A Level Physics 9702: Damping and Resonance in Oscillating Systems

An engineering student sets up a demonstration rig: a metal block hangs from a vertical spring, with the block partly submerged in a cylinder of oil so that its oscillations can be damped by varying the oil's viscosity. In each case below, the block is pulled down by the same amount and released from rest.

(a) With no oil in the cylinder, the block oscillates with negligible damping. State what is meant by damping in an oscillating system, and describe how adding light damping would change the displacement-time graph of the block compared with this undamped case. [3]

(b) Oil is added so that the block returns to its equilibrium position in the shortest possible time, without overshooting and oscillating about it. State the name given to this particular case of damping, and describe how heavy damping would differ from it. [3]

(c) A small motor is later attached to the top of the spring, shaking the support up and down at an adjustable frequency. Explain what is meant by resonance, and state the condition on the driving frequency required for resonance to occur in this system. [2]

Part (a): Damping and its effect on the displacement-time graph

Damping is the process by which an oscillating system continuously loses energy, usually to the surroundings as heat, because of resistive (frictional/viscous) forces acting on it as it moves.

With no damping, the block’s displacement-time graph is a smooth, unchanging cosine curve: the same amplitude is repeated, oscillation after oscillation, forever.

Adding light damping changes this graph in one key way: the amplitude of successive oscillations gradually decreases (decays) with time, while the shape of each oscillation and its period remain almost the same as in the undamped case. So the graph still looks like a regular, repeating wave crossing the time axis at (very nearly) the same time intervals, but the height of each peak and the depth of each trough shrink steadily toward zero as time goes on, rather than staying constant.

Part (b): Critical damping versus heavy damping

Returning to equilibrium in the shortest possible time, without overshooting or oscillating about the equilibrium position is the definition of critical damping.

Heavy damping (overdamping) also brings the block back to equilibrium without any oscillation about it, the block does not cross the equilibrium position and swing back, but it does so more slowly than in the critically damped case. So on a displacement-time graph, both the critically damped and heavily damped curves fall smoothly from the initial displacement toward zero without crossing the axis, but the heavily damped curve takes noticeably longer to approach zero.

Part (c): Resonance

When the motor drives the support at a frequency far from the block-spring system’s own natural frequency of oscillation, only a small amplitude of forced oscillation results, because energy is transferred inefficiently from the driver to the block.

Resonance occurs when the amplitude of the block’s forced oscillation becomes a maximum. This happens because, at this particular driving frequency, energy is transferred from the driving motor to the oscillating block at the greatest possible rate, building up a much larger oscillation amplitude than at other driving frequencies.

The condition required for resonance is that the driving frequency must equal the natural frequency of the block-spring system (the frequency at which it would oscillate freely if simply displaced and released).

Final answers

(a) Damping = continuous loss of energy from an oscillating system to resistive forces; light damping leaves the period almost unchanged but the amplitude decays gradually over successive oscillations
(b) Critical damping; heavy damping also shows no oscillation about equilibrium, but returns to equilibrium more slowly than the critical case
(c) Resonance = maximum amplitude of forced oscillation, occurring when the driving frequency equals the natural frequency of the system