Question

A disc rotates about a fixed axis. Its angular velocity $\omega$ varies with time according to the equation $\omega =\text{at + b}$. Initially at t = 0 its angular velocity is 1.0 rad/s and angular position is 2 rad; at the instant t= 2s angular velocity is 5.0 rad/s. Determine angular position $\theta$ and angular acceleration when t = 4s.

Answer 1

We have

$w=at+b$

at $t=Os,w=1rod/s$

So, $1=O+b$

At $t=2$ & $w=5rod/s$

so, $5=2a+1$

$\Rightarrow a=2rod/s^2$

$w=2t+1$

$\alpha =\frac{dw}{dt}=\frac{d(2t+1)}{dt}=2rad/s$

$w=\frac{d\theta }{dt}=(2t+1)dt$

$\Rightarrow d\theta =(2f+1)dt$

$\Rightarrow \int d\theta =\int (2f+1)dt$

$\Rightarrow [\theta {]}_{{\theta }_0}^{\theta }=[t^2+t{]}_0^t$

$\Rightarrow \theta -2=t^2+t$

$\Rightarrow \theta =t^2+t+2$