Q6PE - College Physics (Urone)

Question

A helicopter blade spins at exactly $\mathbf{100}$ revolutions per minute. Its tip is $\mathbf{5}\mathbf{.}\mathbf{00}\mathbf{m}$ from the centre of rotation. 

(a) Calculate the average speed of the blade tip in the helicopter’s frame of reference. 

(b) What is its average velocity over one revolution?

Step-by-Step Solution

Answer

1. The average speed is $\mathbf{52}\mathbf{.}\mathbf{38}\mathbf{ }\mathbf{m}\mathbf{/}\mathbf{s}$.
2. The average velocity is zero.

Step 1: Given Data

• The number of revolutions in one minute = 100.
• Distance of the tip from the center of rotation = 5.00 m.

Step 2: Average speed and average velocity

Average speed is a scalar quantity.  It is the ratio of the total distance covered to that of the total time taken to cover that distance. Hence the distance covered by the blade will be the circumference of the circle.

a)    The distance can be calculated as:

         $\begin{array}{l}\mathrm{D}=2\mathrm{\pi R}\\ \mathrm{D}=2\times \mathrm{\pi }\times 5\\ \mathrm{D}=31.42\text{\hspace{0.17em}m}\end{array}$

Hence the total distance traveled is 31.4 meters in one rotation. 

Such 100 revolutions are made by the blade.

 

So, the total distance can be calculated as:

$\begin{array}{l}=31.42\times 100\\ =3142\text{\hspace{0.17em}m}\end{array}$

$\text{Average\hspace{0.17em}Speed}=\frac{\text{Total\hspace{0.17em}Distance}}{\text{Time}}$

Substituting values in the above equation, we get,

$\begin{array}{c}\text{Average\hspace{0.17em}speed}=\frac{3142}{60}\\ =52.38\text{\hspace{0.17em}m/s}\end{array}$

Hence the average speed of the helicopter blade is $\mathbf{52}\mathbf{.}\mathbf{38}\mathbf{m}\mathbf{/}\mathbf{s}$ .

 

Average velocity is the ratio of the displacement of the body to that of time when the displacement is the shortest distance from the initial and final position.

b.

Here in this, the body is in a circular motion. So the initial and final points will be the same. So the displacement of the body will be zero.

$\begin{array}{c}\text{Average\hspace{0.17em}veloctiy}=\frac{\text{Total\hspace{0.17em}displacement}}{\text{Time}}\\ =\frac{0}{60}\\ =0\text{\hspace{0.17em}m/s}\end{array}$

The average velocity is zero.