Question

A wheel is subjected to uniform angular acceleration about its axis. Initially its angular velocity is zero. In the first two seconds it rotates through angle ${\theta _1}$ . In the next two seconds it rotates through angle ${\theta _2}$ . What is the ratio ${\theta _2}$ / ${\theta _1}$ ?

Answer 1

As an object goes in a circular motion, its velocity is referred to as angular velocity. The angular acceleration is the rate at which the angular velocity changes and is normally denoted by $\alpha$ and represented in radians per second per second.

Complete step by step answer:
The angular acceleration is represented by $\alpha$and its initial angular velocity is given by ${\omega _0}$. The second equation of angular motion which is $\theta = {\omega _0}t + \dfrac{1}{2}\alpha {t^2}$ can be used to solve this question.Since, the initial angular velocity is zero which means that ${\omega _0} = 0$ and the wheel rotates through angle ${\theta _1}$ in the first $2$ second. So,
$\Rightarrow {\theta _1} = (0 \times 2) + \dfrac{1}{2}\alpha {(2)^2}$
$\Rightarrow {\theta _1} = 2\alpha$ $- \left( 1 \right)$

Let the angular momentum of the wheel be ${\omega ^1}$ before the wheel rotates through angle ${\theta _2}$. Here, using the first equation of angular motion, ${\omega ^1}$ can be given by the equation:
${\omega ^1} = {\omega _0} + \alpha t$
As discussed above, ${\omega _0} = 0$ and $t$ here is equal to two. So, the value of ${\omega ^1}$ can be given by the equation:
$\Rightarrow {\omega ^1} = 0 + \alpha \times 2$
$\Rightarrow {\omega ^1} = 2a$

So, as mentioned in the equation the wheel rotates through angle ${\theta _2}$ in the next two seconds. The angle ${\theta _2}$ can be given by the equation:
${\theta _2} = {\omega ^1}t + \dfrac{1}{2}\alpha {t^2}$ ,
(Here $\omega = {\omega ^1}$ because the initial angular velocity after the first two seconds is ${\omega ^1}$)
Since ${\omega ^1} = 2a$ and $t = 2$, we get:
$\Rightarrow {\theta _2} = 2 \times 2\alpha + \dfrac{1}{2} \times \alpha \times{2^2}$ $= 6\alpha - (2)$
So, using $(1)$ and $(2)$ ,
$\dfrac{{{\theta _2}}}{{{\theta _1}}} = \dfrac{{6\alpha }}{{2\alpha }}$ $= 3$

Thus, the ratio is $\dfrac{{{\theta _2}}}{{{\theta _1}}} = 3$.

Note: An object experiences acceleration if its speed changes. The rate at which the angular velocity changes is known as angular acceleration. If a wheel accelerates at a steady rate, the angular acceleration is constant.