Spring Potential/Work Energy Questions

[Sasor]

Question 1:

1. Homework Statement
You have a spring at height d where it is relaxed.
You drop a ball (mass m) from a height (h) so that it lands on the spring with spring constant k.
What is the max compression of the spring in terms of given variables?
Given-

m
g
k
d
h


2. Homework Equations
Find
dmax=max compression distance

3. The Attempt at a Solution
i did-

deltaUgrav+deltaUspring=0

(mg(d-dmax)-mg(d+h))+(.5k(dmax)^2-.5k(d-d))=0

mgd-mgdmax-mgd-mgh+.5k(dmax)^2=0

-mg(dmax)-mg(h)+.5k(dmax)^2=0

.5k(dmax)^2=mg(dmax+h)

Can you solve for dmax or do u have to do quadratic equation?


Question 2:

1. Homework Statement
If you have a spring and an object with mass m
and you put the object on the spring and let go, without giving it any initial velocity, what is the work done by the spring on the object? Answer is symbolic
Given variables-

Fspring with respect to s
m
g
k
s0(= initial length, relaxed length)
sf
2. Homework Equations

Symbollically, what is the work done?

3. The Attempt at a Solution

I did it like this-

Work= Integral(Fspring) evaluated from initial s to final s

so

Integral of ks ds= .5ks^2] sf-s0

=.5k(sf)^2-.5k(s0)^2
=.5k(sf-s0)

Is this the right amount of work?

 

[anathema]

Hello,

For Question 1, your attempt at a solution is correct. To solve for dmax, you will need to use the quadratic equation. The final equation you have is in the form of ax^2 + bx + c = 0, where a = 0.5k, b = -mg, and c = -mgh. Solving for dmax will give you the maximum compression distance of the spring.

For Question 2, your approach is correct. The work done by the spring on the object can be symbolically represented as W = 0.5ks^2 - 0.5ks0^2. This is the same as the work done by a spring when compressed from its initial length (s0) to its final length (sf). Therefore, your final answer is correct.

Hope this helps!