Question

A wheel is subjected to uniform angular acceleration about its axis. Initially its angular velocity is zero. In the first two second it rotate through an angle \(\theta_{1}\), in the next \(2 \mathrm{sec}\). it rotates through an angle \(\theta_{2}\), find the ratio \(\left(\theta_{2} / \theta_{1}\right)=\) \(\\{\mathrm{A}\\} 1\) \(\\{B\\} 2\) \(\\{\mathrm{C}\\} 3\) \(\\{\mathrm{D}\\} 4\)

Short answer

The ratio \(\left(\theta_{2} / \theta_{1}\right) = 3\), which corresponds to option C.

Step-by-step solution

Find the angle θ₁

To find the angle \(\theta_1\) rotated within the first 2 seconds, we use the equation of motion mentioned above. Since the initial angular velocity \(\omega_0\) is zero, we can simplify the equation to: $$\theta_1 = \frac{1}{2} \alpha t^2$$ Where \(\alpha\) is the angular acceleration, and \(t = 2\,\text{sec}\).

Find the angle θ₂

To find the angle \(\theta_2\) rotated within the next 2 seconds, first calculate the angular velocity \(\omega\) at the end of the first 2 seconds using the formula: $$\omega = \omega_0 + \alpha t$$ Since \(\omega_0 = 0\), the formula becomes \(\omega = \alpha t\). Now, we have to calculate the angle rotated in the next 2 seconds with the given angular velocity \(\omega\) and angular acceleration \(\alpha\). Again, we apply the equation of motion mentioned above: $$\theta_2 = \omega t + \frac{1}{2}\alpha t^2$$ Replacing \(\omega\) in the above equation, we get: $$\theta_2 = \alpha t \cdot t + \frac{1}{2}\alpha t^2$$

Find the ratio of angles θ₂ and θ₁

To find the ratio \(\left(\frac{\theta_2}{\theta_1}\right)\), simply divide the equations of Step 2 by Step 1: $$\frac{\theta_2}{\theta_1} = \frac{\alpha t \cdot t + \frac{1}{2}\alpha t^2}{\frac{1}{2} \alpha t^2}$$ By simplification, we get: $$\frac{\theta_2}{\theta_1} = \frac{2 \alpha t \cdot t + \alpha t^2}{\alpha t^2}$$ The term \(\alpha\) can be canceled out from both the numerator and denominator. Further simplification provides: $$\frac{\theta_2}{\theta_1} = \frac{2t \cdot t + t^2}{t^2} = \frac{3t^2}{t^2} = 3$$ Thus, the ratio \(\left(\theta_{2} / \theta_{1}\right) = 3\), which corresponds to option C.