Question

A 70 cm diameter wheel accelerates uniformly about its center from 130 rpm to 280 rpm in 4.0 s. determine (a) its angular acceleration, and (b) the radial and tangential components of the linear acceleration of a point on the edge of the wheel 2.0 s after it has started accelerating

Answer 1

Part a)

as we know that angular speed is given as

$\omega = 2 \pi f$

here we have

$f_2 = 280 rpm = 4.67Hz$

$f_1 = 130 rpm = 2.17 Hz$

now for angular acceleration we have

$\alpha = \frac{\omega_f - \omega_i}{\Delta t}$

$\alpha = \frac{2\pi(f_2 - f_1)}{\Delta t}$

$\alpha = \frac{2\pi(4.67 - 2.17)}{4}$

$\alpha = 3.93 rad/s^2$

Part b)

angular speed of the wheel after t = 2 s

$\omega_f = \omega_i + \alpha t$

$\omega_f = 2\pi (2.17) + 3.93 (2)$

$\omega_f = 21.5 rad/s$

now we have

tangential acceleration

$a_t = r \alpha$

$a_t = 0.70(3.93) = 2.75 m/s^2$

radial acceleration will be

$a_r = \omega^2 r$

$a_r = (21.5)^2(0.70) = 323.6 m/s^2$