Damped Harmonic Motion, Shock Absorbers and Perpetual Motion

Why the bouncing stops

Push down on the corner of a parked car and let go. It rises, maybe dips once, and settles. A playground swing left alone slows to a stop. A guitar note fades. Real oscillators lose mechanical energy, and the motion dies away. This lesson explains damped harmonic motion, how engineers use it in shock absorbers, and why the law of conservation of energy means a perpetual motion machine can never work.

Damped harmonic motion

In ideal SHM, mechanical energy stays at ½kA² forever. In real systems, non-conservative forces such as air resistance, friction and fluid drag do negative work on every cycle. This is called damping. Since E = ½kA², losing energy means the amplitude shrinks.

• Underdamped: light damping. The object oscillates with a gradually shrinking amplitude (a swing, a guitar string). With light fluid drag, the amplitude typically drops by about the same fraction each cycle, so it decays exponentially (as in the MHF4U unit Exponential and Logarithmic Functions).
• Critically damped: just enough damping that the object returns to equilibrium as quickly as possible without overshooting.
• Overdamped: heavy damping. The object creeps back to equilibrium slowly without oscillating, like a door closer that is set too stiff.

The period of a lightly damped oscillator is almost the same as without damping; it is the amplitude that changes.

Shock absorbers

A car's suspension has two parts at each wheel. The coil spring supports the car and stores energy when the wheel hits a bump. On its own, the spring would let the car bounce up and down for many cycles (try the period calculation in Simple Harmonic Motion of a Mass on a Spring). The shock absorber (damper) is a piston in a sealed cylinder of oil. When the suspension moves, oil is forced through small holes in the piston. That fluid friction does negative work and turns the oscillation's mechanical energy into thermal energy in the oil, which is then lost to the surrounding air.

Designers aim close to critical damping, usually staying slightly underdamped: firm enough to stop bouncing within about one cycle, but not so stiff that every bump is passed straight to the passengers. When shocks wear out, the car keeps bouncing after bumps, which reduces tire grip and control.

The same idea protects tall buildings. Taipei 101, a skyscraper in Taiwan, has a huge steel pendulum near its top: a tuned mass damper that swings against the building's sway and, with hydraulic dampers, turns its energy into thermal energy.

Why perpetual motion is impossible

A perpetual motion machine is a device that would run forever and keep delivering useful work with no energy input. Inventors have proposed overbalanced wheels with rolling weights, water wheels that pump their own water back up, and magnet motors for centuries. None has ever worked, and the conservation laws explain why:

1. Energy cannot be created. A machine that delivers work must take that energy from somewhere. If nothing is put in, the stored energy runs down. A machine claiming to put out more energy than it takes in breaks the law of conservation of energy.
2. Real machines have non-conservative forces. Friction, air resistance and electrical resistance turn some energy into thermal energy on every cycle. Even a machine that only kept itself moving would slowly run down, like a damped oscillator.
3. Even with zero friction, a machine could at best keep its energy. The moment you take useful work out of it, its energy decreases.

(University physics adds a second barrier from the second law of thermodynamics: you cannot turn thermal energy completely back into work. That is beyond this course.)