Question

The equation x(t) = A sin(ωt + φ) describes the simple harmonic motion of a block attached to a spring with spring constant k = 20 N/m. At t = 0 s, x = 0.5 m and the block is at rest.

a (5 points) After 0.5 s, the potential energy stored in the spring first reaches zero and the velocity of the block is in the negative x direction. What is the period of the oscillation?

b (5 points) What are A and φ?

c (5 points) What is the maximum potential energy stored in the spring?

d (5 points) What is the mass of the block?

e (5 points) When x = 0 m, what is the speed of the block?

Answer 1

x(t) = A sin(ωt + φ) describes the simple harmonic motion

Given k=20 N/m, at t=0 x = 0.5m and the body is at rest.

This indicates at t=0, the body is at positive extreme hence the amplitude becomes A=0.5m

at t=0, 0.5 = 0.5 sin(ωt + φ) => 0.5 = 0.5 sin(0 + φ) => sin(φ) = 1 => φ =900

a) The potential energy becomes zero at x = 0 and it happens for the first time at t = 0.5s. This indicates the time taken by the block to move from right extreme to mean position is 0.5s. Hence the time period is T= 2s because minimum time taken by the block from extreme to mean is T/4.

b) A is amplitude and φ is phase constant. A=0.5m and φ =900

c) The maximum potential energy stored in the spring is kA2 /2 = 20 x (0.5)2 /2 = 2.5 J

d) since   then m = 2 kg (if  )

The mass of the block is 2kg.

e) when x=0 the block is at mean position and hence the speed of the block is maximum and is given by V= = 0.5 = as =

The speed of the block at x=0 is 1.57 m/s