Problem 664 - JEE Physics

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 \(\\{\mathrm{A}\\}(\pi / 2)\) \(\\{\mathrm{B}\\}(\pi / 4)\) \(\\{\mathrm{C}\\}(\pi / 6)\) \(\\{\mathrm{D}\\}(\pi / 8)\)

Step-by-Step Solution

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 1: 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

Step 2: 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: \(\boxed{(\mathrm{A})(\pi / 2)}\).

Key Concepts

Angular Motion Equations

Angular motion equations are fundamental in analyzing movements involving rotations. One of the most crucial equations is the angular analog of one of the straight-line motion formulas. It states:\[ \omega_f = \omega_i + \alpha t \]Here,

• \( \omega_f \): final angular speed,
• \( \omega_i \): initial angular speed,
• \( \alpha \): angular acceleration,
• \( t \): time.

These equations give us insights into how rotational speed changes over time.
If a rotating object slows down, like in our problem, the angular acceleration will be negative. We call this angular deceleration or retardation. These equations help us calculate such quantities efficiently.
By rearranging the equation above, we can solve for any unknown variable, such as angular acceleration when given the initial and final speeds and time.

Conversion of Units (rpm to rad/s)

Often in physics, it is necessary to convert units to ensure efficiency and clarity. One common conversion necessary when studying mechanics is from revolutions per minute (rpm) to radians per second (rad/s). RPM is a measure of the speed of rotation, but in physics, using radians per second offers more precision.
To convert, we use the relationships:

• 1 revolution = \( 2\pi \) radians
• 1 minute = 60 seconds

The conversion involves multiplying the rpm value by \( 2\pi \) and then dividing by 60. For example, converting 900 rpm:\[900\, \mathrm{rpm} \times \frac{2\pi\, \mathrm{rad}}{1\, \mathrm{rev}} \times \frac{1\, \mathrm{min}}{60\, \mathrm{s}} = 30\pi\, \mathrm{rad/s}\]This formula converts initial angular speeds efficiently. Always make sure to use consistent units, as it makes the calculations straightforward and accurate.

Angular Speed Calculations

Angular speed is a measure of how quickly an object rotates. It is crucial in solving mechanical problems involving rotating bodies. The unit typically used for angular speed in equations is radians per second (rad/s). Calculations include converting units and applying formulas.
Let's breakdown the calculation of initial angular speed in this context.

• Given: 900 rotations per minute (rpm)
• Conversion: Use the formula and facts: 1 revolution = \( 2\pi \) rad and 1 minute = 60 s
• Result: Initial speed in rad/s: \( 30\pi \)

Angular speed can change due to influences like gravity or mechanical influences, affecting how a rotating body behaves over time.
Understanding angular speed concepts helps solve more complex physics problems, especially those involving rotation and torque.

Mechanics in Physics

Mechanics is a core branch of physics that deals with motion and the forces affecting that motion. When dealing with rotational motion, as in this exercise, we examine how objects rotate around an axis.
This branch includes crucial concepts such as:

• Rotational Kinematics: dealing with angular displacement, speed, and acceleration
• Torque: the cause of rotational motion
• Angular Momentum: a measure of rotational motion consistency

Angular mechanics plays a significant role in understanding everything from the spin of wheels to the rotation of celestial bodies.
Having a grasp on basics of these concepts provides a foundation for more advanced topics in physics. Mechanics helps explain real-world applications, like how cutting power from a machine affects its rotation, aligning perfectly with the problem at hand.