Question

Propeller blades in an aeroplane are $2m$ long. When a propeller is rotating at $1800rev/\min $, compute the tangential velocity of the tip of the blade.

Answer 1

Hint: This problem can be solved by using the direct formula for the tangential velocity of a body in uniform circular motion in terms of the angular frequency and the radius of the circular path. We are given the angular frequency and the radius of the circle (the length of the propeller).

Formula used:
$v=r\omega $

Complete step-by-step answer:
For a body in circular motion, the tangential velocity can be written in terms of the angular frequency of motion and the radius of the circular motion.
The tangential velocity $v$ of a body in uniform circular motion on a circular path of radius $r$ and with angular frequency $\omega $ is given by
$v=r\omega $ --(1)
Now, let us analyze the calculation.
The tip of the propeller can be considered to be a particle in uniform circular motion.
The radius of the circular path in this case will be nothing but the length of the propeller. Therefore, the radius of the path undertaken by the tip is $r=2m$
The angular frequency of the motion is $1800rev/\min =\dfrac{1800\times 2\pi }{60}rad/s=60\pi rad/\sec $
$\left( \because 1rev/\min =\dfrac{2\pi }{60}rad/\sec \right)$
Let the tangential velocity of the tip of the propeller be $v$.
Therefore, using (1), we get
$v=r\omega $
$\therefore v=2\times 60\pi =377m/s$
Therefore, we have got the tangential velocity of the tip of the propeller as $377m/s$.

Note: This problem could also have been solved by converting the given angular frequency to the total distance covered in a minute by using the formula for the circumference of a circle in terms of the radius. That would have given the speed of the tip of the propeller. However, that is a lengthy process which we avoided and employed the approach of using the angular velocity which was readily available to us from the question.