# Pendulums and deriving the work energy equation

Obviously deriving the work energy equation is not a demonstration (other than a demonstration of one’s memory and math ability). This demonstration was used by Ben Sill to illustrate the terms in the work energy equation that can then form the basis for the derivation.

Equipment

You will need either a string pendulum with a weight on the end or a spring with some mass on the end. I use the spring mass system that I also use for dimensional analysis.

Demonstration

Start the pendulum or spring moving with some initial deflection. Talk to the students about what they observe. Perhaps ask them to predict what will eventually happen (maybe a clicker question). Maybe get them to reflect on why the amplitude of the oscillation decreases over time. Once there is a substantial, noticeable reduction in the amplitude stop the demonstration.

Analysis

Write down a generic work energy equation

Total energy at time (2) = Total energy at time (1) + work done on the system – losses (i.e. work done by the system)

I keep the losses separate even though they are work done by the system and hence work done with a minus sign. This just keeps things a little more physically accessible as students seem to be able to think in terms of losses rather than in terms of the pendulum or spring doing work on the surroundings. The work done is the energy spent initially raising the pendulum or pulling down the spring. The total energy at time (1) is zero if you take this time to be before you touch it. The losses are due to drag on the object and internal damping. There is no need to get into details of how to model these terms, it is the bulk equation that you want to demonstrate.

From this point it is possible to derive the work energy equation with the caveat that the energy terms are energy fluxes and the work and loss terms are rates of work and loss.

An index of all the demonstrations posted on this blog can be found here. Don’t forget to follow @nbkaye on twitter for updates to this blog. If you have a demonstration that you use in class that you would like to share on this blog please email me (nbkaye@clemson.edu). I also welcome comments (through the comments section or via email) on improving the demonstrations.