Question

· [4 pts ] A fan blade is rotating, as shown. Two points, A and B, are at different distances 21 0 from the axis of the blade. Which of the two points has (a) greater centripetal acceleration, (b) greater linear speed, and (c) greater angular velocity. 3r blode is 3c cn tolot ) AN Che res, jus ru

Answer 1

Vt acent

The angular velocity is the "rate of the change of angle per unit time". for every point on the fan blade, this rate of change of Angele with respect to time is same

$\frac{\mathrm{d} \theta}{\mathrm{d} t}=\omega$ .........................................equation (1)

the linear speed any point is given by

$v=r\omega$  .........................................equation (2)

$\omega$ is the angular velocity and r is the distance of the point from the center.

The centripetal acceleration is the rate of change of the tangential velocity., and is always directed along the radius vector, towards the center of the circular motion.

$|\vec a_{ct}|= \frac{v_t^2}{r}$  

or

$|\vec a_{ct}|= \frac{(r\omega)^2}{r}=\omega^2 r$ .........................................equation (3)

where v_t is the magnitude of tangential velocity or the linear speed, and r is the distance of the point from the center.

a)

From equation (3) The centripetal acceleration is directly proportional to the distance from the center (as $\omega$ is the same for all the points). Hence the point farther away from the center experience more centripetal acceleration.

Hence the point having more centripetal acceleration is 'A'

b)

From Equation (2) that is $v=r\omega$ , The linear speed is proportional to the distance from the center (as $\omega$ is constant for all the points.). Hence the point farther from the center have more linear speed than the points nearer to the centre.

Hence the point having larger linear speed is 'A'.

c)

The change in the angle the position vector of a point(vector joining the point to the center) for a unit change in time is same for all the points, the value of the angular speed is same for all the points, irrespective of its distance from the center.

Hence both the points 'A' and 'B' will have an identical angular speed.