Question

A wheel starting from rest is uniformly acceleration at $4rad/s^2$ for 10 seconds. It is allowed to rotate uniformly for the next 10 seconds and is finally brought to rest in the next 10 seconds. Find the total angle rotated by the wheel.

Answer 1

Initial angular velocity, ${\omega }_o=0$

Displacement in first 10 sec

${\theta }_1={\omega }_0{t}_1+\frac{1}{2}{\alpha }_1{t}_1^2=0+\frac{1}{2}\times 4\times {10}^2=200rad$

Angular velocity after 10 sec

${\omega }_1={\omega }_o+{\alpha }_1{t}_2$

${\omega }_1=0+4\times 10=40rad{s}^{-1}$

Displacement in next 10 sec

${\theta }_2={\omega }_{1}t=40\times 10=400rad$

Deceleration in later 10 sec

${\omega }_2={\omega }_1+{\alpha }_2{t}_3$

${\alpha }_2=\frac{{\omega }_2-{\omega }_1}{t_3}=\frac{0-40}{10}=-4rad{s}^{-1}$

Displacement in later 10 sec

${\theta }_2={\omega }_1{t}_3+\frac{1}{2}{\alpha }_2{t}_3$

${\theta }_2=40\times 10+\frac{1}{2}\times (-4)\times 10$

${\theta }_3=200rad$

Total angular displacement, ${\theta }_{total}={\theta }_1+{\theta }_2+{\theta }_3=200+400+200=800rad$

Hence, total angular displacement is $800rad$