Name: An Equation for Velocity and Acceleration in SHM
Uploaded: 2019-07-19
Duration: 14 min 10 s
Description: Velocity and acceleration in SHM are the foundations of many concepts in physics. Watch this video to find equations for the velocity and acceleration of a p...

Question

Present graphically the variations of displacement, velocity and acceleration with respect to time for a particle in S.H.M.

Accepted Answer

The equation of particle undergoing SHM can be given as $x = A \cos (ω t + ϕ)$ , where $A$ is the amplitude, $ω$ is the angular frequency and $ϕ$ is the phase constant.

Let us consider $ϕ = 0$ ,

Thus, The equation of motion of SHM is:
$x (t) = A \cos ω t$ .......(i),
Differentiate the above equation with respect to time, to calculate the velocity, we get:
$v (t) = - ω A \sin ω t$ .......(ii)
Differentiate the equation(ii) with respect to time, to calculate the acceleration, we get:
$a (t) = - ω^{2} A \cos ω t$ ..........(iii)

The plots of displacement vs time, velocity vs time and acceleration vs time are shown in figure. All quantities vary sinusoidally with time.

Since the range of sine function is: $- 1 ⩽ \sin θ ⩽ 1$
Thus, displacement: x(t) varies between: $- A ⩽ x (t) ⩽ A$
Velocity: v(t) varies between: $- ω A ⩽ v (t) ⩽ ω A$
Acceleration: a(t) varies between: $- ω^{2} A ⩽ a (t) ⩽ ω^{2} A$

With respect to displacement plot, velocity plot has a phase difference of $π / 2$ and acceleration plot has a phase difference of $π$ .

Present graphically the variations of displacement, velocity and acceleration with respect to time for a particle in S.H.M.

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