10.3

Simple Versus Damped Harmonic Motion

Student Notes

You can see a block resting on a frictionless horizontal surface. When damping is off, the spring force is the only horizontally directed force on the block. Initially the spring is stretched to a length longer than its relaxed length, so that the spring force is initially to the left. How will the block move? Make a prediction of the shape of the position vs. time graph, and then check your prediction using the simulation.
The spring applies a linear restoring force. In the equilibrium position the spring is relaxed.
The motion you observed is called simple harmonic motion. The time that it takes the block to complete one cycle of back-and-forth motion is called the period.
How will the acceleration be changed if the mass of the block is doubled? Would the period be increased or decreased? Set the mass to twice its current value, and use the simulation to check your answer.
How will the acceleration be changed if the spring constant is doubled? Would the period be increased or decreased? Remember that the spring constant is a measure of the spring's stiffness. For two springs stretched the same amount, the one with the larger spring constant would apply a larger force. Set the spring constant to twice its current value, and use the simulation to check your answer.
The simulation allows you to turn on damping. Damping is a resistive force that gradually removes energy from the spring-mass system. Turn on damping now. How would the position vs. time graph change if the total energy of the system decreases in time? Check your prediction by running the simulation.
The damping rate can be increased with the slider. How would increasing the damping affect the shape of the position vs. time graph?
For further exploration of this simulation, look up the relations for the dependence of the period on the mass and spring constant. Is the simulation consistent with those expressions? Try several combinations of values. Turn damping off for these investigations.

Self-test

How can you obtain the shortest possible period given the values of m and k available in the simulation?

Copyright © 2006 John Wiley & Sons, Inc. All rights reserved.