Energy analysis of a damped mechanical oscillator. Attach an aluminum rod to a dynamics cart, and set up a magnet above the rod to induce eddy currents inside the rod, thus exerting a magnetic drag force on the rod which is anti-proportional to the cart’s velocity (i.e. a viscous magnetic drag force).
The demo visually demonstrates that the amplitude of the cart’s oscillations decay exponentially.
It is also possible to collect data to highlight this exponential decay graphically. By plotting the energy dissipated by the magnet over time (i.e. the force from the magnet (measured directly) times the velocity of the cart (calculated by taking the integral of acceleration, which is measured directly by an accelerometer on the cart, over time)), we see that the energies dissipated over two consecutive half-periods of the cart’s motion are always proportional by the same constant factor, which is the definition of exponential decay. Since the total energy of the cart-spring system is proportional to the square of the amplitude of the cart’s oscillations (E = (1/2)*k*A^2) it follows that amplitude also decreases exponentially (at a rate half as fast as energy does).
Location: Aluminum bar E4-4, Dynamics Track: Wall A, Dynamics Cart B4-3, Force Sensors L5-3, Magnet E4-4, Springs in the Dynamics Track spring set on F4-3, WDSS (accelerometer) L5-3