Consider a spring hanging at rest from a rod.
If left to itself, it has some rest length.
We could define a coordinate system based
on vertical distance **y** above this rest length.

Suppose that we attach a block of mass **m**
to the spring, and then we push this block
up a distance **y**,
compressing the spring.

- What is the spring potential energy now?
- What is the gravitational potential energy of the block now?
- Write down the total potential energy of the system.

The total potential energy of the system is an expression with two terms --- right?

Now, let us allow the block of mass **m**
to pull the spring down until it reaches
an equilibrium point.
At this point,
the end of the spring is a distance **L** below
the rest length.

- Solve for the magnitude of the equlibrium distance
**L**in terms of**m**,**g**and the spring constant**k**.

Now, suppose that we instead decide to measure
distances from this equilibrium position.
(It will turn out to make our calculations simpler if we do this.)
We'll use the coordinate **z** to denote
"distance above the equilibrium position."

- Write an expression for
**y**in terms of**z**.

Now, we lift the block up from its equilibrium
position by a distance **z**.
What is its total potential energy now?

- Use the expression for total potential energy you wrote in step 3,
but convert from using the variable
**y**to the variable**z**, using the expression you derived above.

This expression should have one term that changes with the position **z**,
plus one term that is a constant.

Copyright © Michael Richmond. This work is licensed under a Creative Commons License.