# Hooke's Law vs. Conservation of Energy

#### keroberous

Homework Statement
I can never quite remember when solving problems with elastic materials whether I should be using Hooke's Law or conservation of energy. Below are two sample problems.
Homework Equations
Hooke's Law: $F=kx$
Elastic Potential Energy: $E_e=\frac{1}{2}kx^2$
Gravitational Potential Energy: $E_g=mgh$
Here are the two questions I want to compare:

1. A student of mass 62 kg stands on an upholstered chair containing springs, each of force constant 2.4 × 103 N/m. If the student is supported equally by six springs, what is the compression of each spring?

2. A 0.20-kg ball attached to a vertical spring of force constant 28 N/m is released from rest from the unstretched equilibrium position of the spring. Determine how far the ball falls, under negligible air resistance, before being brought to a momentary stop by the spring.

I can get the right answer for both questions, by using Hooke's law for number 1 and conservation of energy for question 2. My question is if I didn't know what the answers were, how would I choose the method? By using the incorrect method I'm off by a factor of two in both cases (bigger in 1 and smaller in 2).

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#### tnich

Homework Helper
Here are the two questions I want to compare:

1. A student of mass 62 kg stands on an upholstered chair containing springs, each of force constant 2.4 × 103 N/m. If the student is supported equally by six springs, what is the compression of each spring?

2. A 0.20-kg ball attached to a vertical spring of force constant 28 N/m is released from rest from the unstretched equilibrium position of the spring. Determine how far the ball falls, under negligible air resistance, before being brought to a momentary stop by the spring.

I can get the right answer for both questions, by using Hooke's law for number 1 and conservation of energy for question 2. My question is if I didn't know what the answers were, how would I choose the method? By using the incorrect method I'm off by a factor of two in both cases (bigger in 1 and smaller in 2).
In question 1, you are asked to determine the equilibrium position of the system (student plus springs). In question 2, you are asked how far the ball would fall before it reverses direction. This is not the same as the equilibrium position and would in fact be twice as far from the initial position as the equilibrium position.
Each method, used correctly, should give you the right answer to each question. If you were to post your solutions, we could comment on where you went wrong.

#### haruspex

Homework Helper
Gold Member
2018 Award
Each method, used correctly, should give you the right answer to each question.
But... in 1, it is a static arrangement. A conservation law is only directly useful when something has changed.
Conversely, in 2, it is not obvious how to find the force when the velocity is zero.
To apply energy in 1 or Hooke in 2 seems to require performing the conversion, i.e. integrating Hooke or differentiating energy.

#### CWatters

Homework Helper
Gold Member
Any reason you can't turn 1) into a dynamic situation and equate the PE lost by the man with the PE gained by the spring...

mgh=0.5kh^2

Solve for h.

#### haruspex

Homework Helper
Gold Member
2018 Award
Any reason you can't turn 1) into a dynamic situation and equate the PE lost by the man with the PE gained by the spring...
Yes, there is a reason. That will give twice the correct answer, as @keroberous found and @tnich explained.

#### CWatters

Homework Helper
Gold Member
Palm -> Face

"Hooke's Law vs. Conservation of Energy"

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