Deck 12: Oscillatory Motion

Question 1

Ellen says that whenever the acceleration is directly proportional to the displacement of an object from its equilibrium position,the motion of the object is simple harmonic motion.Mary says this is true only if the acceleration is opposite in direction to the displacement.Which one,if either,is correct?

A)Ellen,because $\omega$ ² is directly proportional to the constant multiplying the displacement and to the mass.
B)Ellen,because $\omega$ ² is directly proportional to the mass.
C)Mary,because $\omega$ ² is directly proportional to the constant multiplying the displacement and to the mass.
D)Mary,because $\omega$ ² is directly proportional to the mass.
E)Mary,because the second derivative of an oscillatory function like sin( $\omega$ t)or cos( $\omega$ t)is always proportional to the negative of the original function.

Answer 1

Mary,because the second derivative of an oscillatory function like sin( $\omega$ t)or cos( $\omega$ t)is always proportional to the negative of the original function.

Question 2

Exhibit 15-1
A graph of position versus time for an object oscillating at the free end of a horizontal spring is shown below.A point or points at which the object has positive velocity and zero acceleration is(are) Use this exhibit to answer the following question.
Refer to Exhibit 15-1.The point at which the object has zero velocity and positive acceleration is

A)A
B)B
C)C
D)D
E)E

Answer 2

D

Question 3

For the wave described by y = 0.02 sin (kx)at t = 0 s,the first maximum at a positive x coordinate occurs where x = 4 m.Where on the positive x axis does the second maximum occur?

A)20 m
B)18 m
C)24 m
D)28 m
E)16 m

Answer 3

20 m

Question 4

A body oscillates with simple harmonic motion along the x axis.Its displacement varies with time according to the equation x = 5.0 cos ( $\pi$ t).The magnitude of the acceleration (in m/s²)of the body at t = 1.0 s is approximately

A)3.5
B)49
C)14
D)43
E)4.3

Question 5

Exhibit 15-1
A graph of position versus time for an object oscillating at the free end of a horizontal spring is shown below.A point or points at which the object has positive velocity and zero acceleration is(are) Use this exhibit to answer the following question.
Refer to Exhibit 15-1.A point or points at which the object has positive velocity and zero acceleration is(are)

A)B
B)C
C)D
D)B and D
E)A and E

Question 6

If y = 0.02 sin (30x - 400t)(SI units),the wavelength of the wave is

A) $\pi$ /15 m
B)15/ $\pi$ m
C)60 $\pi$ m
D)4.2 m
E)30 m

Question 7

The equation $y = A \sin \left( k x - ? t + \frac { \pi } { 2 } \right)$ is the same as

A) $y = - A \sin \left( k x - \omega t + \frac { \pi } { 2 } \right)$
B) $y = A \cos ( k x - ? t )$
C) $y = - A \cos ( k x - \omega t )$
D) $y = - A \sin \left( k x - \omega t - \frac { \pi } { 2 } \right)$
E) $y = A \sin \left( k x - \omega t + \frac { 3 \pi } { 2 } \right)$

Question 8

A weight of mass m is at rest at O when suspended from a spring,as shown.When it is pulled down and released,it oscillates between positions A and B.Which statement about the system consisting of the spring and the mass is correct?

A)The gravitational potential energy of the system is greatest at A.
B)The elastic potential energy of the system is greatest at O.
C)The rate of change of momentum has its greatest magnitude at A and B.
D)The rate of change of gravitational potential energy is smallest at O.
E)The rate of change of gravitational potential energy has its greatest magnitude at A and B.

Question 9

If y = 0.02 sin (30x - 400t)(SI units),the frequency of the wave is

A)30 Hz
B)15/ $\pi$ Hz
C)200/ $\pi$ Hz
D)400 Hz
E)800 $\pi$ Hz

Question 10

Write the equation of a wave,traveling along the +x axis with an amplitude of 0.02 m,a frequency of 440 Hz,and a speed of 330 m/sec.

A)y = 0.02 sin [880 $\pi$ (x/330 - t)]
B)y = 0.02 cos [880 $\pi$ x/330 -440t]
C)y = 0.02 sin [880 $\pi$ (x/330 + t)]
D)y = 0.02 sin [2 $\pi$ (x/330 + 440t)]
E)y = 0.02 cos [2 $\pi$ (x/330 + 440t)]

Question 11

A body oscillates with simple harmonic motion along the x axis.Its displacement varies with time according to the equation x = 5.0 sin ( $\pi$ t + $\pi$ /3).The velocity (in m/s)of the body at t = 1.0 s is

A)+7.9
B)-7.9
C)-14
D)+14
E)-5.0

Question 12

A mass m = 2.0 kg is attached to a spring having a force constant k = 290 N/m as in the figure.The mass is displaced from its equilibrium position and released.Its frequency of oscillation (in Hz)is approximately

A)12
B)0.50
C)0.010
D)1.9
E)0.080

Question 13

A body oscillates with simple harmonic motion along the x axis.Its displacement varies with time according to the equation x = 5 sin ( $\pi$ t + $\pi$ /3).The phase (in rad)of the motion at t = 2 s is

A)7 $\pi$ /3
B) $\pi$ /3
C) $\pi$
D)5 $\pi$ /3
E)2 $\pi$

Question 14

For the wave described by $y = 0.15 \sin \left[ \frac { \pi } { 16 } ( 2 x - 64 t ) \right] \text { (SI units) }$ ,determine the first positive x coordinate where y is a maximum when t = 0.

A)16 m
B)8 m
C)4 m
D)2 m
E)13 m

Question 15

Exhibit 15-1
A graph of position versus time for an object oscillating at the free end of a horizontal spring is shown below.A point or points at which the object has positive velocity and zero acceleration is(are) Use this exhibit to answer the following question.
Refer to Exhibit 15-1.The point at which the object has negative velocity and zero acceleration is

A)A
B)B
C)C
D)D
E)E

Question 16

For the wave described by $y = 0.15 \sin \left[ \frac { \pi } { 16 } ( 2 x - 64 t ) \right] \text { (SI units) }$ ,determine the x coordinate of the second maximum when t = 0.

A)20 m
B)18 m
C)24 m
D)28 m
E)16 m

Question 17

A body of mass 5.0 kg is suspended by a spring which stretches 10 cm when the mass is attached.It is then displaced downward an additional 5.0 cm and released.Its position as a function of time is approximately

A)y = -0.10 sin 9.9t
B)y = 0.10 cos 9.9t
C)y = -0.10 cos (9.9t + .1)
D)y = 0.10 sin (9.9t + 5)
E)y = -0.05 cos 9.9t

Question 18

If y = 0.02 sin (30x - 400t)(SI units),the angular frequency of the wave is

A)30 rad/s
B)30/2 $\pi$ rad/s
C)400/2 $\pi$ rad/s
D)400 rad/s
E)40/3 rad/s

Question 19

The mass in the figure slides on a frictionless surface.If m = 2 kg,k₁ = 800 N/m and k₂ = 500 N/m,the frequency of oscillation (in Hz)is approximately

A)6
B)2
C)4
D)8
E)10

Question 20

Exhibit 15-1
A graph of position versus time for an object oscillating at the free end of a horizontal spring is shown below.A point or points at which the object has positive velocity and zero acceleration is(are) Use this exhibit to answer the following question.
Refer to Exhibit 15-1.The point at which the object has zero velocity and negative acceleration is

A)A
B)B
C)C
D)D
E)E

Question 21

To double the total energy of a mass oscillating at the end of a spring with amplitude A,we need to

A)increase the angular frequency by $\sqrt { 2 }$ .
B)increase the amplitude by $\sqrt { 2 }$ .
C)increase the amplitude by 2.
D)increase the angular frequency by 2.
E)increase the amplitude by 4 and decrease the angular frequency by $\frac { 1 } { \sqrt { 2 } }$ .

Question 22

John says that the value of the function cos[ $\omega$ (t + T)+ $\varphi$ ],obtained one period T after time t,is greater than cos( $\omega$ t + $\varphi$ )by 2 $\pi$ .Larry says that it is greater by the addition of 1.00 to cos( $\omega$ t + $\varphi$ ).Which one,if either,is correct?

A)John,because $\omega$ T = 2 $\pi$ .
B)John,because $\omega$ T = 1 radian.
C)Larry,because $\omega$ T = 2 $\pi$ .
D)Larry,because $\omega$ T = 1 radian.
E)Neither,because cos( $\theta$ + 2 $\omega$ )= cos $\theta$ .

Question 23

The amplitude of a system moving with simple harmonic motion is doubled.The total energy will then be

A)4 times as large
B)3 times as large
C)2 times as large
D)the same as it was
E)half as much

Question 24

The figure shows a uniform rod (length L = 1.0 m,mass = 2.0 kg)suspended from a pivot a distance d = 0.25 m above its center of mass.The angular frequency (in rad/s)for small oscillations is approximately

A)1.0
B)2.5
C)1.5
D)4.1
E)3.5

Question 25

In the figure below,a hoop (radius R = 1.0 m,mass = 2.0 kg)having four spokes of negligible mass is suspended from a pivot a distance d = .25 m above its center of mass.The angular frequency (in rad/s)for small oscillations is approximately

A)4.0
B)2.5
C)1.5
D)1.0
E)0.5

Question 26

In the figure below,a disk (radius R = 1.0 m,mass = 2.0 kg)is suspended from a pivot a distance d = 0.25 m above its center of mass.For a circular disk, $I _ { \mathrm { cm } } = \frac { 1 } { 2 } m R ^ { 2 }$ .The angular frequency (in rad/s)for small oscillations is approximately

A)4.2
B)2.1
C)1.5
D)1.0
E)3.8

Question 27

Three pendulums with strings of the same length and bobs of the same mass are pulled out to angles $\theta$ ₁, $\theta$ ₂ and $\theta$ ₃ respectively and released.The approximation sin $\theta$ = $\theta$ holds for all three angles,with $\theta$ ₃ > $\theta$ ₂ > $\theta$ ₁.How do the angular frequencies of the three pendulums compare?

A) $\omega$ ₃ > $\omega$ ₂ > $\omega$ ₁
B)Need to know amplitudes to answer this question.
C)Need to know $\sqrt { g / L }$ to answer this question.
D) $\omega$ ₁ > $\omega$ ₂ > $\omega$ ₃
E) $\omega$ ₁ = $\omega$ ₂ = $\omega$ ₃

Question 28

A torsional pendulum consists of a solid disk (mass = 2.0 kg,radius = 1.0 m)suspended by a wire attached to a rigid support.The body oscillates about the support wire.If the torsion constant is 16 N.m/rad.What is the angular frequency (in rad/s)?

A)2
B)4
C)6
D)8
E)7

Question 29

Which of the following combinations of variables results in the greatest period for a pendulum?

A)length = L,mass = M,and maximum angular displacement = 3 degrees
B)length = 2L,mass = M/2,and maximum angular displacement = 1 degree
C)length = 1.5L,mass = 2M,and maximum angular displacement = 2 degrees
D)length = $\sqrt { 2 }$ L,mass = $\sqrt { 2 }$ M,and maximum angular displacement = $\sqrt { 2 }$ degrees
E)length = $\sqrt { 3 }$ L,mass = 4M,and maximum angular displacement = 4 degrees

Question 30

In an inertia balance,a body supported against gravity executes simple harmonic oscillations in a horizontal plane under the action of a set of springs.If a 1.00 kg body vibrates at 1.00 Hz,a 2.00 kg body will vibrate at

A)0.500 Hz.
B)0.707 Hz.
C)1.00 Hz.
D)1.41 Hz.
E)2.00 Hz.

Question 31

A uniform rod (mass m = 1.0 kg and length L = 2.0 m)pivoted at one end oscillates in a vertical plane as shown below.The period of oscillation (in s)is approximately

A)4.0
B)1.6
C)3.2
D)2.3
E)2.0

Question 32

Simple harmonic oscillations can be modeled by the projection of circular motion at constant angular velocity onto a diameter of the circle.When this is done,the analog along the diameter of the acceleration of the particle executing simple harmonic motion is

A)the displacement from the center of the diameter of the projection of the position of the particle on the circle.
B)the projection along the diameter of the velocity of the particle on the circle.
C)the projection along the diameter of tangential acceleration of the particle on the circle.
D)the projection along the diameter of centripetal acceleration of the particle on the circle.
E)meaningful only when the particle moving in the circle also has a non-zero tangential acceleration.

Question 33

A 2.00 m-long 6.00 kg ladder pivoted at the top hangs down from a platform at the circus.A 42.0 kg trapeze artist climbs to a point where her center of mass is at the center of the ladder and swings at the system's natural frequency.The angular frequency (in s ^{- 1})of the system of ladder and woman is

A)1.01.
B)3.07.
C)4.03.
D)8.05.
E)16.2.

Question 34

The motion of a particle connected to a spring is described by x = 10 sin ( $\pi$ t).At what time (in s)is the potential energy equal to the kinetic energy?

A)0
B)0.25
C)0.50
D)0.79
E)1.0

Question 35

A horizontal plank (m = 2.0 kg,L = 1.0 m)is pivoted at one end.A spring (k = 1.0* 10³ N/m)is attached at the other end,as shown in the figure.Find the angular frequency (in rad/s)for small oscillations.

A)39
B)44
C)55
D)66
E)25

Question 36

An object of mass m is attached to string of length L.When it is released from point A,the object oscillates between points A and B.Which statement about the system consisting of the pendulum and the Earth is correct?

A)The gravitational potential energy of the system is greatest at A and B.
B)The kinetic energy of mass m is greatest at point O.
C)The greatest rate of change of momentum occurs at A and B.
D)All of the above are correct.
E)Only (a)and (b)above are correct.

Question 37

A hoop,a solid cylinder,and a solid sphere all have the same mass m and the same radius R.Each is mounted to oscillate about an axis a distance 0.5 R from the center.The axis is perpendicular to the circular plane of the hoop and the cylinder and to an equatorial plane of the sphere as shown below.Which is the correct ranking in order of increasing angular frequency $\omega$ ?

A)hoop,cylinder,sphere
B)cylinder,sphere,hoop
C)sphere,cylinder,hoop
D)hoop,sphere,cylinder
E)sphere,hoop,cylinder

Question 38

At sea level,at a latitude where $g = 9.80 \frac { \mathrm { m } } { \mathrm { s } ^ { 2 } }$ ,a pendulum that takes 2.00 s for a complete swing back and forth has a length of 0.993 m.What is the value of g in m/s² at a location where the length of such a pendulum is 0.970 m?

A)0.098 3
B)3.05
C)9.57
D)10.0
E)38.3

Question 39

The oscillation of the 2.0-kg mass on a spring is described by $x = 3.0 \cos ( 4.0 t + 0.80 )$ where x is in centimeters and t is in seconds.What is the force constant of the spring?

A)4.0 N/m
B)0.80 N/m
C)16 N/m
D)32 N/m
E)2.0 $\pi$ N/m

Question 40

Two circus clowns (each having a mass of 50 kg)swing on two flying trapezes (negligible mass,length 25 m)shown in the figure.At the peak of the swing,one grabs the other,and the two swing back to one platform.The time for the forward and return motion is

A)10 s
B)50 s
C)15 s
D)20 s
E)25 s

Question 41

The motion of a piston in an auto engine is simple harmonic.If the piston travels back and forth over a distance of 10 cm,and the piston has a mass of 1.5 kg,what is the maximum speed of the piston and the maximum force acting on the piston when the engine is running at 4 200 rpm?

Question 42

When a damping force is applied to a simple harmonic oscillator which has angular frequency $\omega$ ₀ in the absence of damping,the new angular frequency $\omega$ is such that

A) $\omega$ < $\omega$ ₀.
B) $\omega$ = $\omega$ ₀.
C) $\omega$ > $\omega$ ₀.
D) $\omega$ T < $\omega$ ₀T₀.
E) $\omega$ T > $\omega$ ₀T₀.

Question 43

When a damping force is applied to a simple harmonic oscillator which has period T₀ in the absence of damping,the new period T is such that

A)T < T₀.
B)T = T₀.
C)T > T₀.
D) $\omega$ T < $\omega$ ₀T₀.
E) $\omega$ T > $\omega$ ₀T₀.

Question 44

A damped oscillator is released from rest with an initial displacement of 10.00 cm.At the end of the first complete oscillation the displacement reaches 9.05 cm.When 4 more oscillations are completed,what is the displacement reached?

A)7.41 cm
B)6.71 cm
C)6.07 cm
D)5.49 cm
E)5.25 cm