Name: Simple Harmonic Motion: Hooke's Law
Uploaded: 2019-07-19
Duration: 17 min 48 s
Description: View this video to learn simple harmonic motion and its associated laws using the oscillation of a spring. Who doesn't love a good oscillation now and again?

Question

Obtain the expression for the period of a simple pendulum performing S.H.M.

Accepted Answer

Simple harmonic motion of an object is defined as a motion where the restoring force is directly proportional to the displacement of the object from its mean position.

Time period of oscillation of an object executing S.H.M. is defined as the total time taken by the object to complete one oscillation.

The displacement of an object executing S.H.M. at any time $t$ is given by

$x = A \sin (ω t + ϕ) . . . . . . . . . . . . . . . . . . . . (1)$

After a time $t = t + \frac{2 π}{ω}$ , the displacement of the object is given by

$\begin{array}{rcl} x' & = & A \sin [ω (t + \frac{2 π}{ω}) + ϕ] \\ = & A \sin (ω t + ϕ) . . . . . . . . . . . . . . . . . . . . . . (2) \end{array}$

Equations (1) and (2) indicate that after a time of $\frac{2 π}{ω}$ , the object will be in the same initial position if its motion. Also, we know that

$ω^{2} = \frac{k}{m} . . . . . . . . . . . . . . . . . . . . . . . . . . . . (3)$

where, $k, m$ represent the force constant and the mass of the object.

Hence, the time period $(T)$ is given by

$\begin{array}{rcl} T & = & \frac{2 π}{ω} \\ = & \frac{2 π}{\sqrt{\frac{k}{m}}} \\ T & = & 2 π \sqrt{\frac{m}{k}} \end{array}$

Obtain the expression for the period of a simple pendulum performing S.H.M.

Important Questions on Oscillations

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