Q.A pendulum oscillates in air, so it experiences air resistance (a damping force). Consider how its total mechanical energy E changes with time. Which of the following E-versus-time curves correctly represents this variation?
(a) A curve that oscillates about the time axis with steadily decreasing amplitude (crossing zero repeatedly).
(b) A curve that starts at a high value and falls to zero at a finite time along a concave arc.
(c) A curve that starts at a high value and decreases gradually, decaying asymptotically toward zero.
(d) A sinusoidal curve of roughly constant amplitude oscillating about zero.
Imagine pushing a child on a swing. You give one big push, then step back. The swing goes high, then lower, then lower still, until eventually it stops. That is a damped oscillation in everyday life. The swing wants to keep swinging forever — that would be an ideal, undamped oscillation — but something is stealing its energy. Air resistance, friction at the pivot, even the slight bending of the ropes all act as a brake.
The key intuition: the system still oscillates, but each swing is a little smaller than the last. The amplitude does not drop suddenly; it shrinks in a smooth, predictable way — exponentially.
The Physics: Where Does the Energy Go?
In an ideal oscillator (like a mass on a spring with no friction), the total mechanical energy is constant. Kinetic energy converts to potential energy and back, forever. The equation of motion is:
mdt2d2x+kx=0
where m is mass, k is spring constant, and x is displacement.
Now add a resistive force. The simplest model is a force proportional to velocity, like air drag at low speeds or the friction in a dashpot (a piston in oil). That force is:
Fdamping=−bv=−bdtdx
where b is the damping coefficient — a positive number that measures how strong the resistive force is. The minus sign means the force always opposes the motion.
Newton's second law then becomes:
mdt2d2x+bdtdx+kx=0
That is the damped harmonic oscillator equation. It is the precise statement.
The Solution: Exponential Decay of Amplitude
The solution to this differential equation depends on how strong the damping is. For the most common case — underdamping — the system still oscillates, and the displacement is:
x(t)=A0e−2mbtcos(ω′t+ϕ)
Here is what each piece means:
A0 is the initial amplitude.
e−2mbt is the exponential decay factor. As time t increases, this factor shrinks from 1 toward 0. The quantity 2mb is often written as γ (the damping constant) or β.
cos(ω′t+ϕ) is the oscillatory part, with a new angular frequency ω′ that is slightly less than the natural frequency ω0=k/m:
ω′=ω02−(2mb)2
Important
The amplitude of a damped oscillation decays as A(t)=A0e−2mbt. The energy, which is proportional to amplitude squared, decays as E(t)=E0e−mbt.
Three Regimes of Damping
Not all damped systems oscillate. The value of b relative to the critical value bc=2km decides the behaviour:
Regime
Condition
Behaviour
Underdamped
b<2km
Oscillates with decaying amplitude
Critically damped
b=2km
Returns to equilibrium fastest, no oscillation
Overdamped
b>2km
Returns slowly, no oscillation
Note
Critical damping is the sweet spot for things like door closers and car shock absorbers — you want the system to settle to zero as quickly as possible without bouncing.
Air resistance continuously drains energy from the swinging pendulum, so its total mechanical energy only decreases with time — smoothly and gradually, never oscillating and never negative. This is the exponential-type decay of option (C).
Reasoning
The kinetic and potential energies individually oscillate as the bob swings, but their sum (total mechanical energy) is what the question asks about. A resistive (damping) force does negative work every cycle, so:
dtdE<0at all times
The energy therefore decreases monotonically, tending to zero as the oscillations die out — an exponential-type decay.