy with the simple pendulum and see Chapter 14. Assume the air applies a buoyant force on the balloon but does not otherwise affect its motion.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 66

Consider the damped oscillator illustrated in Figure $15.20 .$ The mass of the object is 375 $\mathrm{g}$ , the spring constant is $100 \mathrm{N} / \mathrm{m},$ and $b=0.100 \mathrm{N} \cdot \mathrm{s} / \mathrm{m} .$ (a) Over what time interval does the amplitude drop to half its initial value? (b) What If? Over what time interval does the mechanical energy drop to half its initial value? (c) Show that, in general, the fractional rate at which the amplitude decreases in a damped harmonic oscillator is one-half the fractional rate at which the mechanical energy decreases.

Prashant Bana

Numerade Educator

Problem 67

A block of mass m is connected to two springs of force constants k1 and k2 in two ways as shown in Figure P15.67. In both cases, the block moves on a frictionless table after it is displaced from equilibrium and released. Show that in the two cases the block exhibits simple harmonic motion with periods
(a) $T=2 \pi \sqrt{\frac{m\left(k_{1}+k_{2}\right)}{k_{1} k_{2}}} \quad$ and $\quad(\mathrm{b}) T=2 \pi \sqrt{\frac{m}{k_{1}+k_{2}}}$

Matt Braby

Numerade Educator

Problem 68

Your thumb squeaks on a plate you have just washed. Your sneakers squeak on the gym floor. Car tires squeal when you start or stop abruptly. You can make a goblet sing by wiping your moistened finger around its rim. When chalk squeaks on a blackboard, you can see that it makes a row of regularly spaced dashes. As these examples suggest, vibration commonly results when friction acts on a moving elastic object. The oscillation is not simple harmonic motion, but is called stick-and-slip. This problem models stick-and-slip motion.A block of mass m is attached to a fixed support by a
horizontal spring with force constant k and negligible mass (Fig. P15.68). Hooke’s law describes the spring both in extension and in compression. The block sits on a long horizontal board, with which it has coefficient of static friction ms and a smaller coefficient of kinetic friction mk. The board moves to the right at constant speed v. Assume the block spends most of its time sticking to the board and moving to the right with it, so the speed v is small in comparison to the average speed the block has as it slips back toward the left. (a) Show that the maximum extension of the spring from its unstressed position is very nearly given by $\mu_{s} m g / k .$ (b) Show that the block oscillates around
an equilibrium position at which the spring is stretched by $\mu_{k} m g / k .$ (c) Graph the block's position versus time. (d) Show that the amplitude of the block's motion is
$$A=\frac{\left(\mu_{s}-\mu_{k}\right) m g}{k}$$
(e) Show that the period of the block's motion is
$$T=\frac{2\left(\mu_{s}-\mu_{k}\right) m g}{v k}+\pi \sqrt{\frac{m}{k}}$$
It is the excess of static over kinetic friction that is important for the vibration. "The squeaky wheel gets the grease" because even a viscous fluid cannot exert a force of static friction.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 69

A lobsterman’s buoy is a solid wooden cylinder of radius r and mass M. It is weighted at one end so that it floats upright in calm seawater, having density r. A passing shark tugs on the slack rope mooring the buoy to a lobster trap, pulling the buoy down a distance x from its equilibrium position and releasing it. (a) Show that the buoy will execute simple harmonic motion if the resistive effects of the water are ignored. (b) Determine the period of the oscillations.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 70

Why is the following situation impossible? Your job involves building very small damped oscillators. One of your designs involves a spring–object oscillator with a spring of force constant k 5 10.0 N/m and an object of mass m 5 1.00 g. Your design objective is that the oscillator undergo many oscillations as its amplitude falls to 25.0% of its initial value in a certain time interval. Measurements on your latest design show that the amplitude falls to the 25.0% value in 23.1 ms. This time interval is too long for what is needed in your project. To shorten the time interval, you double the damping constant b for the oscillator. This doubling allows you to reach your design objective.

Keshav Singh

Numerade Educator

Problem 71

Two identical steel balls, each of mass 67.4 g, are moving in opposite directions at 5.00 m/s. They collide head-on and bounce apart elastically. By squeezing one of the balls in a vise while precise measurements are made of the resulting amount of compression, you find that Hooke’s law is a good model of the ball’s elastic behavior. A force of 16.0 kN exerted by each jaw of the vise reduces the diameter by 0.200 mm. Model the motion of each ball, while the balls are in contact, as one-half of a cycle of simple harmonic motion. Compute the time interval for which the balls are in contact. (If you solved Problem 57 in Chapter 7, com- pare your results from this problem with your results from that one.)

Matt Braby

Numerade Educator

Problem 72

A smaller disk of radius $r$ and mass $m$ is attached rigidly to the face of a second larger disk of radius $R$ and mass $M$ as shown in Figure P15.72. The center of the small disk is located at the edge of the large disk. The large disk is mounted at its center on a frictionless axle. The assembly is rotated through a small angle $\theta$ from its equilibrium position and released. (a) Show that the speed of the center of the small disk as it passes through the equilibrium position is
$$v=2\left[\frac{R g(1-\cos \theta)}{(M / m)+(r / R)^{2}+2}\right]^{1 / 2}$$
(b) Show that the period of the motion is
$$T=2 \pi\left[\frac{(M+2 m) R^{2}+m r^{2}}{2 m g R}\right]^{1 / 2}$$

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 73

An object of mass $m_{1}=9.00 \mathrm{kg}$ is in equilibrium when connected to a light spring of constant $k=100 \mathrm{N} / \mathrm{m}$ that is fastened to a wall as shown in Figure $\mathrm{P} 15.73 \mathrm{a}$ . A second object, $m_{2}=7.00 \mathrm{kg},$ is slown pushed up against $m_{1},$ compressing the spring by the amount $A=0.200 \mathrm{m}$ (see Fig. Pl5.73b). The system is then released, and both objects start moving to the right on the frictionless surface. (a) When $m_{1}$ reaches the equilibrium point, $m_{2}$ loses contact with $m_{1}$ (see Fig. P15.73 c) and moves to the right with speed $v .$ Determine the value of $v$ . (b) How far apart are the objects when the spring is fully stretched for the first time (the distance $D$ in Fig. $\mathrm{P} 15.48 \mathrm{d} ) ?$

Matt Braby

Numerade Educator

Problem 74

Why is the following situation impossible? You are in the high-speed package delivery business. Your competitor in the next building gains the right-of-way to build an evacuated tunnel just above the ground all the way around the Earth. By firing packages into this tunnel at just the right speed, your competitor is able to send the packages into orbit around the Earth in this tunnel so that they arrive on the exact opposite side of the Earth in a very short time interval. You come up with a competing idea. Figuring that the distance through the Earth is shorter than the distance around the Earth, you obtain permits to build an evacuated tunnel through the center of the Earth (Fig. P15.74). By simply dropping packages into this tunnel, they fall down-ward and arrive at the other end of your tunnel, which is
in a building right next to the other end of your competitor’s tunnel. Because your packages arrive on the other side of the Earth in a shorter time interval, you win the competition and your business flourishes. Note: An object at a distance r from the center of the Earth is pulled toward the center of the Earth only by the mass within the sphere of radius r (the reddish region in Fig. P15.74). Assume the
Earth has uniform density.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 75

A block of mass M is connected to a spring of mass m and oscillates in simple harmonic motion on a frictionless, horizontal track (Fig. P15.75). The force constant of the spring is k, and the equilibrium length is ,. Assume all portions of the spring oscillate in phase and the velocity of a segment of the spring of length dx is proportional to the distance $x$ from the fixed end; that is, $v_{x}=(x / \ell) v$ . Also,
notice that the mass of a segment of the spring is $d m=$ $(m / \ell) d x$ . Find $(a)$ the kinetic energy of the system when the block has a speed $v$ and $(b)$ the period of oscillation.

Matt Braby

Numerade Educator

Problem 76

A system consists of a spring with force constant $k=1250 \mathrm{N} / \mathrm{m}$ length $L=1.50 \mathrm{m},$ and an object of mass $m=5.00 \mathrm{kg}$ attached to the end $(\mathrm{Fig} \text { . }$ placed at the level of the position $y_{i}=L,$ and then it is released so that it swings like a pendulum.
(a) Find the $y$ position of the object at the lowest point.
(b) Will the pendulum's period be greater or less than the period of a simple pendulum with the same mass $m$ and length $L$ ? Explain.

Khoobchandra Agrawal

Khoobchandra Agrawal

Numerade Educator

Problem 77

A light, cubical container of volume a3 is initially filled with a liquid of mass density r as shown in Figure P15.77a. The cube is initially supported by a light string to form a simple pendulum of length $L_{i}$ measured from the center of mass of the filled container, where $L_{i}>>a$. The liquid is allowed to flow from the bottom of the container at a constant rate $(d M / d t) .$ At any time $t,$ the level of the liquid in the container is $h$ and the length of the pendulum is $L$ (measured relative to the instantaneous center of mass) as shown in Figure PI5. 77 $\mathrm{b}$ . (a) Find the period of the pendulum as a function of time. (b) What is the period of the pendulum after the liquid completely runs out of the container?

Matt Braby

Numerade Educator