Question

A block with mass m =6.9 kg is hung from a vertical spring. When the mass hangs in equilibrium, the spring stretches x = 0.24 m. While at this equilibrium position, the mass is then given an initial push downward at v = 5.1 m/s. The block oscillates on the spring without friction.
1) What is the spring constant of the spring? A: 282.04 N/m (Correct)
2) What is the oscillation frequency? A: 1.02 Hz (Correct)
3) After t = 0.32 s what is the speed of the block? A:? m/s
4) What is the magnitude of the maximum acceleration of the block? A:? m/s^2
5) At t = 0.32 s what is the magnitude of the net force on the block? A:? N
6) Where is the potential energy of the system the greatest?
a) At the highest point of the oscillation.
b) At the new equilibrium position of the oscillation.
c) At the lowest point of the oscillation.
(I can choose multiple options for question 6).

 NOTE: I've had a lot of trouble figuring out the answer to Q3 - I know it's not 5.1, as that is the maximum velocity. I've tried many more but I'm stuck.

Answer 1

U need to set up n solve the general eqn for simple harmonic motion:
x" = -(k/m)x

solution is x(t) = (x0)*cos(wt) + (v0/w)*sin(wt)
where w=sqrt(k/m), x0 is x-position at t=0 and v0 is vel at t=0
u already calculated f in Q.2 and w = 2*pi*f
x0 is 0 as it starts at eqm
v0 is given at 5.1

so u have x(t)

vel is given by x'(t) = (x0)*(-w)*sin(wt) + (v0/w)*w*cos(wt)

substitute t=0.32, x0=0, v0=5.1 n w in the above, u can solve for v at t=0.32.