Forced Simple Harmonic Motion

When we displace a pendulum from its equilibrium position, it oscillates to and fro about its mean position. Eventually, its motion dies out due to the opposing forces in the medium. But can we force the pendulum to oscillate continuously? Yes. This type of motion is known as forced simple harmonic motion. Let’s find out what forced simple harmonic motion is.

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Definition of Forced Simple Harmonic Motion

When we displace a system, say a simple pendulum, from its equilibrium position and then release it, it oscillates with a natural frequency ω and these oscillations are free oscillations. But all free oscillations eventually die out due to the everpresent damping forces in the surrounding.

However, an external agency can maintain these oscillations. These oscillations are known as forced or driven oscillations.  The motion that the system performs under this external agency is known as Forced Simple Harmonic Motion. The external force is itself periodic with a frequency ωd which is known as the drive frequency.

A very important point to note is that the system oscillates with the driven frequency and not its natural frequency in Forced Simple Harmonic Motion. If it oscillates with its natural frequency, the motion will die out. A good example of forced oscillations is when a child uses his feet to move the swing or when someone else pushes the swing to maintain the oscillations.

Browse more Topics under Oscillations

Download the Cheat Sheet of Oscillations below

Expression of Forced Simple Harmonic Motion

Consider an external force F(t) of amplitude F0 that varies periodically with time. This force is applied to a damped oscillator. Therefore, we can represent it as,

F(t) = Fcos ωdt                               (I)

Thus, at this time, the forces acting on the oscillator are its restoring force, the external force and a time-dependent driving force. Therefore,

ma(t) = -kx(t) – bυ(t) + Fcos ωdt                         (II)

We know that acceleration = d2x/dt2. Substituting this value of acceleration in equation II , we get,

m(d2x/dt2) + b(dx/dt) + kx = Fcos ωdt                          (III)

Equation III is the equation of an oscillator of mass m on which a periodic force of frequency ωis applied. Obviously, the oscillator first oscillates with its natural frequency. When we apply the external periodic force, the oscillations with natural frequency die out and the body then oscillates with the driven frequency. Therefore, its displacement after the natural oscillations die out is given by:

x(t) = Acos(ω+ ø )                                    (IV)

where t is the time from the moment we apply external periodic force.


The phenomenon of increase in amplitude when the driving force is close to the natural frequency of the oscillator is known as resonance. To understand the phenomenon of resonance, let us consider two pendulums of nearly equal (but not equal) lengths (therefore, different amplitudes) suspended from the same rigid support.

When we swing the first pendulum which is greater in length, it oscillates with its natural frequency. The energy of this pendulum transfers through the rigid support to the second pendulum which is slightly smaller in length. Therefore, the second pendulum starts oscillating with its natural frequency first.

At one point, the frequency with the second pendulum vibrates becomes nearly equal to the first one. Therefore, the second pendulum now starts with the frequency of the first one, which is the driven frequency. When this happens, the amplitude of the oscillations is maximum. Thus, resonance takes place.

simple harmonic motion

Here’s a Solved Question for You:

Q: Resonance takes place only when the natural frequency of the oscillator is _________ the driven frequency of the external periodic force.

a) Less than    b) More than    c) Equal to    d) None of the above

Solution: c) When the natural frequency is equal to the driven frequency, the amplitude of the oscillations is the maximum. Therefore, the phenomenon takes place.

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