Question

A spring-mass system undergoes simple harmonic motion. Which of the following statements are True/False?

The kinetic energy is maximum at the point of zero displacement. (True or False)

The potential energy is maximum at the point of maximum displacement. (True or False)

The force exerted by the spring is zero at the point of maximum displacement. (True or False )

The acceleration is greatest at the point of zero displacement. (True or False )

If the amplitude of the oscillations is doubled, then

The total energy of the system (increases by a factor of 2, increases by a factor of 4, stays the same)

The maximum restoring force (increases by a factor of 2, increases by a factor of 4, stays the same)

The frequency of the oscillations (increases by a factor of 2, increases by a factor of 4, stays the same)

The maximum speed (increases by a factor of 2, increases by a factor of 4, stays the same)

Answer 1

The kinetic energy is maximum at the point of zero displacement, True, By conservation of energy , at center the complete elastic potential energy is converted in to kinetic energy so the statement is True.

The potential energy is maximum at the point of maximum displacement.True, at maximum displacement the velocity become zero , to satisfy the conservation of energy, that is total Mechanical energy is constant, the potential energy is maximum , when the displacement is maximum.

The force exerted by the spring is zero at the point of maximum displacement, False, F = - kx , so force is directly proportional to the displacement , but acts in opposite direction which is also called Restoring force.

False, Because at the center the acceleration is zero ( a = -w^2*A cos (wt)), where A is the amplitude if A = 0 , means a= 0 , That is the magnitude of the acceleration is greatest when the displacement is maximum, that is F = -kx = ma ==> a depends on 'x', where m,k are constants

The Energy of the system stays the same , by conservation of energy so that there is no friction and other dissipative forces

Maximum restoring force is increases by a factor of 2

Frequency stays the same

Maximum speed increases by a factor of 2. Because

0.5*m*v^2 = 0.5*k*x^2

if x = 2x, then 0.5*m*v^2 = 0.5*k*(2x)^2 = 0.5*k4x^2 ==> v is proportional to 2x