Question

The shaft of an electric motor starts from rest and on the application of a torque, it gains an angular acceleration given by $\alpha =3t-t^2$ during the first 2 seconds after it starts after which $a=0$. The angular velocity after $6sec$ will be

• A. $\frac{4}{30}rad/sec$
• B. $\frac{10}{3}rad/sec$
• C. $\frac{3}{10}rad/sec$
• D. $\frac{30}{4}rad/sec$

Answer 1

The correct option is B $\frac{10}{3}rad/sec$

Step 1:Find the angular velocity of the shaft.
Formula Used: $\alpha =\frac{d\omega }{dt}$
Given: $\alpha =3t-t^2$
$\frac{d\omega }{dt}=3t-t^2$
$d\omega =(3t-t^2)dt$
$\omega =\frac{3t^2}{2}-\frac{t^3}{3}+c$
at $t=0,\omega =0\therefore c=0$,
$\omega =\frac{3t^2}{2}-\frac{t^5}{3}$

Step 2:Find the angular velocity of the shaft at $t=2sec$.
Angular velocity at
$t=2sec$,
$(\omega {)}_{t=2sec}=\frac{3}{2}(4)-\frac{8}{3}=\frac{10}{3}rad/sec$
Since there is no angular acceleration after $2sec$.
The angular velocity after $6sec$ remains the same.

Final Answer: (a)