Question

1. A technician carried out a test on a piece of rotating equipment and established that it turned through 550 rotations per minute. What is the angular velocity of the rotating part in radians per second? 

 

 

      b) Calculate the area and arc length of the major sector of the circle shown below 

Answer 1

1.

$$\nu=550\ min^{-1}=\dfrac{550}{60}\ s^{-1}$$

The angular velocity of the rotating part $w$

$$w=2\pi\nu=2\pi(\dfrac{550}{60}) \ rad/s=\dfrac{55\pi}{3} rad/s$$

$$w=\dfrac{55\pi}{3} rad/s\approx57.6\ rad/s$$

b)

The formula for the length of an arc is

$$L_1=r\theta$$

where $L_1$ represents the arc length, $r$ represents the radius of the circle and $\theta$ represents the angle in radians made by the arc at the centre of the circle.

Then the arc length of the major sector of the circle is

$$L_2=2\pi r-L_1=2\pi r-r\theta=r(2\pi-\theta)$$

If the value of angle is given in degrees

$$L_2=2\pi(1-\dfrac{\theta\degree}{360\degree})$$

The total area of a circle is $A=\pi r^2.$

The area of the minor sector is $A_1=\pi r^2(\dfrac{\theta}{2\pi})$

The area of the major sector is

$$A_2=A-A_1$$

$$=\pi r^2-\pi r^2(\dfrac{\theta}{2\pi})=r^2(\pi-\dfrac{\theta}{2})$$

If the value of angle is given in degrees

$$A_2=\pi r^2(1-\dfrac{\theta\degree}{360\degree})$$