Question 1

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

Accepted Answer

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

Question 2

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

Accepted Answer

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

Question 3

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

Accepted Answer

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

Question 4

At sea level, at a latitude where   , 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?

Accepted Answer

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

Question 5

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.

Accepted Answer

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

Question 6

The mass in the figure below slides on a frictionless surface. When the mass is pulled out, spring 1 is stretched a distance x1 from its equilibrium position and spring 2 is stretched a distance x2. The spring constants are k1 and k2 respectively. The force pulling back on the mass is:

Accepted Answer

A)  −k2x1. 
B)  −k2x2. 
C)  −(k1x1 + k2x2). 
D) 
  . 
E) 
  . 
A)  −k2x1. 
B)  −k2x2. 
C)  −(k1x1 + k2x2). 
D) 
  . 
E) 
  .

Question 7

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(s).
-Refer to Exhibit 15-1. The point at which the object has zero velocity and negative acceleration is

Accepted Answer

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

Question 8

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(s).
-Refer to Exhibit 15-1. A point or points at which the object has positive velocity and zero acceleration is(are)

Accepted Answer

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

Question 9

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

Accepted Answer

A)  increase the angular frequency by
  . 
B)  increase the amplitude by
  . 
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
  . 
A)  increase the angular frequency by
  . 
B)  increase the amplitude by
  . 
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
  .

Question 10

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

Accepted Answer

A)  T 0. 
B)  T = T0. 
C)  T > T0. 
D)  ωT 0T0. 
E)  ωT > ω0T0. 
A)  T 0. 
B)  T = T0. 
C)  T > T0. 
D)  ωT 0T0. 
E)  ωT > ω0T0.

Question 11

An ore car of mass 4 000 kg starts from rest and rolls downhill on tracks from a mine. A spring with k = 400 000 N/m is located at the end of the tracks. At the spring's maximum compression, the car is at an elevation 10 m lower than its elevation at the starting point. How much is the spring compressed in stopping the ore car? Ignore friction.

Accepted Answer

The answer of An ore car of mass 4 000...

Question 12

Three pendulums with strings of the same length and bobs of the same mass are pulled out to angles θ1, θ2 and θ3 respectively and released. The approximation sin θ = θ holds for all three angles, with θ3 > θ2 > θ1. How do the angular frequencies of the three pendulums compare?

Accepted Answer

A)  ω3 > ω2 > ω1 
B)  Need to know amplitudes to answer this question. 
C)  Need to know
  to answer this question. 
D)  ω1 > ω2 > ω3 
E)  ω1 = ω2 = ω3 
A)  ω3 > ω2 > ω1 
B)  Need to know amplitudes to answer this question. 
C)  Need to know
  to answer this question. 
D)  ω1 > ω2 > ω3 
E)  ω1 = ω2 = ω3

Question 13

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

Accepted Answer

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

Question 14

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

Accepted Answer

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. 
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 15

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 ω?

Accepted Answer

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

Question 16

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(s).
-Refer to Exhibit 15-1. The point at which the object has zero velocity and positive acceleration is

Accepted Answer

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

Question 17

When a damping force is applied to a simple harmonic oscillator which has angular frequency ω0 in the absence of damping, the new angular frequency ω is such that

Accepted Answer

A)  ω 0. 
B)  ω = ω0. 
C)  ω > ω0. 
D)  ωT 0T0. 
E)  ωT > ω0T0. 
A)  ω 0. 
B)  ω = ω0. 
C)  ω > ω0. 
D)  ωT 0T0. 
E)  ωT > ω0T0.

Question 18

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?

Accepted Answer

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

Question 19

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?

Accepted Answer

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. 
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 20

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?

Accepted Answer

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. 
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.

Exam 15: Oscillatory Motion

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

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

At sea level, at a latitude where , 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 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.

The mass in the figure below slides on a frictionless surface. When the mass is pulled out, spring 1 is stretched a distance x1 from its equilibrium position and spring 2 is stretched a distance x2. The spring constants are k1 and k2 respectively. The force pulling back on the mass is:

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

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

Three pendulums with strings of the same length and bobs of the same mass are pulled out to angles θ1, θ2 and θ3 respectively and released. The approximation sin θ = θ holds for all three angles, with θ3 > θ2 > θ1. How do the angular frequencies of the three pendulums compare?

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

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

When a damping force is applied to a simple harmonic oscillator which has angular frequency ω0 in the absence of damping, the new angular frequency ω is such that

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?

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 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?

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

At sea level, at a latitude where , 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 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.

The mass in the figure below slides on a frictionless surface. When the mass is pulled out, spring 1 is stretched a distance x₁ from its equilibrium position and spring 2 is stretched a distance x₂. The spring constants are k₁ and k₂ respectively. The force pulling back on the mass is:

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

Three pendulums with strings of the same length and bobs of the same mass are pulled out to angles θ₁, θ₂ and θ₃ respectively and released. The approximation sin θ = θ holds for all three angles, with θ₃ > θ₂ > θ₁. How do the angular frequencies of the three pendulums compare?

When a damping force is applied to a simple harmonic oscillator which has angular frequency ω₀ in the absence of damping, the new angular frequency ω is such that