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 aboutA. 2370 $ms^{-2}$B. 5055 $ms^{-2}$C. 1000 $ms^{-2}$D. 4740 $ms^{-2}$

Hint: 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:
$a = \omega^2 r$

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). $4740 ms^{-2}$.

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.
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.