Question

A fly wheel is accelerated uniformly from rest and rotates through 5 rad in the first second. The angle rotated by the fly wheel in the next second, will be:

• A. 7.5 rad
• B. 15 rad
• C. 20 rad
• D. 30 rad

Answer 1

The Correct Option is B

To solve this problem, we need to calculate the angle rotated by the flywheel in the second second after it starts accelerating uniformly. This is a common problem involving rotational motion and uniform acceleration. Let's break down the steps:

1. First, understand that the flywheel starts from rest and has uniformly accelerated rotational motion. The angular displacement in the first second is given as 5 radians.
2. We use the equation for angular displacement in uniformly accelerated rotational motion: \(\theta = \omega_0 t + \frac{1}{2} \alpha t^2\). Here: - \(\theta\) is the angular displacement. - \(\omega_0\) is the initial angular velocity. - \(\alpha\) is the angular acceleration. - \(t\) is time.
3. Since the flywheel starts from rest, \(\omega_0 = 0\). Therefore, the equation simplifies to: \(\theta = \frac{1}{2} \alpha t^2\).
4. We know that \(\theta = 5 \, \text{rad}\) when \(t = 1 \, \text{s}\). Therefore, \(\frac{1}{2} \alpha (1)^2 = 5\). Solving for \(\alpha\), we get: \(\alpha = 10 \, \text{rad/s}^2\).
5. We now need to find the angular displacement during the second second, which is from \(t = 1\) s to \(t = 2\) s.
6. First, calculate the total angular displacement at \(t = 2\) s. Using the same formula, \(\theta = \frac{1}{2} \alpha (2)^2 = \frac{1}{2} \times 10 \times 4 = 20 \, \text{rad}\).
7. To find only the angle rotated during the second second, subtract the angle rotated in the first second from the angle rotated in two seconds: \( \theta_{2^\text{nd}\, \text{second}} = \theta_{\text{total at } 2 \, \text{s}} - \theta_{\text{total at } 1 \, \text{s}} = 20 \, \text{rad} - 5 \, \text{rad} = 15 \, \text{rad} \).

Therefore, the angle rotated by the flywheel in the next second is 15 rad.