Question

A wheel starting from rest is uniformly accelerated at 4 rads/s² 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

Given in the question :-

The rotation in this case is divide into three parts , uniform acceleration, uniform rotation without acceleration & at last uniform deceleration.

Time duration t = 10 sec.
angle rotation = θ
Here the rotation in the first and third part will be same.

Hence Total angle rotation,
{  $\theta +\theta' +\theta"$ }
=θ+θ'+θ      {here θ=θ''}
=2θ+θ'. ----→(a)

 
We know the equation of kinematics,
 $\theta=\omega t + \frac{1}{2} \alpha t^2$
Put all the values in this equation 
=0 x 10 +(1/2 x 4 x 10 x 10)
θ=200 radians

Now, 
$\theta' = \omega' * t$
= $( \omega+\alpha t )t$
=(0+4 x 10)10
θ'=400 radians

Hence the rotation of total angle , by equation (a)
=2θ+θ' = (2 x 200 +400)

=800 rad .


Hope it Helps :-)

Answer 2

Area under the curve will decide the total angle rotated

maximum angular velocity = 4 × 10 = 40 rad/s

Therefore, area under the curve = 1/2 × 10 × 40 + 40 × 10 + 1/2 × 40 × 10

= 800 rad

Total angle rotated = 800 rad.