Name: Experiment: First Law of Simple Pendulum
Uploaded: 2019-07-19
Duration: 5 min 16 s
Description: Do you know how to measure the time period using a simple pendulum? The time period of a pendulum is independent of its amplitude when the amplitude is small...

Question

Obtain formulae for the displacement velocity and acceleration and time period executing simple harmonic motion. When is the velocity of the particle is maximum and zero acceleration.

Accepted Answer

Displacement

Displacement is the shortest distance of the particle from its mean position

Then, in triangle $O P N$ ,

$ON = OP \sin θ$ or $y = a \sin (ωt)$ where $θ = ωt$

Since, $OP = a$ , the radius of the circle, the displacement of the vibrating particle is The amplitude of the vibrating particle is defined as its maximum displacement from the mean position.

Velocity

The rate of change of displacement is the velocity of the vibrating particle.

As the displacement is given by

$y = a \sin ωt$

Differentiating equation with respect to time $(t)$

$v = \frac{dy}{dt} (a \sin (ωt)) = a ω \cos (ωt)$

Acceleration

Acceleration is the rate of change of velocity i.e $A = \frac{d^{2} y}{{dt}^{2}} = \frac{dv}{dt} = \frac{d}{dt} (a ω \cos (ωt)) = - a ω^{2} \sin (ωt) = - ω^{2} y$ Since, $y = a \sin (ωt)$

For maximum velocity as velocity is given by : $v = a ω \cos (ωt)$

Here, the velocity is maximum when $\cos (ωt)$ is maximum that is $\cos (ωt) = 1$ or $v = a ω$ when $y = 0$ i.e at the mean position.

At the extreme position when $y = a v = 0$ that is minimum.

Acceleration will me maximum at extreme positions when $y = a$ then $A = - ω^{2} a$

and acceleration is minimum at mean position $y = 0$ then $A = 0$ .

Obtain formulae for the displacement velocity and acceleration and time period executing simple harmonic motion. When is the velocity of the particle is maximum and zero acceleration.

Important Questions on Simple Harmonic Motion

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