Harmonic Motion

The graph below is a plot of displacement versus time of a mass oscillating on a spring.

At which point on the graph is the acceleration of the mass zero?

• A

• B

• C

• D

Answer explanation

As the object moves closer to the equilibrium position (position C), the speed increases and the force decreases. The speed is its maximum and the force is zero at the equilibrium position. In the equilibrium position, the upward force balances the object's weight, therefore, there is no acceleration.

A mass is attached to a vertical spring and bobs up and down between points A and B. Where is the mass located when its kinetic energy is a minimum?

• at either A or B

  
  

• 
  

  one-fourth of the way between A and B

• midway between A and B

• 
  

  none of the above

If you take a given pendulum to the Moon, where the acceleration of gravity is less than on Earth, the resonant frequency of the pendulum will

• 
  

  not change.

• 
  

  decrease.

  
  

• 
  

  increase.

• either increase or decrease; it depends on its length to mass ratio.

A pendulum with a period of 1 s on Earth, where the acceleration due to gravity is g, is taken to another planet, where its period is 2 s. The acceleration due to gravity on the other planet is most nearly

• 
  

  g/4

  
  

• 
  

  g/2

• 
  

  g

• 
  

  2g

• 
  

  4g

A mass is attached to a vertical spring and bobs up and down between points A and B. Where is the mass located when its kinetic energy is a maximum?

• 
  

  one-fourth of the way between A and B

• at either A or B

• 
  

  midway between A and B

  
  

• 
  

  none of the above

  
  

A mass is attached to a vertical spring and bobs up and down between points A and B. Where is the mass located when its potential energy is a minimum?

• midway between A and B

  
  

• 
  

  at either A or B

• one-fourth of the way between A and B

• 
  

  none of the above

Increasing the mass M of a mass-and-spring system causes what kind of change in the resonant frequency of the system? (Assume no change in the system's spring constant k.)

• 
  

  The frequency increases.

• 
  

  The frequency decreases.

  
  

• The frequency increases if the ratio k/m is greater than or equal to 1 and decreases if the ratio k/m is less than 1.

• There is no change in the frequency.

The graph below is a plot of displacement versus time of a mass oscillating on a spring.

At which point on the graph is the velocity of the mass zero?

• A

• B

• C

• D

Answer explanation

At the object's maximum displacement (A), the net force is at its maximum and the motion of the object has stopped (v = 0 m/s) for an instant before moving in the opposite direction. As the block moves closer to the equilibrium position, the speed increases and the force decreases. The speed is its maximum and the restoring force is zero at the equilibrium position (C). As the block continues to move past its equilibrium position, the force acting on it tends to slow it down.

Doubling only the amplitude of a vibrating mass-and-spring system produces what effect on the system's mechanical energy?

• increases the energy by a factor of two

• 
  

  increases the energy by a factor of four

• increases the energy by a factor of three

• 
  

  produces no change

Increasing the spring constant k of a mass-and-spring system causes what kind of change in the resonant frequency of the system? (Assume no change in the system's mass m.)

• 
  

  There is no change in the frequency.

• 
  

  The frequency increases.

  
  

• 
  

  The frequency decreases.

• The frequency increases if the ratio k/m is greater than or equal to 1 and decreases if the ratio k/m is less than 1.