University Physics with Modern Physics

Wolfgang Bauer, Gary D. Westfall

Chapter 14

Oscillations - all with Video Answers

Educators

Chapter Questions

Problem 1

Two children are on adjacent playground swings with chains of the same length. They are given pushes by an adult and then left to swing. Assuming that each child on a swing can be treated as a simple pendulum and that friction is negligible, which child takes the longer time for one complete swing (has a longer period)?
a) the bigger child
b) the lighter child
c) neither child
d) the child given the bigger push

Prabhu Ramji

Prabhu Ramji

Numerade Educator

Problem 2

Identical blocks oscillate on the end of a vertical spring, one on Earth and one on the Moon. Where is the period of the oscillations greater?
a) on Earth
b) on the Moon
c) same on both Earth and Moon
d) cannot be determined from the information given.

Prabhu Ramji

Numerade Educator

Problem 3

A mass that can oscillate without friction on a horizontal surface is attached to a horizontal spring that is pulled to the right $10.0 \mathrm{~cm}$ and is released from rest. The period of oscillation for the mass is $5.60 \mathrm{~s}$. What is the speed of the mass at $t=2.50 \mathrm{~s} ?$
a) $-2.61 \cdot 10^{-1} \mathrm{~m} / \mathrm{s}$
b) $-3.71 \cdot 10^{-2} \mathrm{~m} / \mathrm{s}$
c) $-3.71 \cdot 10^{-1} \mathrm{~m} / \mathrm{s}$
d) $-2.01 \cdot 10^{-1} \mathrm{~m} / \mathrm{s}$

Prabhu Ramji

Numerade Educator

Problem 4

The spring constant for a spring-mass system undergoing simple harmonic motion is doubled. If the total energy remains unchanged, what will happen to the maximum amplitude of the oscillation? Assume that the system is underdamped.
a) It will remain unchanged.
b) It will be multiplied by 2 .
c) It will be multiplied by $\frac{1}{2}$
d) It will be multiplied by $1 / \sqrt{2}$.

Prabhu Ramji

Numerade Educator

Problem 5

With the right choice of parameters, a damped and driven physical pendulum can show chaotic motion, which is sensitively dependent on the initial conditions. Which statement about such a pendulum is true?
a) Its long-term behavior can be predicted.
b) Its long-term behavior is not predictable.
c) Its long-term behavior is like that of a simple pendulum of equivalent length.
d) Its long-term behavior is like that of a conical pendulum.
e) None of the above is true.

Prabhu Ramji

Numerade Educator

Problem 6

A spring is hanging from the ceiling with a mass attached to it. The mass is pulled downward, causing it to oscillate vertically with simple harmonic motion. Which of the following will decrease the frequency of the oscillation?
a) adding a second, identical spring with one end attached to the mass and the other to the ceiling
b) adding a second, identical spring with one end attached to the mass and the other to the floor
c) increasing the mass
d) adding both springs, as described in (a) and (b).

Prabhu Ramji

Numerade Educator

Problem 7

A child of mass $M$ is swinging on a swing of length $L$ to a maximum deflection angle of $\theta .$ A man of mass $4 M$ is swinging on a similar swing of length $L$ to a maximum angle of $2 \theta .$ Each swing can be treated as a simple pendulum undergoing simple harmonic motion. If the period for the child's motion is $T$, then the period for the man's motion is
a) $T$.
b) $2 T$.
c) $T / 2$.
d) $T / 4$.

Prabhu Ramji

Numerade Educator

Problem 8

Object A is four times heavier than object B. Each object is attached to a spring, and the springs have equal spring constants. The two objects are then pulled from their equilibrium positions and released from rest. What is the ratio of the periods of the two oscillators if the amplitude of $\mathrm{A}$ is half that of $\mathrm{B}$ ?
a) $T_{\mathrm{A}}: T_{\mathrm{B}}=1: 4$
b) $T_{\mathrm{A}}: T_{\mathrm{B}}=4: 1$
c) $T_{\mathrm{A}}: T_{\mathrm{B}}=2: 1$
d) $T_{\mathrm{A}}: T_{\mathrm{B}}=1: 2$

Prabhu Ramji

Numerade Educator

Problem 9

Rank the simple harmonic oscillators shown in the figure in order of their intrinsic frequencies, from highest to lowest. All the springs have identical spring constants, and all the blocks have identical masses.
a) $1,2,3,4=5$
b) $4=5,3,2,1$
c) $1,2,3=4,5$
d) 3,4,2,1,5
e) $3,2=5,4,1$

Prabhu Ramji

Numerade Educator

Problem 10

A pendulum is suspended from the ceiling of an elevator. When the elevator is at rest, the period of the pendulum is $T$. The elevator accelerates upward, and the period of the pendulum is then
a) still $T$.
b) less than $T$.
c) greater than $T$.

Prabhu Ramji

Numerade Educator

Problem 11

A 36-kg mass is placed on a horizontal frictionless surface and then connected to walls by two springs with spring constants $k_{1}=3.0 \mathrm{~N} / \mathrm{m}$ and $k_{2}=4.0 \mathrm{~N} / \mathrm{m},$ as shown in the figure. What is the period of oscillation for the 36 -kg mass if it is displaced slightly to one side?
a) $11 \mathrm{~s}$
b) $14 \mathrm{~s}$
c) $17 \mathrm{~s}$
d) $20 .$ s
e) $32 \mathrm{~s}$
f) 38 s

Prabhu Ramji

Numerade Educator

Problem 12

A spring with $k=12.0 \mathrm{~N} / \mathrm{m}$ has a mass $m=3.00 \mathrm{~kg}$ attached to its end. The mass is pulled $+10.0 \mathrm{~cm}$ from the equilibrium position and released from rest. What is the velocity of the mass as it passes the equilibrium position?
a) $-0.125 \mathrm{~m} / \mathrm{s}$
b) $+0.750 \mathrm{~m} / \mathrm{s}$
c) $-0.200 \mathrm{~m} / \mathrm{s}$
d) $+0.500 \mathrm{~m} / \mathrm{s}$
e) $-0.633 \mathrm{~m} / \mathrm{s}$

Prabhu Ramji

Numerade Educator

Problem 13

A grandfather clock keeps time using a pendulum consisting of a light rod connected to a small, heavy mass. How long should the rod be to make the period of the oscillations $1.00 \mathrm{~s} ?$
a) $0.0150 \mathrm{~m}$
b) $0.145 \mathrm{~m}$
c) $0.248 \mathrm{~m}$
d) $0.439 \mathrm{~m}$
e) $0.750 \mathrm{~m}$

Prabhu Ramji

Numerade Educator

Problem 14

An automobile with a mass of $1640 \mathrm{~kg}$ is lifted into the air. During the lift, the suspension spring on each wheel lengthens by $30.0 \mathrm{~cm} .$ What damping constant is required for the shock absorber on each wheel to produce critical damping?
a) $101 \mathrm{~kg} / \mathrm{s}$
b) $234 \mathrm{~kg} / \mathrm{s}$
c) $1230 \mathrm{~kg} / \mathrm{s}$
d) $2310 \mathrm{~kg} / \mathrm{s}$
e) $4690 \mathrm{~kg} / \mathrm{s}$

Prabhu Ramji

Numerade Educator

Problem 15

Suppose you start another weakly damped oscillator with the same initial conditions as in Figure 14.34 and with all parameters unchanged except for a larger mass. How will the trajectory in the phase space change?
a) It remains unchanged.
b) It remains basically unchanged but spirals inward to a different point.
c) It spirals inward to the same point but does so faster; that is, it crosses the $x$ -axis fewer times on its way in.
d) It spirals inward to the same point but does so more slowly; that is, it crosses the $x$ -axis more times on its way in.

Narayan Hari

Numerade Educator

Problem 16

You sit in your SUV in a traffic jam. Out of boredom, you rock your body from side to side and then sit still. You notice that your SUV continues rocking for about $2 \mathrm{~s}$ and makes three side-to-side oscillations, before subsiding. You estimate that the amplitude of the SUV's motion must be less than $5 \%$ of the amplitude of your initial rocking. Knowing that your SUV has a mass of $2200 \mathrm{~kg}$ (including the gas, your stuff, and yourself), what can you say about the effective spring constant and damping of your SUV's suspension?

Narayan Hari

Numerade Educator

Problem 17

An astronaut, taking his first flight on a space shuttle, brings along his favorite miniature grandfather clock. At launch, the clock and his digital wristwatch are synchronized, and the grandfather clock is pointed in the direction of the shuttle's nose. During the boost phase, the shuttle has an upward acceleration whose magnitude is several times the value of the gravitational acceleration at the Earth's surface. As the shuttle reaches a constant cruising speed after completion of the boost phase, the astronaut compares his grandfather clock's time to that of his watch. Are the timepieces still synchronized? If not, which is ahead of the other? Explain.

Prabhu Ramji

Numerade Educator

Problem 18

A small cylinder of mass $m$ can slide without friction on a shaft that is attached to a turntable, as shown in the figure. The shaft also passes through the coils of a spring with spring constant $k$, which is attached to the turntable at one end and to the cylinder at the other end. The equilibrium length of the spring (unstretched and uncompressed) matches the radius of the turntable; thus, when the turntable is not rotating, the cylinder is at equilibrium at the center of the turntable. The cylinder is given a small initial displacement, and, at the same time, the turntable is set into uniform circular motion with angular speed $\omega$ Calculate the period of the oscillations performed by the cylinder. Discuss the result.

Narayan Hari

Numerade Educator

Problem 19

A door closer on a door allows the door to close by itself, as shown in the figure. The door closer consists of a return spring attached to an oil-filled damping piston. As the spring pulls the piston to the right, teeth on the piston rod engage a gear, which rotates and causes the door to close. Should the system be underdamped, critically damped, or overdamped for the best performance (where the door closes quickly without slamming into the frame)?

Averell Hause

Carnegie Mellon University

Problem 20

When the amplitude of the oscillation of a mass on a stretched string is increased, why doesn't the period of oscillation also increase?

Narayan Hari

Numerade Educator

Problem 21

Mass-spring systems and pendulum systems can both be used in mechanical timing devices. What are the advantages of using one type of system rather than the other in a device designed to generate reproducible time measurements over an extended period of time?

Averell Hause

Carnegie Mellon University

Problem 22

You have a linear (following Hooke's Law) spring with an unknown spring constant, a standard mass, and a timer. Explain carefully how you could most practically use these to measure masses in the absence of gravity. Be as quantitative as you can. Regard the mass of the spring as negligible.

Averell Hause

Carnegie Mellon University

Problem 23

Pendulum A has a bob of mass $m$ hung from a string of length $L$; pendulum $\mathrm{B}$ is identical to $\mathrm{A}$ except its bob has mass $2 m .$ Compare the frequencies of small oscillations of the two pendulums.

Prabhu Ramji

Numerade Educator

Problem 24

A sharp spike in a plotted curve of frequency can be represented as a sum of sinusoidal functions of all possible frequencies, with equal amplitudes. A bell struck with a hammer rings at its natural frequency, that is, the frequency at which it vibrates as a free oscillator. Explain why, as clearly and concisely as you can.

Averell Hause

Carnegie Mellon University

Problem 25

A mass $m=5.00 \mathrm{~kg}$ is suspended from a spring and oscillates according to the equation of motion $x(t)=0.500 \cos (5.00 t+\pi / 4)$. What is the spring constant?

Prabhu Ramji

Numerade Educator

Problem 26

Determine the frequency of oscillation of a $200 .-g$ block that is connected by a spring to a wall and sliding on a frictionless surface, for the three conditions shown in the figure.

Prabhu Ramji

Numerade Educator

Problem 27

A mass of $10.0 \mathrm{~kg}$ is hanging by a steel wire $1.00 \mathrm{~m}$ long and $1.00 \mathrm{~mm}$ in diameter. If the mass is pulled down slightly and released, what will be the frequency of the resulting oscillations? Young's modulus for steel is $2.0 \cdot 10^{11} \mathrm{~N} / \mathrm{m}^{2}$

Prabhu Ramji

Numerade Educator

Problem 28

A 100 .-g block hangs from a spring with $k=5.00 \mathrm{~N} / \mathrm{m}$. At $t=0 \mathrm{~s}$, the block is $20.0 \mathrm{~cm}$ below the equilibrium position and moving upward with a speed of $200 . \mathrm{cm} / \mathrm{s}$. What is the block's speed when the displacement from equilibrium is $30.0 \mathrm{~cm} ?$

Narayan Hari

Numerade Educator

Problem 29

A block of wood of mass 55.0 g floats in a swimming pool, oscillating up and down in simple harmonic motion with a frequency of $3.00 \mathrm{~Hz}$.
a) What is the value of the effective spring constant of the water?
b) A partially filled water bottle of almost the same size and shape as the block of wood but with mass $250 . \mathrm{g}$ is placed on the water's surface. At what frequency will the bottle bob up and down?

Prabhu Ramji

Numerade Educator

Problem 30

When a mass is attached to a vertical spring, the spring is stretched a distance $d$. The mass is then pulled down from this position and released It undergoes 50.0 oscillations in $30.0 \mathrm{~s}$. What was the distance $d$ ?

Narayan Hari

Numerade Educator

Problem 31

A cube of density $\rho_{c}$ floats in a liquid of density $\rho_{1}$, as shown in the figure. At rest, an amount $h$ of the cube's height is submerged in liquid. If the cube is pushed down, it bobs up and down like a spring and oscillates about its equilibrium position. Show that the frequency of its oscillations is given by $f=(2 \pi)^{-1} \sqrt{g / h}$

Prabhu Ramji

Numerade Educator

Problem 32

A U-shaped glass tube with a uniform crosssectional area, $A$, is partly filled with fluid of density $\rho$. Increased pressure is applied to one of the arms, resulting in a difference in elevation, $h$, between the two arms of the tube, as shown in the figure. The increased pressure is then removed, and the fluid oscillates in the tube. Determine the period of the oscillations of the fluid column. (You have to determine what the known quantities are.)

Prabhu Ramji

Numerade Educator

Problem 33

The figure shows a mass $m_{2}=20.0$ g resting on top of a mass $m_{1}=20.0 \mathrm{~g},$ which is attached to a spring with $k=10.0 \mathrm{~N} / \mathrm{m} .$ The coefficient of static friction between the two masses is 0.600 . The masses are oscillating with simple harmonic motion on a frictionless surface. What is the maximum amplitude the oscillation can have without $m_{2}$ slipping off $m_{1} ?$

Prabhu Ramji

Numerade Educator

Problem 34

Consider two identical oscillators, each with spring constant $k$ and mass $m$, in simple harmonic motion. One oscillator is started with initial conditions $x_{0}$ and $v_{0}$; the other starts with slightly different conditions, $x_{0}+\delta x$ and $v_{0}+\delta v$
a) Find the difference in the oscillators' positions, $x_{1}(t)-x_{2}(t),$ for all $t$.
b) This difference is bounded; that is, there exists a constant $C$, independent of time, for which $\left|x_{1}(t)-x_{2}(t)\right| \leq C$ holds for all $t$. Find an expression for
C. What is the best bound, that is, the smallest value of $C$ that works? (Note:
An important characteristic of chaotic systems is exponential sensitivity to initial conditions; the difference in position of two such systems with slightly different initial conditions grows exponentially with time. You have just shown that an oscillator in simple harmonic motion is not a chaotic system.

Averell Hause

Carnegie Mellon University

Problem 35

What is the period of a simple pendulum that is $1.00 \mathrm{~m}$ long in each situation?
a) in the physics lab
b) in an elevator accelerating at $2.10 \mathrm{~m} / \mathrm{s}^{2}$ upward
c) in an elevator accelerating $2.10 \mathrm{~m} / \mathrm{s}^{2}$ downward
d) in an elevator that is in free fall

Prabhu Ramji

Numerade Educator

Problem 36

Suppose a simple pendulum is used to measure the acceleration due to gravity at various points on the Earth. If $g$ varies by $0.16 \%$ over the various locations where it is sampled, what is the corresponding variation in the period of the pendulum? Assume that the length of the pendulum does not change from one site to another.

Prabhu Ramji

Numerade Educator

Problem 37

A 1.00 -kg mass is connected to a 2.00 -kg mass by a massless $\operatorname{rod} 30.0 \mathrm{~cm}$ long, as shown in the figure. A hole is then drilled in the rod $10.0 \mathrm{~cm}$ away from the 1.00 - $\mathrm{kg}$ mass, and the rod and masses are free to rotate about this pivot point, $P$. Calculate the period of oscillation for the masses if they are displaced slightly from the stable equilibrium position.

Prabhu Ramji

Numerade Educator

Problem 38

A torsional pendulum is a vertically suspended wire with a mass attached to one end that is free to rotate. The wire resists
twisting as well as elongation and compression, so it is subject to a rotational equivalent of Hooke's Law: $\tau=-\kappa \theta .$ Use this information to find the frequency of a torsional pendulum.

Prabhu Ramji

Numerade Educator

Problem 39

A physical pendulum consists of a uniform rod of mass $M$ and length
L. The pendulum is pivoted at a point that is a distance $x$ from the center of the rod, so the period for oscillation of the pendulum depends on $x: T(x)$.
a) What value of $x$ gives the maximum value for $T$ ?
b) What value of $x$ gives the minimum value for $T ?$

Prabhu Ramji

Numerade Educator

Problem 40

Shown in the figure are four different pendulums:
(a) a rod has a mass $M$ and a length $L$;
(b) a rod has a mass $2 M$ and a length $L$; (c) a mass $M$ is attached to a massless string of a length $L$; and (d) a mass $M$ is attached to a massless string of length $\frac{1}{2} L$. Find the period of each pendulum when it is pulled $20^{\circ}$ to the right and then released.

Narayan Hari

Numerade Educator

Problem 41

A grandfather clock uses a physical pendulum to keep time. The pendulum consists of a uniform thin rod of mass $M$ and length $L$ that is pivoted freely about one end, with a solid sphere of the same mass, $M$, and a radius of $L / 2$ centered about the free end of the rod.
a) Obtain an expression for the moment of inertia of the pendulum about its pivot point as a function of $M$ and $L$.
b) Obtain an expression for the period of the pendulum for small oscillations.

Averell Hause

Carnegie Mellon University

Problem 42

A standard CD has a diameter of $12 \mathrm{~cm}$ and a hole that is centered on the axis of symmetry and has a diameter of $1.5 \mathrm{~cm} .$ The CD's thickness is $2.0 \mathrm{~mm} .$ If a (very thin) pin extending through the hole of the CD suspends it in a vertical orientation, as shown in the figure, the CD may oscillate about an axis parallel to the pin, rocking back and forth. Calculate the oscillation frequency.

Narayan Hari

Numerade Educator

Problem 43

A massive object of $m=5.00 \mathrm{~kg}$ oscillates with simple harmonic motion. Its position as a function of time varies according to the equation $x(t)=2 \sin ([\pi / 2] t+\pi / 6),$ where $x$ is in meters.
a) What are the position, velocity, and acceleration of the object at $t=0$ s?
b) What is the kinetic energy of the object as a function of time?
c) At which time after $t=0$ s is the kinetic energy first at a maximum?

Prabhu Ramji

Numerade Educator

Problem 44

The mass $m$ shown in the figure is displaced a distance $x$ to the right from its equilibrium position.
a) What is the net force acting on the mass, and what is the effective spring constant?
b) What will the frequency of the oscillation be when the mass is released?
c) If $x=0.100 \mathrm{~m},$ what is the total energy of the mass-spring system after the mass is released, and what is the maximum velocity of the mass?

Prabhu Ramji

Numerade Educator

Problem 45

A 2.00 -kg mass attached to a spring is displaced $8.00 \mathrm{~cm}$ from the equilibrium position. It is released and then oscillates with a frequency of $4.00 \mathrm{~Hz}$
a) What is the energy of the motion when the mass passes through the equilibrium position?
b) What is the speed of the mass when it is $2.00 \mathrm{~cm}$ from the equilibrium position?

Prabhu Ramji

Numerade Educator

Problem 46

A Foucault pendulum is designed to demonstrate the effect of the Earth's rotation. A Foucault pendulum displayed in a museum is typically quite long, making the effect easier to see. Consider a Foucault pendulum of length $15.0 \mathrm{~m}$ with a $110 .-\mathrm{kg}$ brass bob. It is set to swing with an amplitude of $3.50^{\circ}$
a) What is the period of the pendulum?
b) What is the maximum kinetic energy of the pendulum?
c) What is the maximum speed of the pendulum?

Prabhu Ramji

Numerade Educator

Problem 47

A mass, $m_{1}=8.00 \mathrm{~kg},$ is at rest on a frictionless horizontal surface and connected to a wall by a spring with $k=70.0 \mathrm{~N} / \mathrm{m},$ as shown in the figure. A second mass, $m_{2}=5.00 \mathrm{~kg}$, is moving to the right at $v_{0}=17.0 \mathrm{~m} / \mathrm{s}$. The two masses collide and stick together.
a) What is the maximum compression of the spring?
b) How long will it take after the collision to reach this maximum compression?

Prabhu Ramji

Numerade Educator

Problem 48

A mass $M=0.460 \mathrm{~kg}$ moves with an initial speed $v=3.20 \mathrm{~m} / \mathrm{s}$ on a level frictionless air track. The mass is initially a distance $D=0.250 \mathrm{~m}$ away from a spring with $k=840 \mathrm{~N} / \mathrm{m},$ which is mounted rigidly at one end of the air track. The mass compresses the spring a maximum distance $d$, before reversing direction. After bouncing off the spring, the mass travels with the same speed $v$, but in the opposite direction.
a) Determine the maximum distance that the spring is compressed.
b) Find the total elapsed time until the mass returns to its starting point. (Hint: The mass undergoes a partial cycle of simple harmonic motion while in contact with the spring.)

Prabhu Ramji

Numerade Educator

Problem 49

The relative motion of two atoms in a molecule can be described as the motion of a single body of mass $m$ moving in one dimension, with potential energy $U(r)=A / r^{12}-B / r^{6},$ where $r$ is the separation between the atoms and $A$ and $B$ are positive constants.
a) Find the equilibrium separation, $r_{0},$ of the atoms, in terms of the constants $A$ and $B$.
b) If moved slightly, the atoms will oscillate about their equilibrium separation. Find the angular frequency of this oscillation, in terms of $A, B,$ and $m$.

Narayan Hari

Numerade Educator

Problem 50

A $3.00-\mathrm{kg}$ mass attached to a spring with $k=140 . \mathrm{N} / \mathrm{m}$ is oscillating in a vat of oil, which damps the oscillations.
a) If the damping constant of the oil is $b=10.0 \mathrm{~kg} / \mathrm{s},$ how long will it take the amplitude of the oscillations to decrease to $1.00 \%$ of its original value?
b) What should the damping constant be to reduce the amplitude of the oscillations by $99.0 \%$ in $1.00 \mathrm{~s} ?$

Narayan Hari

Numerade Educator

Problem 51

A vertical spring with a spring constant of $2.00 \mathrm{~N} / \mathrm{m}$ has a $0.300-\mathrm{kg}$ mass attached to it, and the mass moves in a medium with a damping constant of $0.0250 \mathrm{~kg} / \mathrm{s}$. The mass is released from rest at a position $5.00 \mathrm{~cm}$ from the equilibrium position. How long will it take for the amplitude to decrease to $2.50 \mathrm{~cm} ?$

Narayan Hari

Numerade Educator

Problem 52

A mass of $0.404 \mathrm{~kg}$ is attached to a spring with a spring constant of $206.9 \mathrm{~N} / \mathrm{m} .$ Its oscillation is damped, with damping constant $b=14.5 \mathrm{~kg} / \mathrm{s} .$ What is the frequency of this damped oscillation?

Narayan Hari

Numerade Educator

Problem 53

Cars have shock absorbers to damp the oscillations that would otherwise occur when the springs that attach the wheels to the car's frame are compressed or stretched. Ideally, the shock absorbers provide critical damping. If the shock absorbers fail, they provide less damping, resulting in an underdamped motion. You can perform a simple test of your shock absorbers by pushing down on one corner of your car and then quickly releasing it. If this results in an up-and-down oscillation of the car, you know that your shock absorbers need changing. The spring on each wheel of a car has a spring constant of $4005 \mathrm{~N} / \mathrm{m}$, and the car has a mass of $851 \mathrm{~kg}$, equally distributed over all four wheels. Its shock absorbers have gone bad and provide only $60.7 \%$ of the damping they were initially designed to provide. What will the period of the underdamped oscillation of this car be if the pushing-down shock absorber test is performed?

Averell Hause

Carnegie Mellon University

Problem 54

In a lab, a student measures the unstretched length of a spring as $11.2 \mathrm{~cm} .$ When a 100.0 -g mass is hung from the spring, its length is $20.7 \mathrm{~cm} .$ The mass-spring system is set into oscillatory motion, and the student observes that the amplitude of the oscillation decreases by about a factor of 2 after five complete cycles.
a) Calculate the period of oscillation for this system, assuming no damping.
b) If the student can measure the period to the nearest $0.05 \mathrm{~s}$, will she be able to detect the difference between the period with no damping and the period with damping?

Manish Jain

Numerade Educator

Problem 55

An 80.0 -kg bungee jumper is enjoying an afternoon of jumps. The jumper's first oscillation has an amplitude of $10.0 \mathrm{~m}$ and a period of $5.00 \mathrm{~s}$. Treating the bungee cord as a spring with no damping, calculate each of the following:
a) the spring constant of the bungee cord,
b) the bungee jumper's maximum speed during the oscillation, and
c) the time for the amplitude to decrease to $2.00 \mathrm{~m}$ (with air resistance providing the damping of the oscillations at $7.50 \mathrm{~kg} / \mathrm{s}$ ).

Narayan Hari

Numerade Educator

Problem 56

A small mass, $m=50.0 \mathrm{~g}$, is attached to the end of a massless rod that is hanging from the ceiling and is free to swing. The rod has length $L=1.00 \mathrm{~m} .$ The rod is displaced $10.0^{\circ}$ from the vertical and released at time $t=0$. Neglect air resistance. What is the period of the rod's oscillation? Now suppose the entire system is immersed in a fluid with a small damping constant, $b=0.0100 \mathrm{~kg} / \mathrm{s}$, and the rod is again released from an initial displacement angle of $10.0^{\circ} .$ What is the time for the amplitude of the oscillation to reduce to $5.00^{\circ}$ ? Assume that the damping is small. Also note that since the amplitude of the oscillation is small and all the mass of the pendulum is at the end of the rod, the motion of the mass can be treated as strictly linear, and you can use the substitution $R \theta(t)=x(t)$, where $R=1.0 \mathrm{~m}$ is the length of the pendulum rod.

Narayan Hari

Numerade Educator

Problem 57

A 3.00 -kg mass is vibrating on a spring. It has a resonant angular speed of $2.40 \mathrm{rad} / \mathrm{s}$ and a damping angular speed of $0.140 \mathrm{rad} / \mathrm{s} .$ If the sinusoidal driving force has an amplitude of $2.00 \mathrm{~N}$, find the maximum amplitude of the vibration if the driving angular speed is (a) $1.20 \mathrm{rad} / \mathrm{s}$, (b) $2.40 \mathrm{rad} / \mathrm{s}$, and
(c) $4.80 \mathrm{rad} / \mathrm{s}$.

Narayan Hari

Numerade Educator

Problem 58

A mass of $0.404 \mathrm{~kg}$ is attached to a spring with a spring constant of $204.7 \mathrm{~N} / \mathrm{m}$ and hangs at rest. The spring hangs from a piston. The piston then moves up and down, driven by a force given by $(29.4 \mathrm{~N}) \cos [(17.1 \mathrm{rad} / \mathrm{s}) t]$
a) What is the maximum displacement from its equilibrium position that the mass can reach?
b) What is the maximum speed that the mass can attain in this motion?

Narayan Hari

Numerade Educator

Problem 59

A mass, $M=1.60 \mathrm{~kg}$, is attached to a wall by a spring with $k=578 \mathrm{~N} / \mathrm{m} .$ The mass slides on a frictionless floor. The spring and mass are immersed in a fluid with a damping constant of $6.40 \mathrm{~kg} / \mathrm{s}$. A horizontal force, $F(t)=F_{\mathrm{d}} \cos \left(\omega_{\mathrm{d}} t\right),$ where $F_{\mathrm{d}}=52.0 \mathrm{~N},$ is applied to the mass through a knob,
causing the mass to oscillate back and forth. Neglect the mass of the spring and of the knob and rod. At what frequency will the amplitude of the mass's oscillation be greatest, and what is the maximum amplitude? If the driving frequency is reduced slightly (but the driving amplitude remains the same), at what frequency will the amplitude of the mass's oscillation be half of the maximum amplitude?

Manish Jain

Numerade Educator

Problem 60

When the displacement of a mass on a spring is half of the amplitude of its oscillation, what fraction of the mass's energy is kinetic energy?

Narayan Hari

Numerade Educator

Problem 61

A mass $m$ is attached to a spring with a spring constant of $k$ and set into simple harmonic motion. When the mass has half of its maximum kinetic energy, how far away from its equilibrium position is it, expressed as a fraction of its maximum displacement?

Averell Hause

Carnegie Mellon University

Problem 62

If you kick a harmonic oscillator sharply, you impart to it an initial velocity but no initial displacement. For a weakly damped oscillator with mass $m$, spring constant $k$, and damping force $F_{\gamma}=-b v$, find $x(t)$, if the total impulse delivered by the kick is $J_{0}$.

Narayan Hari

Numerade Educator

Problem 63

A mass $m=1.00 \mathrm{~kg}$ in a spring-mass system with $k=1.00 \mathrm{~N} / \mathrm{m}$ is observed to be moving to the right, past its equilibrium position with a speed of $1.00 \mathrm{~m} / \mathrm{s}$ at time $t=0$
a) Ignoring all damping, determine the equation of motion.
b) Suppose the initial conditions are such that at time $t=0,$ the mass is at $x=0.500 \mathrm{~m}$ and moving to the right with a speed of $1.00 \mathrm{~m} / \mathrm{s}$. Determine the new equation of motion. Assume the same spring constant and mass.

Averell Hause

Carnegie Mellon University

Problem 64

The motion of a block-spring system is described by $x(t)=A \sin (\omega t)$ Find $\omega,$ if the potential energy equals the kinetic energy at $t=1.00 \mathrm{~s}$.

Prabhu Ramji

Numerade Educator

Problem 65

A hydrogen gas molecule can be thought of as a pair of protons bound together by a spring. If the mass of a proton is $1.7 \cdot 10^{-27} \mathrm{~kg}$ and the period of oscillation is $8.0 \cdot 10^{-15} \mathrm{~s}$, what is the effective spring constant for the bond in a hydrogen molecule?

Prabhu Ramji

Numerade Educator

Problem 66

A shock absorber that provides critical damping with $\omega_{\gamma}=72.4 \mathrm{rad} / \mathrm{s}$ is compressed by $6.41 \mathrm{~cm} .$ How far from the equilibrium position is it after $0.0247 \mathrm{~s} ?$

Averell Hause

Carnegie Mellon University

Problem 67

Imagine you are an astronaut who has landed on another planet and wants to determine the free-fall acceleration on that planet. In one of the experiments you decide to conduct, you use a pendulum $0.500 \mathrm{~m}$ long and find that the period of oscillation for this pendulum is $1.50 \mathrm{~s}$. What is the acceleration due to gravity on that planet?

Narayan Hari

Numerade Educator

Problem 68

A horizontal tree branch is directly above another horizontal tree branch. The elevation of the higher branch is $9.65 \mathrm{~m}$ above the ground, and the elevation of the lower branch is $5.99 \mathrm{~m}$ above the ground. Some children decide to use the two branches to hold a tire swing. One end of the tire swing's rope is tied to the higher tree branch so that the bottom of the tire swing is $0.470 \mathrm{~m}$ above the ground. This swing is thus a restricted pendulum. Starting with the complete length of the rope at an initial angle of $14.2^{\circ}$ with respect to the vertical, how long does it take a child of mass $29.9 \mathrm{~kg}$ to complete one swing back and forth?

Narayan Hari

Numerade Educator

Problem 69

Two pendulums with identical lengths of $1.000 \mathrm{~m}$ are suspended from the ceiling and begin swinging at the same time. One is at Manila, in the Philippines, where $g=9.784 \mathrm{~m} / \mathrm{s}^{2},$ and the other is at Oslo, Norway, where $g=9.819 \mathrm{~m} / \mathrm{s}^{2}$. After how many oscillations of the Manila pendulum will the two pendulums be in phase again? How long will it take for them to be in phase again?

Averell Hause

Carnegie Mellon University

Problem 70

Two springs, each with $k=125 \mathrm{~N} / \mathrm{m},$ are hung vertically, and $1.00-\mathrm{kg}$ masses are attached to their ends. One spring is pulled down $5.00 \mathrm{~cm}$ and released at $t=0$; the other is pulled down $4.00 \mathrm{~cm}$ and released at $t=0.300 \mathrm{~s} .$ Find the phase difference, in degrees, between the oscillations of the two masses and the equations for the vertical displacements of the masses, taking upward to be the positive direction.

Manish Jain

Numerade Educator

Problem 71

A piston that has a small toy car sitting on it is undergoing simple harmonic motion vertically, with amplitude of $5.00 \mathrm{~cm}$, as shown in the figure. When the frequency of oscillation is low, the car stays on the piston. However, when the frequency is increased sufficiently, the car leaves the piston. What is the maximum frequency at which the car will remain in place on the piston?

Narayan Hari

Numerade Educator

Problem 72

The period of a pendulum is $0.24 \mathrm{~s}$ on Earth. The period of the same pendulum is found to be 0.48 s on planet $X$, whose mass is equal to that of Earth.
(a) Calculate the gravitational acceleration at the surface of planet X.
(b) Find the radius of planet $X$ in terms of that of Earth.

Prabhu Ramji

Numerade Educator

Problem 73

A grandfather clock uses a pendulum and a weight. The pendulum has a period of $2.00 \mathrm{~s},$ and the mass of the bob is $250 . \mathrm{g} .$ The weight slowly falls, providing the energy to overcome the damping of the pendulum due to friction. The weight has a mass of $1.00 \mathrm{~kg},$ and it moves down $25.0 \mathrm{~cm}$ every day. Find $Q$ for this clock. Assume that the amplitude of the oscillation of the pendulum is $10.0^{\circ}$.

Narayan Hari

Numerade Educator

Problem 74

A cylindrical can of diameter $10.0 \mathrm{~cm}$ contains some ballast so that it floats vertically in water. The mass of can and ballast is $800.0 \mathrm{~g}$, and the density of water is $1.00 \mathrm{~g} / \mathrm{cm}^{3}$. The can is lifted $1.00 \mathrm{~cm}$ from its equilibrium position and released at $t=0 .$ Find its vertical displacement from equilibrium as a function of time. Determine the period of the motion. Ignore the damping effect due to the viscosity of the water.

Narayan Hari

Numerade Educator

Problem 75

The period of oscillation of an object in a frictionless tunnel running through the center of the Moon is $T=2 \pi / \omega_{0}=6485 \mathrm{~s},$ as shown in Example 14.2. What is the period of oscillation of an object in a similar tunnel through the Earth $\left(R_{\mathrm{E}}=6.37 \cdot 10^{6} \mathrm{~m} ; R_{\mathrm{M}}=1.74 \cdot 10^{6} \mathrm{~m} ; M_{\mathrm{E}}=5.98 \cdot 10^{24} \mathrm{~kg}\right.$ $\left.M_{\mathrm{M}}=7.35 \cdot 10^{22} \mathrm{~kg}\right) ?$

Prabhu Ramji

Numerade Educator

Problem 76

The motion of a planet in a circular orbit about a star obeys the equations of simple harmonic motion. If the orbit is observed edge-on, so the planet's motion appears to be one-dimensional, the analogy is quite direct:
The motion of the planet looks just like the motion of an object on a spring.
a) Use Kepler's Third Law of planetary motion to determine the "spring constant" for a planet in circular orbit around a star with period $T$.
b) When the planet is at the extremes of its motion observed edge-on, the analogous "spring" is extended to its largest displacement. Using the "spring" analogy, determine the orbital velocity of the planet.

Narayan Hari

Numerade Educator

Problem 77

An object in simple harmonic motion is isochronous, meaning that the period of its oscillations is independent of their amplitude. (Contrary to a common assertion, the operation of a pendulum clock is not based on this principle. A pendulum clock operates at fixed, finite amplitude. The gearing of the clock compensates for the anharmonicity of the pendulum. Consider an oscillator of mass $m$ in one-dimensional motion, with a restoring force $F(x)=-c x^{3},$ where $x$ is the displacement from equilibrium and $c$ is a constant with appropriate units. The motion of this oscillator is periodic but not isochronous.
a) Write an expression for the period of the undamped oscillations of this oscillator. If your expression involves an integral, it should be a definite integral. You do not need to evaluate the expression.
b) Using the expression of part (a), determine the dependence of the period of oscillation on the amplitude.
c) Generalize the results of parts (a) and (b) to an oscillator of mass $m$ in one-dimensional motion with a restoring force corresponding to the potential energy $U(x)=\gamma|x|^{\alpha} / \alpha,$ where $\alpha$ is any positive value and $\gamma$ is a constant.

Manish Jain

Numerade Educator

Problem 78

A block of mass $1.605 \mathrm{~kg}$ is attached to a horizontal spring with spring constant $14.55 \mathrm{~N} / \mathrm{m}$ and rests on a frictionless surface at the equilibrium position of the spring. The block is then pulled $12.09 \mathrm{~cm}$ from the equilibrium position and released. What is the distance of the block from the equilibrium position after $2.834 \cdot 10^{-1} \mathrm{~s} ?$

Prabhu Ramji

Numerade Educator

Problem 79

A block of mass $1.833 \mathrm{~kg}$ is attached to a horizontal spring with spring constant $14.97 \mathrm{~N} / \mathrm{m}$ and rests on a frictionless surface at the equilibrium position of the spring. The block is then pulled $13.37 \mathrm{~cm}$ from the equilibrium position and released. At what time is the block located $4.990 \mathrm{~cm}$ from the equilibrium position?

Prabhu Ramji

Numerade Educator

Problem 80

A block of mass $1.061 \mathrm{~kg}$ is attached to a horizontal spring with spring constant $15.39 \mathrm{~N} / \mathrm{m}$ and rests on a frictionless surface at the equilibrium position of the spring. The block is then pulled from the equilibrium position and released. After $3.900 \cdot 10^{-1} \mathrm{~s}$, the block is located $1.25 \mathrm{~cm}$ from the equilibrium position. How far away from that position was the block pulled?

Prabhu Ramji

Numerade Educator

Problem 81

A pendulum consisting of a small sphere of mass $1.145 \mathrm{~kg}$ and a string of length $71.57 \mathrm{~cm}$ is suspended from a ceiling. Its motion is restricted by a peg that is sticking out of the wall $40.95 \mathrm{~cm}$ directly below the pivot point. What is the period of oscillation?

Prabhu Ramji

Numerade Educator

Problem 82

A pendulum consisting of a small sphere of mass $1.261 \mathrm{~kg}$ and a string of length $72.39 \mathrm{~cm}$ is suspended from a ceiling. Its motion is restricted by a peg that is sticking out of the wall directly below the pivot point. The period of oscillation is $1.404 \mathrm{~s}$. How far below the ceiling is the restricting peg?

Prabhu Ramji

Numerade Educator

Problem 83

A vertical spring with spring constant $23.31 \mathrm{~N} / \mathrm{m}$ is hanging from a ceiling. A small object with a mass of $1.375 \mathrm{~kg}$ is attached to the lower end of the spring, and the spring stretches to its equilibrium length. The object is then pulled down a distance of $18.51 \mathrm{~cm}$ and released. What is the speed of the object when it is $1.849 \mathrm{~cm}$ from the equilibrium position?

Prabhu Ramji

Numerade Educator

Problem 84

A vertical spring with spring constant $23.51 \mathrm{~N} / \mathrm{m}$ is hanging from a ceiling. A small object is attached to the lower end of the spring, and the spring stretches to its equilibrium length. The object is then pulled down a distance of $19.79 \mathrm{~cm}$ and released. The speed of the object a distance $7.417 \mathrm{~cm}$ from the equilibrium point is $0.7286 \mathrm{~m} / \mathrm{s}$. What is the mass of the object?

Prabhu Ramji

Numerade Educator

Problem 85

A vertical spring with spring constant $23.73 \mathrm{~N} / \mathrm{m}$ is hanging from a ceiling. A small object with mass $1.103 \mathrm{~kg}$ is attached to the lower end of the spring, and the spring stretches to its equilibrium length. The object is then pulled down and released. The speed of the object when it is $4.985 \mathrm{~cm}$ from the equilibrium position is $0.4585 \mathrm{~m} / \mathrm{s}$. How far down was the object pulled?

Prabhu Ramji

Numerade Educator