0.20 Phy1180: energy -- potential energy  (Page 6/7)

Potential energy versus stretch or compression

There is an equation for springs that relates the amount of elastic potential energy to the amount of stretch (or compression) and the spring constant. The equation is

PEs = 0.5 * k * x^2

where

• PEs represents the elastic potential energy stored in the spring
• k is the spring constant for the material from which the spring is made
• x is the amount of compression or stretch relative to the equilibrium state.

Summary

Gravitational potential energy can be stored in an object by moving it further away from the center of the earth. As a practical matter, this usuallymeans moving it to a higher position relative to the ground, the floor, or a tabletop.

Elastic potential energy can be stored in an object by deforming it within its elastic limit. Usually, but not always, this involves stretching orcompressing the object, but it could also mean twisting it or deforming it in some other way. If the deformation doesn't exceed the elastic limit of theobject, it will return to its original shape when the load is removed.

Some materials, such as the spring material in a fisherman's scale, can sustain considerable deformation before reaching the elastic limit. Othermaterials, such as toasted bread, not only reach their elastic limit, but also reach their structural limit and break at the slightest amount of deformation.

Other materials, such as fresh bread and modeling clay have a low elastic limit but a relatively high structural limit. In other words, they doesn'treturn to their original shape when deformed, but they also don't easily break when deformed.

Sample Calculations

The rock and the tree

You pick a flat rock with a 10kg mass off the ground and balance it on a tree limb 2 meters above the ground.

What was the gravitational potential energy of the rock before your arrival and what change in gravitational potential energy did you impart to the rock byyour actions?

Answer:

Assuming that the rock was flat and the ground was flat and there was essentially no chance the the rock could roll downhill, the gravitationalpotential energy of the rock before you picked it up was zero.

Using the Google calculator to do the arithmetic and handle the units, after you balanced the rock on the tree limb, the potential energy was:

(10 kg) * (9.8 (m / (s^2))) * (2 m) = 196 joules

Therefore, the change in gravitational potential energy was 196 joules.

A crate and a ramp

You push a 100 kg mass up a 5 meter-long ramp and onto a platform at the upper end of the ramp. The ramp makes an angle of 36.9 degrees relative to theground on which it is setting.

What change in gravitational potential energy did you impart to the crate?

Answer:

First compute the height of the platform at the upper end of the ramp.

height = 5 * sin(36.9 degrees) = 3m

Now compute the change in gravitational potential energy.

(100 kg) * (9.8 (m / (s^2))) * (3 m) = 2940 joules

All that matters insofar as the change in gravitational potential energy is concerned is that the crate was lifted 3 meters higher than its initial position.How that lift was accomplished doesn't matter. It could be accomplished with a ramp, a pulley, or with brute strength and the change in gravitational potential energy would be the same.