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# Exploration 16.6: Damped and Forced Motion

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A mass can be driven by an external force in addition to an internal restoring force and friction. Restart. Specifically, F_{net} = F_{restore} + F_{friction} + F_{driving}, where the default values are

F_{restore} = -2*y, F_{friction} = -0.2*vy, and F_{driving}= sin(t).

You can change these default values as you see fit*. Remember to use the proper syntax such as -10+0.5*t, -10+0.5*t*t, and -10+0.5*t^2. Revisit Exploration 1.3 to refresh your memory.*

- Find the mass. Hint: consider a linear restoring force.
- Change the restoring force to -y-0.1*y*y. Is the motion periodic? Is it harmonic? What about -y-2.0?
- Design your own force that produces periodic, but not necessarily harmonic, motion.
- Drive the mass at resonance and explain the behavior of the position graph. How does the behavior change with and without friction?

Drive the system (use a linear restoring force of -1*y and initially no friction) with a function that switches a constant force on and off. This can be achieved with the step function: step(sin(t/4)). The step function is zero if the argument is negative and one if the argument is positive. The given function, step(sin(t/4)), will therefore produce a square wave with amplitude of one and an angular frequency of one quarter. Note that the total force you should use is -1*y+step(sin(t/4)). Start the mass in its original position; do not drag it.

- Draw a graph of the force vs. time superimposed on the position vs. time graph.
- Why does the system oscillate, stop, and oscillate again?
- Does this behavior occur at any other frequencies? For example, notice that the function step(sin(t/4.5)) produces qualitatively different behavior. Why is this?

Note that the mass is not allowed to oscillate past about 22 m.

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