Deck 15: Oscillations

Question 1

A particle moves in simple harmonic motion according to x = 2cos(50t), where x is in meters and t is in seconds. Its maximum velocity is:

A) 100 sin(50t) m/s
B) 100 cos(50t) m/s
C) 100 m/s
D) 200 m/s
E) none of these

Answer 1

100 m/s

Question 2

The amplitude and phase constant of an oscillator are determined by:

A) the frequency
B) the angular frequency
C) the initial displacement alone
D) the initial velocity alone
E) both the initial displacement and velocity

Answer 2

both the initial displacement and velocity

Question 3

An object attached to one end of a spring makes 20 vibrations in 10 seconds. Its frequency is:

A) 2 Hz
B) 10 s
C) 0.05 Hz
D) 2 s
E) 0.50 s

Answer 3

2 Hz

Question 4

A particle is in simple harmonic motion with period T. At time t=0 it is halfway between the equilibrium point and an end point of its motion, travelling toward the end point. The next time it is at the same place is:

A) t = T
B) t = T/2
C) t = T/3
D) t = T/4
E) none of the above

Question 5

Two identical undamped oscillators have the same amplitude of oscillation only if:

A) they are started with the same displacement x₀
B) they are started with the same velocity v₀
C) they are started with the same phase
D) they are started so the combination is the same

Question 6

A particle oscillating in simple harmonic motion is:

A) never in equilibrium because it is in motion
B) never in equilibrium because there is a force
C) in equilibrium at the ends of its path because its velocity is zero there
D) in equilibrium at the center of its path because the acceleration is zero there
E) in equilibrium at the ends of its path because the acceleration is zero there

Question 7

A particle is in simple harmonic motion with period T. At time t = 0 it is at the equilibrium point. At the times listed below it is at various points in its cycle. Which of them is farthest away from the equilibrium point?

A) 0.5T
B) 0.7T
C) T
D) 1.4T
E) 1.5T

Question 8

In simple harmonic motion, the displacement is maximum when the:

A) acceleration is zero
B) velocity is maximum
C) velocity is zero
D) kinetic energy is maximum
E) momentum is maximum

Question 9

A particle moves back and forth along the x axis from x = -x_m to x = +x_m, in simple harmonic motion with period T. At time t = 0 it is at x = +x_m. When t = 0.75T:

A) it is at x = 0 and is traveling toward x = +x_m
B) it is at x = 0 and is traveling toward x = -x_m
C) it is at x = +x_m and is at rest
D) it is between x = 0 and x = +x_m and is traveling toward x = -x_m
E) it is between x = 0 and x = -x_m and is traveling toward x = -x_m

Question 10

It is impossible for two particles, each executing simple harmonic motion, to remain in phase with each other if they have different:

A) masses
B) periods
C) amplitudes
D) spring constants
E) kinetic energies

Question 11

Frequency f and angular frequency $\omega$ are related by

A) f = $\pi$ $\omega$
B) f = 2 $\pi$ $\omega$
C) f = $\omega$ / $\pi$
D) f = $\omega$ /2 $\pi$
E) f = 2 $\omega$ / $\pi$

Question 12

When a body executes simple harmonic motion, its acceleration at the ends of its path must be:

A) zero
B) less than g
C) more than g
D) suddenly changing in sign
E) none of these

Question 13

An object attached to one end of a spring makes 20 complete vibrations in 10s. Its period is:

A) 2 Hz
B) 10 s
C) 0.5 Hz
D) 2 s
E) 0.50 s

Question 14

The amplitude of any oscillator will be doubled by:

A) doubling only the initial displacement
B) doubling only the initial speed
C) doubling the initial displacement and halving the initial speed
D) doubling the initial speed and halving the initial displacement
E) doubling both the initial displacement and the initial speed

Question 15

An object is undergoing simple harmonic motion. Throughout a complete cycle it:

A) has constant speed
B) has varying amplitude
C) has varying period
D) has varying acceleration
E) has varying mass

Question 16

An oscillatory motion must be simple harmonic if:

A) the amplitude is small
B) the potential energy is equal to the kinetic energy
C) the motion is along the arc of a circle
D) the acceleration varies sinusoidally with time
E) the derivative, dU/dx, of the potential energy is negative

Question 17

A weight suspended from an ideal spring oscillates up and down with a period T. If the amplitude of the oscillation is doubled, the period will be:

A) T
B) 1.5 T
C) 2T
D) T/2
E) 4T

Question 18

A block attached to a spring oscillates in simple harmonic motion along the x axis. The limits of its motion are x = 10 cm and x = 50 cm and it goes from one of these extremes to the other in 0.25 s. Its amplitude and frequency are:

A) 40 cm, 2 Hz
B) 20 cm, 4 Hz
C) 40 cm, 4 Hz
D) 25 cm, 4 Hz
E) 20 cm, 2 Hz

Question 19

An object attached to one end of a spring makes 20 vibrations in 10 seconds. Its angular frequency is:

A) 0.79 rad/s
B) 1.57 rad/s
C) 2.0 rad/s
D) 6.3 rad/s
E) 12.6 rad/s

Question 20

This plot shows a mass oscillating as x = x_m cos (ωt + φ). What are x_m and φ?

A) 1 m, 0°
B) 2 m, 0°
C) 2 m, 90°
D) 4 m, 0°
E) 4 m, 90°

Question 21

A 0.20-kg object mass attached to a spring whose spring constant is 500 N/m executes simple harmonic motion. If its maximum speed is 5.0 m/s, the amplitude of its oscillation is:

A) 0.0020 m
B) 0.10 m
C) 0.20 m
D) 25 m
E) 250 m

Question 22

A 0.25-kg block oscillates on the end of the spring with a spring constant of 200 N/m. If the oscillation is started by elongating the spring 0.15 m and giving the block a speed of 3.0 m/s, then the amplitude of the oscillation is:

A) 0.13 m
B) 0.18 m
C) 3.7 m
D) 5.2 m
E) 13 m

Question 23

Let U be the potential energy (with the zero at zero displacement) and K be the kinetic energy of a simple harmonic oscillator. U_avg and K_avg are the average values over a cycle. Then:

A) K_avg > U_avg
B) K_avg < U_avg
C) K_avg = U_avg
D) K = 0 when U = 0
E) K + U = 0

Question 24

The displacement of an object oscillating on a spring is given by x(t) = x_mcos( $\omega$ t + $\phi$ ). If the initial displacement is zero and the initial velocity is in the negative x direction, then the phase constant $\phi$ is:

A) 0 radians
B) $\pi$ /2 radians
C) $\pi$ radians
D) 3 $\pi$ /2 radians
E) 2 $\pi$ radians

Question 25

A certain spring elongates 9 mm when it is suspended vertically and a block of mass M is hung on it. The angular frequency of this mass-spring system:

A) is 0.088 rad/s
B) is 33 rad/s
C) is 200 rad/s
D) is 1140 rad/s
E) cannot be computed unless the value of M is given

Question 26

The displacement of an object oscillating on a spring is given by x(t) = x_mcos( $\omega$ t + $\phi$ ). If the object is initially displaced in the negative x direction and given a negative initial velocity, then the phase constant $\phi$ is between:

A) 0 and $\pi$ /2 radians
B) $\pi$ /2 and $\pi$ radians
C) $\pi$ and 3 $\pi$ /2 radians
D) 3 $\pi$ /2 and 2 $\pi$ radians
E) none of the above ( $\phi$ is exactly 0, $\pi$ /2, $\pi$ , or 3 $\pi$ /2 radians)

Question 27

In simple harmonic motion, the restoring force must be proportional to the:

A) amplitude
B) frequency
C) velocity
D) displacement
E) displacement squared

Question 28

An object on the end of a spring is set into oscillation by giving it an initial velocity while it is at its equilibrium position. In the first trial the initial velocity is v₀ and in the second it is 4v₀. In the second trial:

A) the amplitude is half as great and the maximum acceleration is twice as great
B) the amplitude is twice as great and the maximum acceleration is half as great
C) both the amplitude and the maximum acceleration are twice as great
D) both the amplitude and the maximum acceleration are four times as great
E) the amplitude is four times as great and the maximum acceleration is twice as great

Question 29

A 0.25-kg block oscillates on the end of the spring with a spring constant of 200 N/m. If the system has an energy of 6.0 J, then the maximum speed of the block is:

A) 0.06 m/s
B) 0.17 m/s
C) 0.24 m/s
D) 4.9 m/s
E) 6.9 m/s

Question 30

A particle is in simple harmonic motion along the x axis. The amplitude of the motion is x_m. At one point in its motion its kinetic energy is K = 5 J and its potential energy (measured with U = 0 at x = 0) is U = 3 J. When it is at x = x_m, the kinetic and potential energies are:

A) K = 5 J and U = 3 J
B) K = 5 J and U = -3 J
C) K = 8 J and U = 0 J
D) K = 0 J and U = 8 J
E) K = 0 J and U = -8 J

Question 31

A 0.25-kg block oscillates on the end of the spring with a spring constant of 200 N/m. If the system has an energy of 6.0 J, then the amplitude of the oscillation is:

A) 0.06 m
B) 0.17 m
C) 0.24 m
D) 4.9 m
E) 6.9 m

Question 32

A block attached to a spring undergoes simple harmonic motion on a horizontal frictionless surface. Its total energy is 50 J. When the displacement is half the amplitude, the kinetic energy is:

A) 0 J
B) 12.5 J
C) 25 J
D) 37.5 J
E) 50 J

Question 33

A 3-kg block, attached to a spring, executes simple harmonic motion according to x = 2cos(50t) where x is in meters and t is in seconds. The spring constant of the spring is:

A) 1 N/m
B) 100 N/m
C) 150 N/m
D) 7500 N/m
E) none of these

Question 34

A 1.2-kg mass is oscillating without friction on a spring whose spring constant is 3400 N/m. When the mass's displacement is 7.2 cm, what is its acceleration?

A) −3.8 m/s²
B) −200 m/s²
C) −240 m/s²
D) −2.0 x 10⁴ m/s²
E) cannot be calculated without more information

Question 35

A 0.25-kg block oscillates on the end of the spring with a spring constant of 200 N/m. If the oscillation is started by elongating the spring 0.15 m and giving the block a speed of 3.0 m/s, then the maximum speed of the block is:

A) 0.13 m/s
B) 0.18 m/s
C) 3.7 m/s
D) 5.2 m/s
E) 13 m/s

Question 36

A particle is in simple harmonic motion along the x axis. The amplitude of the motion is x_m. When it is at x = x₁, its kinetic energy is K = 5 J and its potential energy (measured with U = 0 at x = 0) is U = 3 J. When it is at x = -1/2 x_m, the kinetic and potential energies are:

A) K = 6 J and U = 2 J
B) K = 6 J and U = -2 J
C) K = 8 J and U = 0 J
D) K = 0 J and U = 8 J
E) K = 0 J and U = -8 J

Question 37

The acceleration of a body executing simple harmonic motion leads the velocity by what phase?

A) 0 rad
B) $\pi$ /8 rad
C) $\pi$ /4 rad
D) $\pi$ /2 rad
E) $\pi$ rad

Question 38

In simple harmonic motion, the magnitude of the acceleration is greatest when:

A) the displacement is zero
B) the displacement is maximum
C) the speed is maximum
D) the force is zero
E) the speed is between zero and its maximum

Question 39

In simple harmonic motion:

A) the acceleration is greatest at the maximum displacement
B) the velocity is greatest at the maximum displacement
C) the period depends on the amplitude
D) the acceleration is constant
E) the acceleration is greatest at zero displacement

Question 40

In simple harmonic motion, the magnitude of the acceleration is:

A) constant
B) proportional to the displacement
C) inversely proportional to the displacement
D) greatest when the velocity is greatest
E) never greater than g

Question 41

An angular simple harmonic oscillator:

A) oscillates at an angle to the x axis
B) oscillates along the y axis
C) involves an angular displacement and a restoring torque
D) involves an angular displacement and a restoring force
E) involves a linear displacement and a restoring torque

Question 42

A simple pendulum consists of a small ball tied to a string and set in oscillation. As the pendulum swings the tension in the string is:

A) constant
B) a sinusoidal function of time
C) the square of a sinusoidal function of time
D) the reciprocal of a sinusoidal function of time
E) none of the above

Question 43

An angular simple harmonic oscillator is displaced 5.2 x 10^-2 rad from its equilibrium position. If the torsion constant is 1200 N∙m/rad, what is the torque?

A) 12 N∙m
B) 23 N∙m
C) 43 N∙m
D) 52 N∙m
E) 62 N∙m

Question 44

A disk whose rotational inertia is 450 kg∙m² hangs from a wire whose torsion constant is 2300 N∙m/rad. When its angular displacement is −0.23 rad, what is its angular acceleration?

A) 1.0 x 10^-2 rad/s²
B) 4.5 x 10^-2 rad/s²
C) 0.23 rad/s²
D) 0.52 rad/s²
E) 1.2 rad/s²

Question 45

The angular displacement of a simple pendulum is given by θ(t) = θ_m cos (ωt + φ). If the pendulum is 45 cm in length, and is given an angular speed dθ/dt = 3.4 rad/s at time t = 0, when it is hanging vertically, what is θ_m?

A) 4.6 rad
B) 3.4 rad
C) 1.4 rad
D) 0.73 rad
E) 0.45 rad

Question 46

Five hoops are each pivoted at a point on the rim and allowed to swing as physical pendulums. The masses and radii are Order the hoops according to the periods of their motions, smallest to largest.

A) 1, 2, 3, 4, 5
B) 5, 4, 3, 2, 1
C) 1, 2, 3, 5, 4
D) 1, 2, 5, 4, 3
E) 5, 4, 1, 2, 3

Question 47

A simple pendulum has length L and period T. As it passes through its equilibrium position, the string is suddenly clamped at its mid-point. The period then becomes:

A) 2T
B) T
C) T/2
D) T/4
E) none of the above

Question 48

Which of the following is the difference between a simple pendulum and a physical pendulum?

A) The physical pendulum does not rotate around its center of mass.
B) The physical pendulum does not depend on the acceleration of gravity.
C) The physical pendulum has an extended mass.
D) The simple pendulum has a small amplitude.
E) The physical pendulum rotates around its center of mass.

Question 49

A particle is in simple harmonic motion along the x axis. The amplitude of the motion is x_m. When it is at x = x₁, its kinetic energy is K = 5 J and its potential energy (measured with U = 0 at x = 0) is U = 3 J. When it is at x = -1/2 x_m, its total energy is:

A) 0 J
B) 3 J
C) 4 J
D) 5 J
E) 8 J

Question 50

Which of the following is NOT required for a simple pendulum undergoing simple harmonic oscillation?

A) a point mass
B) a massless string
C) a small amplitude
D) gravitational force
E) a large spring constant

Question 51

A meter stick is pivoted at a point a distance a from its center and swings as a physical pendulum. Of the following values for a, which results in the shortest period of oscillation?

A) a = 0.1 m
B) a = 0.2 m
C) a = 0.3 m
D) a = 0.4 m
E) a = 0.5 m

Question 52

The amplitude of oscillation of a simple pendulum is increased from 1 $\degree$ to 4 $\degree$ . Its maximum acceleration changes by a factor of:

A) 1/4
B) 1/2
C) 2
D) 4
E) 16

Question 53

A simple pendulum of length L and mass M has frequency f. To increase its frequency to 2f:

A) increase its length to 4L
B) increase its length to 2L
C) decrease its length to L/2
D) decrease its length to L/4
E) decrease its mass to < M/4

Question 54

A particle is in simple harmonic motion along the x axis. The amplitude of the motion is x_m. When it is at x = x₁, its kinetic energy is K = 5 J and its potential energy (measured with U = 0 at x = 0) is U = 3 J. When its kinetic energy is 8 J, it is at:

A) x = 0
B) x = x₁
C) x = x_m/2
D) x = x_m/
E) x = x_m

Question 55

If the length of a simple pendulum is doubled, its period will:

A) halve
B) increase by a factor of
C) decrease by a factor of
D) double
E) remain the same

Question 56

A disk whose rotational inertia is 450 kg∙m² hangs from a wire whose torsion constant is 2300 N∙m/rad. What is the angular frequency of its torsional oscillations?

A) 0.20 rad/s
B) 0.44 rad/s
C) 1.0 rad/s
D) 2.3 rad/s
E) 5.1 rad/s

Question 57

Two uniform spheres are pivoted on horizontal axes that are tangent to their surfaces. The one with the longer period of oscillation is the one with:

A) the larger mass
B) the smaller mass
C) the larger rotational inertia
D) the smaller rotational inertia
E) the larger radius

Question 58

A particle is in simple harmonic motion along the x axis. The amplitude of the motion is x_m. When it is at x = x₁, its kinetic energy is K = 5 J and its potential energy (measured with U = 0 at x = 0) is U = 3 J. When its potential energy is 8 J, it is at:

A) x = 0
B) x = x₁
C) x = x_m/2
D) x = x_m/
E) x = x_m

Question 59

Three physical pendulums, with masses m₁, m₂ = 2m₁, and m₃ = 3m₁, have the same shape and size and are suspended at the same point. Rank them according to their periods, from shortest to longest.

A) 1, 2, 3
B) 3, 2, 1
C) 2, 3, 1
D) 2, 1, 3
E) All three are the same

Question 60

At the instant its angular displacement is 0.32 rad, the angular acceleration of a physical pendulum is -630 rad/s². What is its angular frequency of oscillation?

A) 6.6 rad/s
B) 14 rad/s
C) 20 rad/s
D) 44 rad/s
E) 200 rad/s

Question 61

The period of a simple pendulum is 1 s on Earth. When brought to a planet where g is one-tenth that on Earth, its period becomes:

A) 1 s
B)
C) 1/10 s
D)
E) 10 s

Question 62

The rotational inertia of a uniform thin rod about its end is ML²/3, where M is the mass and L is the length. Such a rod is hung vertically from one end and set into small amplitude oscillation. If L = 1.0 m this rod will have the same period as a simple pendulum of length:

A) 33 cm
B) 50 cm
C) 67 cm
D) 100 cm
E) 150 cm

Question 63

For an oscillator subjected to a damping force proportional to its velocity:

A) the displacement is a sinusoidal function of time
B) the velocity is a sinusoidal function of time
C) the frequency is a decreasing function of time
D) the mechanical energy is constant
E) none of the above is true

Question 64

Both the x and y coordinates of a point execute simple harmonic motion. The result might be a circular orbit if:

A) the amplitudes are the same but the frequencies are different
B) the amplitudes and frequencies are both the same
C) the amplitudes and frequencies are both different
D) the phase constants are the same but the amplitudes are different
E) the amplitudes and the phase constants are both different

Question 65

Below are sets of values for the spring constant k, damping constant b, and mass m for a particle in damped harmonic motion. Which of the sets takes the longest time for its mechanical energy to decrease to one-fourth of its initial value?

Question 66

A block on a spring is subjected to an applied sinusoidal force AND to a damping force that is proportional to its velocity. The energy dissipated by damping is supplied by:

A) the potential energy of the spring
B) the kinetic energy of the mass
C) gravity
D) friction
E) the applied force

Question 67

A particle undergoes damped harmonic motion. The spring constant is 100 N/m; the damping constant is 8.0 x 10^-3 kg∙m/s, and the mass is 0.050 kg. If the particle starts at its maximum displacement, x = 1.5 m, at time t = 0, what is the angular frequency of the oscillations?

A) 4.0 rad/s
B) 8.0 rad/s
C) 12 rad/s
D) 23 rad/s
E) 45 rad/s

Question 68

Five particles undergo damped harmonic motion. Values for the spring constant k, the damping constant b, and the mass m are given below. Which leads to the smallest rate of loss of mechanical energy?

A) k = 100 N/m, m = 50 g, b = 8 g∙m/s
B) k = 150 N/m, m = 50 g, b = 5 g∙m/s
C) k = 150 N/m, m = 10 g, b = 8 g∙m/s
D) k = 200 N/m, m = 8 g, b = 6 g∙m/s
E) k = 100 N/m, m = 2 g, b = 4 g∙m/s

Question 69

A particle undergoes damped harmonic motion. The spring constant is 100 N/m; the damping constant is 8.0 x 10^-3 kg∙m/s, and the mass is 0.050 kg. If the particle starts at its maximum displacement, x = 1.5 m, at time t = 0, what is the amplitude of the motion at t = 5.0 s?

A) 1.5 m
B) 1.3 m
C) 1.0 m
D) 0.67 m
E) 0.24 m

Question 70

An oscillator is driven by a sinusoidal force. The frequency of the applied force:

A) must be equal to the natural frequency of the oscillator
B) becomes the natural frequency of the oscillator
C) must be less than the natural frequency of the oscillator
D) must be greater than the natural frequency of the oscillator
E) is independent of the natural frequency of the oscillator

Question 71

A particle undergoes damped harmonic motion. The spring constant is 100 N/m; the damping constant is 8.0 x 10^-3 kg∙m/s, and the mass is 0.050 kg. If the particle starts at its maximum displacement, x = 1.5 m, at time t = 0, what is the particle's position at t = 5.0 s?

A) -1.5 m
B) -0.73 m
C) 0 m
D) 0.73 m
E) 1.5 m

Question 72

An oscillator is subjected to a damping force that is proportional to its velocity. A sinusoidal force is applied to it. After a long time:

A) its amplitude is an increasing function of time
B) its amplitude is a decreasing function of time
C) its amplitude is constant
D) its amplitude is a decreasing function of time only if the damping constant is large
E) its amplitude increases over some portions of a cycle and decreases over other portions

Question 73

Both the x and y coordinates of a point execute simple harmonic motion. The frequencies are the same but the amplitudes are different. The resulting orbit might be:

A) an ellipse
B) a circle
C) a parabola
D) a hyperbola
E) a square

Question 74

A sinusoidal force with a given amplitude is applied to an oscillator. To maintain the largest amplitude oscillation the frequency of the applied force should be:

A) half the natural frequency of the oscillator
B) the same as the natural frequency of the oscillator
C) twice the natural frequency of the oscillator
D) unrelated to the natural frequency of the oscillator
E) determined from the maximum speed desired

Question 75

A sinusoidal force with a given amplitude is applied to an oscillator. At resonance the amplitude of the oscillation is limited by:

A) the damping force
B) the initial amplitude
C) the initial velocity
D) the force of gravity
E) none of the above