Question

An electric fan has blades of length 30cm as measured from the axis of rotation. If the fan is rotating at 1200rpm, the acceleration of a point on the tip of the blade is about
A. 2370 $ms^{-2}$
B. 5055 $ms^{-2}$
C. 1000 $ms^{-2}$
D. 4740 $ms^{-2}$

Answer 1

rpm is read as rotations per minute. This is a unit of frequency and the unit of angular frequency is rad/s. The angular frequency and acceleration of a rotating body are related. As the distance from the axis varies, velocity (or acceleration) varies, but angular frequency remains constant.

Formula used:
Acceleration of a rotating body has a magnitude given by:
$a = \omega^2 r$

Complete step by step answer:
We are given distance from the axis, i.e., r= 30 cm or 0.3m.
Also the fan is making a rotation with frequency 1200 rpm (or rotations per minute):
$\nu = 1200$ rpm.
In terms of S.I. units:
$\nu= \dfrac{1200}{60} =20$ rps or rotations per second.
The angular frequency (in radians) is the total angle that the fan covers in a second. So, when rotating at a rate of 20 rotations in a second (cover $2\pi$ in one rotation):
$\omega = 2 \pi \times \nu$
We get
$\omega = 2 \pi 20$ radians per second.

The formula for acceleration is just:
$a = \omega^2 r$
So, keeping the values in it, we get:
$a = (40 \pi)^2 \times 0.3 ms^{-2}$
Therefore, we get:
$a = 4741.2 ms^{-2}$
after keeping the value of pi to be 22/7.
This is pretty close to option D.

Therefore, the correct answer is option (D). $4740 ms^{-2}$.

Additional Information:
The formula for velocity is $\omega r$ for a rotating body. The closer the body is to the axis, more will be the velocity. The formula for force is $mv^2/r$, equating this with ma, we get the magnitude of the acceleration as $v^2/r$. Thus, even if one doesn't remember the formula, this trick can be used to get the acceleration.

Note:
One might know the formula for acceleration right but here, one might substitute 1200 rpm in place of $\omega$ directly, without much bothering about the rpm part. Therefore, one must remember the unit of $\omega$ at such time. Rpm is clearly not radians per second but it is rotations per second.