Question

A wheel is rotating at \(900 \mathrm{rpm}\) about its axis. When power is cut off it comes to rest in 1 minute, the angular retardation in $\mathrm{rad} / \mathrm{sec}$ is \(\\{\mathrm{A}\\}(\pi / 2)\) \(\\{\mathrm{B}\\}(\pi / 4)\) \(\\{\mathrm{C}\\}(\pi / 6)\) \(\\{\mathrm{D}\\}(\pi / 8)\)

Short answer

The angular retardation, denoted as \(\alpha\), can be calculated using the formula \(\alpha = \cfrac{\omega_{f} - \omega_{i}}{t}\), with the values \(\omega_{i} = 30\pi\,\mathrm{rad/s}\), \(\omega_{f} = 0\,\mathrm{rad/s}\), and \(t = 60\,\mathrm{s}\). Substituting these values gives \(\alpha = -\cfrac{\pi}{2}\,\mathrm{rad/s^2}\), which matches with option \(\boxed{(\mathrm{A})(\pi / 2)}\).

Step-by-step solution

The initial angular speed is given in rpm (revolutions per minute). We need to convert this to radians per second. To do this, first, convert the revolutions to radians, and minutes to seconds: \(900\,\mathrm{rev/min} \times \cfrac{2\pi\,\mathrm{rad}}{1\,\mathrm{rev}} \times \cfrac{1\,\mathrm{min}}{60\,\mathrm{s}} = \cfrac{900\times 2\pi}{60}\,\mathrm{rad/s} = 30\pi\,\mathrm{rad/s}\) Therefore, the initial angular speed, denoted as \(\omega_{i}\), is \(30\pi\,\mathrm{rad/s}\). #Step 2: Identify known values and final angular speed

We know that the wheel comes to rest in 1 minute. This means the final angular speed, denoted as \(\omega_{f}\), is 0. Also, 1 minute is equal to \(60\,\mathrm{s}\), which will be our time, denoted as t. So, we have: \(\omega_{i} = 30\pi\,\mathrm{rad/s}\) \(\omega_{f} = 0\,\mathrm{rad/s}\) \(t = 60\,\mathrm{s}\) #Step 3: Use angular motion formula to find angular deceleration

The formula that relates initial angular speed, final angular speed, angular acceleration (or retardation), and time is: \(\omega_{f} = \omega_{i} + \alpha t\) We need to find angular retardation, denoted as \(\alpha\), for which we rearrange the formula: \(\alpha = \cfrac{\omega_{f} - \omega_{i}}{t}\) Substitute the known values: \(\alpha = \cfrac{0 - 30\pi}{60}\,\mathrm{rad/s^2} = -\cfrac{\pi}{2}\,\mathrm{rad/s^2}\) Note that the negative sign indicates the angular deceleration or retardation. #Step 4: Match the answer with the given options

The calculated angular retardation is \(-\cfrac{\pi}{2}\,\mathrm{rad/s^2}\), which matches with option A. The correct answer is: \(\boxed{(\mathrm{A})(\pi / 2)}\).