A 506 g block is released from rest at height h0 above a vertical spring with spring constant k = 500 N/m and negligible mass. The block sticks to the spring and momentarily stops after compressing the spring 18.4 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 5h0 above the spring, what would be the maximum compression of the spring?
a) I used Ws = -[.5kxf^2 - .5kxi^2] and got 8.464 J
b) Is just the negative of that so -8.464 J
*For both a and b my reference angle was the compressed spring bottomed out. So my initial value was 0 and final value was .184*
c) and d) I don't really know how to go about finding the initial height. How would I do this?
c. THat energy had to come from GPE.
mgh=8.486 solve for h, that is the h above the final position of the spring. So you need to subtract the .184
So tbe work done by the spring is the Potential Energy = mgh?
I get 1.7069 m.
For part d) How would you find the compression of the spring at h0 = 8.5345 m?
I think I screwed up somewhere because my values aren't correct. Am I supposed to set it up like mg(h-.184) = 8.464?
I just tried 8.464 = mg(h-.184) and my answer for the height was still wrong. What am I doing wrong here ...
I should add for part d) I have it set up so that mgh = .5kx^2 to find the max compression at 5h. I think both of these are right yet my answers still ring up as wrong.
To find the initial height (h0) in the given scenario, we can use the principle of conservation of mechanical energy. Before the block is released, it possesses potential energy due to its height above the ground. When the block reaches its maximum compression on the spring, all of its initial potential energy is converted into the potential energy stored in the compressed spring.
Let's consider the conservation of mechanical energy equation:
Potential Energy at height h0 = Potential Energy stored in the compressed spring
The potential energy at height h0 is given by:
Potential Energy at height h0 = mgh0
where
m = mass of the block = 506 g = 0.506 kg
g = acceleration due to gravity = 9.8 m/s^2 (approximate value on the surface of the Earth)
h0 = initial height
The potential energy stored in the compressed spring is given by:
Potential Energy stored in the compressed spring = (1/2)kx^2
where
k = spring constant = 500 N/m
x = compression of the spring = 18.4 cm = 0.184 m (converted to meters)
Equating the two expressions and solving for h0:
mgh0 = (1/2)kx^2
0.506 * 9.8 * h0 = (1/2) * 500 * (0.184)^2
4.9298 * h0 = 8.048
h0 = 8.048 / 4.9298
h0 ≈ 1.63 m
Therefore, the initial height h0 is approximately 1.63 meters.
For part (d), if the block were released from a height of 5h0 above the spring, we can again use the principle of conservation of mechanical energy to find the maximum compression of the spring.
The initial potential energy at height 5h0 is given by:
Potential Energy at height 5h0 = mgh
where
h = 5h0 = 5 * 1.63 m
The maximum potential energy stored in the compressed spring is still given by:
Potential Energy stored in the compressed spring = (1/2)kx^2
Using the conservation of mechanical energy equation:
Potential Energy at height 5h0 = Potential Energy stored in the compressed spring
mgh = (1/2)kx^2
0.506 * 9.8 * 5 * 1.63 = (1/2) * 500 * x^2
x^2 = (0.506 * 9.8 * 5 * 1.63 * 2) / 500
x ≈ 0.471 m
Therefore, the maximum compression of the spring if the block were released from a height of 5h0 above the spring would be approximately 0.471 meters.