Question

Propeller blades in aeroplane are 2m long
A- When the propeller is rotating at 1800 rev/min, compute the tangential velocity of the tip of the blade.
B- What is the tangential velocity at a point on the blade midway between tip and axis?

Answer 1

- Hint: Here we will proceed by using the formula of angular velocity i.e. $\dfrac{\pi }{2}radians{\text{ per minute}}$ to find the tangential velocity of tip of the blade i.e.$V = r\omega $. Then using the calculated tangential velocity of the tip of the blade, we will find the tangential velocity at a point on the blade midway between tip and axis.

Formula used-
1 Angular velocity $ = \dfrac{\pi }{2}radians{\text{ per minute}}$
2 tangential velocity, $V = r\omega $

Complete step-by-step solution -

Here we are given that radius, r = 2m
A- In order to calculate the tangential velocity V, we have to convert given angular velocity in minutes i.e. 1800 rev/min to seconds first.
Using the formula of angular velocity i.e.$\dfrac{\pi }{2}radians{\text{ per minute}}$, we get-
$ \Rightarrow \omega = \dfrac{{1800}}{{60}}rev{\text{ per sec}}$
$ \Rightarrow \omega = 30rev{\text{ per second}}$
$ \Rightarrow \omega = 30 \times 2\pi rad{\text{ per sec}}$
$ \Rightarrow \omega = 60\pi rad{\text{ per second}}$
Now the tangential velocity of the tip of blade will be calculated with the given r = 2m and $\omega = 60\pi rad{\text{ per second}}$-
$ \Rightarrow V = r\omega $
$ \Rightarrow V = 2 \times 60\pi $
$ \Rightarrow V = 120\pi $
$ \Rightarrow V = 120 \times \dfrac{{22}}{7}$
$ \Rightarrow V = 377m{\text{ per second}}$
Therefore, the tangential velocity of the tip of blade will be$V = 377m{\text{ per second}}$.
B- Here tangential velocity at a point on blade midway between tip and axis-
Since the tangential velocity is at a mid-point on blade, then the radius will be halved-
So, we get-
$V = \dfrac{r}{2} \times \omega $
Where r = 2m and $V = 60\pi $
So we get-
$V = \dfrac{2}{2} \times 60\pi $
$ \Rightarrow V = 1 \times 60\pi $
$ \Rightarrow V = 60\pi $
$ \Rightarrow V = 188.49m{\text{ per second}}$
Therefore, the tangential velocity at a mid-point on the blade between tip and axis will be $188.49m{\text{ per second}}$.

Note- While solving this question, we must take care that we have to convert the angular velocity given in minutes to seconds. Also one should not get confused in calculating the tangential velocity at a mid-point on the blade between tip and axis, we have to take half of the radius.