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

Draw graphs of displacement, velocity and acceleration against phase angle, for a particle performing linear S.H.M. from the positive extreme position. Deduce your conclusions from the graph.

Accepted Answer

Consider a particle performing linear S.H.M with amplitude A and period $T = \frac{2 π}{ω}$ , starting from the positive extreme position, where $ω$ is the angular frequency. Its displacement from the mean position $(x)$ , velocity $(y)$ and acceleration $(a)$ at any instant $(t)$ are:

$x = A \cos ω t = A \cos (\frac{2 π}{T} t) (∵ ω = \frac{2 π}{T}) v = - ω A \sin ω t = - ω A \sin (\frac{2 π}{T} t) a = - ω^{2} A \cos ω t = - ω^{2} A \cos (\frac{2 π}{T} t)$

Using these expressions, the values of $x, y$ and $a$ at the end of every quarter of a period,starting from $t = 0$ , are tabulated below.

$t$ $0$ $\frac{T}{4}$ $\frac{T}{2}$ $\frac{3 T}{4}$ $T$

$ω t$ $0$ $\frac{π}{2}$ $π$ $\frac{3 π}{2}$ $2 π$

$x$ $A$ $0$ $- A$ $0$ $A$

$v$ $0$ $- ω A$ $0$ $ω A$ $0$

$a$ $- ω^{2} A$ $0$ $ω^{2} A$ $0$ $- ω^{2} A$

Using these values, we can plot graphs showing the variation of displacement, velocity and acceleration with time.

Conclusions from graphs:

Displacement, velocity and acceleration of S.H.M. are periodic functions of time.

Displacement time curve and acceleration time curves are cosine curves and velocity time curve is a sine curve.

There is phase difference of $\frac{π}{2}$ radian between displacement and velocity.

There is phase difference of $\frac{π}{2}$ radian between velocity and acceleration.

There is phase difference of $π$ radian between displacement and acceleration.

Shapes of all the curves get repeated after $2 π$ radian or after a time $T$ .

Draw graphs of displacement, velocity and acceleration against phase angle, for a particle performing linear S.H.M. from the positive extreme position. Deduce your conclusions from the graph.

Important Questions on Oscillations

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