How Is Spring Compression Calculated When a Mass Is Dropped on It?

[John O' Meara]

Suppose a 300g mass is dropped from a height of 40cm onto a vertical spring with spring constant 200N/m (having a light platform on top) and sticks to the platform (a) How far will the spring compress? (b) How far will the spring be stretched as the mass and spring rebound? use S.I. units.
I can do part (a) ans = .124m. For part (b) I cannot get right answer =.095m

Us at the bottom = Ug at end

where: Us = potential energy of the spring
Ug = potential energy of gravity

this gives:

.5kx^2 = mgx
=> .5kx = mg
= wrong result
what am I missing,
Thanks & regards

 

[OlderDan]

If you have done part a) correctly, you know the lowest point of the motion of the mass attached to the spring. If you find the equilibrium point, you can treat the rest of the problem as a harmonic oscillator with an initial displacement starting with zero velocity.

 

[thepatientmental]

In order to solve for the distance that the spring will be stretched in part (b), you need to consider the conservation of energy in the system. When the mass is dropped onto the spring, it will compress the spring and gain potential energy. This potential energy is then converted into kinetic energy as the mass and spring rebound, and eventually back into potential energy as the mass reaches its maximum height again.

So, in order to solve for the distance the spring will be stretched, you need to equate the initial potential energy gained by compressing the spring to the final potential energy gained when the mass reaches its maximum height.

Initial potential energy: Us = 0.5kx^2 (where x is the distance the spring is compressed)

Final potential energy: Ug = mgh (where h is the maximum height reached by the mass)

Since energy is conserved, we can set these two equations equal to each other:

0.5kx^2 = mgh

Solving for x, we get:

x = √(2mgh/k)

Plugging in the values given in the question, we get:

x = √(2*0.3*9.8*0.4/200) = 0.095m

Therefore, the spring will be stretched by 0.095m as the mass and spring rebound. It is important to note that this is the maximum distance the spring will stretch, as it will gradually decrease as the mass and spring oscillate.