Problem 611 - JEE Physics

Question

The angular momentum of a wheel changes from \(2 \mathrm{~L}\) to \(5 \mathrm{~L}\) in 3 seconds what is the magnitudes of torque acting on it? \(\\{\mathrm{A}\\} \mathrm{L}\) \(\\{\mathrm{B}\\} \mathrm{L} / 2\) \(\\{\mathrm{C}\\} \mathrm{L} / 3\) \(\\{\mathrm{D}\\} \mathrm{L} / 5\)

Step-by-Step Solution

Answer

The correct answer is: \(\boxed{\mathrm{(A)}\, \mathrm{L}}\).

Step 1: Determine the Change in Angular Momentum

To find the change in angular momentum, we will subtract the initial angular momentum from the final angular momentum: ΔL = L_final - L_initial = 5L - 2L = 3L

Step 2: Calculate Torque

Now that we have the change in angular momentum, we can use the formula for torque to find the magnitude of the torque acting on the wheel: Torque = ΔL / Δt = 3L / 3s = L The magnitude of the torque acting on the wheel is L. The correct answer is: \(\boxed{\mathrm{(A)}\, \mathrm{L}}\).

Key Concepts

Angular Momentum

Angular momentum is a fundamental concept in physics that describes the rotational movement of an object. Think of it as a counterpart to linear momentum but for rotating objects. When a wheel spins, for example, it possesses angular momentum due to its rotational mass and velocity. This concept is crucial because it helps us understand how objects behave when they’re spinning or rotating.
An important feature of angular momentum is that it is conserved in a closed system. This means that, unless external forces or torques act on the system, the total angular momentum remains constant. This principle is similar to the conservation of linear momentum but applied to rotational dynamics.
Every rotating object has its own angular momentum, which depends on its moment of inertia (how mass is distributed with respect to the rotational axis) and angular velocity (how fast it is spinning). The unit of angular momentum is typically expressed in terms of the moment of inertia and radians per second, making it a vector quantity that points along the axis of rotation.

Change in Angular Momentum

In physics, changes in angular momentum play a critical role in understanding how forces are influencing a system. The change in angular momentum provides essential information about how much a system's rotation has been altered over time.
If you want to find this change, simply subtract the initial angular momentum from the final angular momentum. This is exactly what was done in the exercise: the change was found by subtracting the initial value (2L) from the final value (5L). Therefore, the change is 3L.
Changes in angular momentum are typically caused by external torques applied to the system. In our problem, an external torque influenced the wheel to change its rate of spin. If no external torque exists, then the change in angular momentum would be zero, maintaining the original state due to conservation laws.

Angular Momentum Formula

To understand the mathematics behind angular momentum, it helps to explore the primary formula used in its calculations. The angular momentum (L) of an object can be calculated using:\[ L = I \cdot \omega \]Where:

• \( I \) is the moment of inertia.
• \( \omega \) is the angular velocity.

This formula shows that angular momentum depends on both how much the object resists changes to its rotation (moment of inertia) and its speed of rotation (angular velocity).
In the context of changing angular momentum, the relevant formula connects the change in angular momentum to torque:\[ \tau = \frac{\Delta L}{\Delta t} \]Where:

• \( \tau \) is the torque.
• \( \Delta L \) is the change in angular momentum.
• \( \Delta t \) is the change in time.

This relationship helps solve many practical problems, such as determining the applied torque when knowing the change in momentum and the time duration.

Physics Problem Solving

Solving physics problems often involves breaking the problem down into clear and manageable steps. This methodical approach ensures that all critical aspects of a problem are addressed, reducing the chance of oversight.
In the exercise provided, the first step was identifying the change in angular momentum. Once this was done, the subsequent task was using the reinforced foundational formula to find the torque. This way of breaking down a problem simplifies complex topics like rotational dynamics.
To solve these problems effectively:

• Clearly identify what is given and what needs to be found.
• Use appropriate formulas at each step.
• Perform algebraic manipulations carefully.
• Check each step with physical intuition; the answer should make sense conceptually.

By following this structure, even the most challenging physics problems can become approachable and understandable.