Question

The angular frequency of a fan increases from $30rpm$ to $60rpm$ in $\pi s.$ A dust particle is present at a distance of $20cm$ from the axis of rotation. The tangential acceleration of the particle is?
(a) $0.8\text{ m}{{\text{s}}^{-2}}$
(b) $0.34\text{ m}{{\text{s}}^{-2}}$
(c) $0.2\text{ m}{{\text{s}}^{-2}}$
(d) $1.2\text{ m}{{\text{s}}^{-2}}$

Answer 1

Hint: Angular frequency $w$ is a scalar measure of rotation rate. It refers to the angular displacement per unit time or rate of change of the phase of a sinusoidal waveform.
Acceleration is a rate of change of velocity.

Formula used:
Angular speed is given by $w=\dfrac{2\pi b,}{60{}^\circ }$
${{w}_{i}}=\dfrac{2\pi {{t}_{1}}}{60{}^\circ }\text{(rad/sec)}$
Angular acceleration:-
$\alpha =\dfrac{wf-wi}{t}$
Complete solution:
Given:-
N(speed)$-\left( 30\text{ to 60} \right)rpm.$
$w=20cm.$
$\therefore $ angular speed $w=\dfrac{2\pi {{b}_{1}}}{60}\text{rad/sec}$
Angular acceleration
$\alpha =\dfrac{{{w}_{f}}-{{w}_{i}}}{t}\text{rad/se}{{\text{c}}^{2}}$
${{w}_{i}}=2\pi \dfrac{30}{60}=2\pi \text{rad/sec}$
${{w}_{f}}=2\pi \dfrac{60}{60}=2\pi \text{rad/sec}$
$\alpha =\dfrac{2\pi -\pi }{314}$
$\alpha =1\text{rad/se}{{\text{c}}^{2}}$

Now the conclusion option (c) is correct.

Additional information:
All the equations Newtonian mechanisms are based on Newton’s Law of Motion. All equations are a derivative and all theories are governed by them.

Note:
The value of tangential acceleration of a particle is given by tangential acceleration formula by calculating angular acceleration.