A 540 g block is released from rest at height h0 above a vertical spring with spring constant k = 320 N/m and negligible mass. The block sticks to the spring and momentarily stops after compressing the spring 19.8 cm. How much work is done (a) by the block on the spring and (b) by the spring on the block? (c) What is the value of h0? (d) If the block were released from height 3h0 above the spring, what would be the maximum compression of the spring?

I just need help with d). I got for a) is 6.27 J, b) is -6.27, and c) is 0.987. All of these are correct, but I am having trouble with d).

To find the maximum compression of the spring when the block is released from a height 3h0 above, we can use the principle of conservation of mechanical energy.

First, let's determine the initial potential energy of the block at height 3h0. The potential energy is given by the equation:

PE = m * g * h

Where m is the mass of the block, g is the acceleration due to gravity, and h is the height above the spring.

Given:
m = 540 g = 0.54 kg
g = 9.8 m/s^2
h = 3h0

Since h0 is not given, we need to find its value first. To do that, we can use the information provided in parts (a) and (b).

The work done by the block on the spring is equal to the change in potential energy of the block:

Work = ΔPE = m * g * h0 - m * g * 0

Work = m * g * h0

Substituting the given values:
6.27 J = 0.54 kg * 9.8 m/s^2 * h0

Solving for h0:
h0 = 6.27 J / (0.54 kg * 9.8 m/s^2)
h0 ≈ 1.196 m

Now, we can find the maximum compression of the spring when the block is released from a height 3h0 above.

Using the principle of conservation of mechanical energy, we can equate the initial potential energy of the block to the maximum potential energy stored in the compressed spring:

PE_initial = PE_spring

m * g * (3h0) = 0.5 * k * x^2

Where x is the maximum compression of the spring.

Substituting the known values:
0.54 kg * 9.8 m/s^2 * (3 * 1.196 m) = 0.5 * 320 N/m * x^2

Simplifying the equation:
x^2 = (0.54 kg * 9.8 m/s^2 * (3 * 1.196 m)) / (0.5 * 320 N/m)

Solving for x:
x ≈ √(5.59 m^2)

Therefore, the maximum compression of the spring when the block is released from height 3h0 above is approximately 2.36 m.