Question

A 350 mm diameter grinding wheel must be driven at a grinding speed (speed at circumference) if 1200 meters per minute. It takes 5 seconds to reach this speed from rest, runs at this speed for the next two minutes and then starts to decelerate uniformly, finally coming to rest after a further 30 seconds. Draw a neat angular velocity-time graph and determine how many revolutions the wheel made in total.

Also clearly show what the following values are:

ω1

θ1

θ2

θ3

θtotal

Answer 1

$\text{diameter} =350mm\\ \text{radius} = 175mm \\ v_1 = 1200m/min = 20m/s\\ \text{but, } v = \omega r\\ 20m/s =\omega_1 × 175 × 10^{-3}m\\ \omega_1 = 114.3\ rad/s$

$\omega_0 = 0\ rad/s \\ \omega_2 = 0\ rad/s$

For first part of the motion

$\theta_1 = (\dfrac{\omega_1+\omega_0}2 )t$

$\theta_1= \dfrac{114.3+0}2×5= 285.75\text{ rad}$

for second part of the motion

$\theta_2= \omega_1 ×t = 114.3 ×2(60)= 13716\text{ rad}$

for third part of the motion

$\theta_3 =(\dfrac{\omega_2+\omega_1}2 )t$

$\theta_3 = \dfrac{0+114.3}{2} × 30=1714.5 \text{ rad}$

$\text{Total angular revolutions} = 285.75+ 13716+1714.5= 15716.25$

$\text{Number of revolutions } = \dfrac{\theta}{2π}=\dfrac{15716.25}{2π} = 2501.32 \text{ rev}$