Problem 60 - Master Resource Book in JEE Main Physics

Question

Angular momentum in absence of any external torque remains constant. If no external torque acts on a system of particles, then angular momentum of the system remains constant, i.e., \(\tau=0\) $$ \begin{aligned} &\therefore \quad \frac{d L}{d t}=0 \\ &\Rightarrow \quad l_{1} \omega_{1}=I_{2} \omega_{2} \\ &\text { Here, } M=2 \mathrm{~kg}, m=0.25 \mathrm{~kg}, r=0.2 \mathrm{~m} \\ &\omega_{1}=30 \mathrm{rad} \mathrm{s}^{-1} \end{aligned} $$ Hence, we get after putting the given values in Eq. (i) $$\frac{1}{2} \times 2 \times(0.2)^{2} \times 30=\frac{1}{2} \times(2+2 \times 0.25)(0.2)^{2} \times \omega_{2}$$ $$\begin{array}{ll}\Rightarrow & 1.2=0.05 \omega_{2} \\ \therefore & \omega_{2}=24 \mathrm{rad} \mathrm{s}^{-1}\end{array}$$

Step-by-Step Solution

Answer

The final angular velocity \(\omega_2\) is 24 rad/s.

Step 1: Understand Angular Momentum Conservation

Angular momentum of a system remains constant when no external torque is applied. The given problem involves using this principle to find the unknown angular velocity \(\omega_2\). The equation representing conservation of angular momentum is \(l_1 \omega_1 = I_2 \omega_2\), where \(l_1\) is the initial angular momentum, \(I_2\) is the final moment of inertia, and \(\omega_2\) is the final angular velocity.

Step 2: Calculate Initial Moment of Inertia

Calculate the initial moment of inertia using the formula \(I = \frac{1}{2} \times M \times r^2\), where \(M = 2\,kg\) and \(r = 0.2\,m\). Substitute these values to get \(I_{initial} = \frac{1}{2} \times 2 \times (0.2)^2 = 0.04\,kg\cdot m^2\).

Step 3: Calculate Initial Angular Momentum

Use the initial angular velocity \(\omega_1 = 30\,rad/s\) to find the initial angular momentum: \(l_1 = I_{initial} \times \omega_1 = 0.04 \times 30 = 1.2\,kg\cdot m^2/s\).

Step 4: Calculate Final Moment of Inertia

Determine the final moment of inertia \(I_2\). The formula \(I = \frac{1}{2} \times (M + 2m) \times r^2\) accounts for the additional mass \(m = 0.25\,kg\). Substitute these values to get \(I_2 = \frac{1}{2} \times (2 + 2 \times 0.25) \times (0.2)^2 = 0.05\,kg\cdot m^2\).

Step 5: Solve for Final Angular Velocity

Set the initial angular momentum equal to the final, using the formula \(l_1 = I_2 \omega_2\). Substitute the known values: \(1.2 = 0.05 \omega_2\). Solve for \(\omega_2\), giving \(\omega_2 = \frac{1.2}{0.05} = 24\,rad/s\).

Key Concepts

Angular Momentum

Angular momentum is a fundamental property of any rotating object or system. It describes the object's rotational motion and plays a crucial role in physics. Essentially, the conservation of angular momentum states that when no external torques (twisting forces) act on a system, the system's total angular momentum remains constant.

Angular momentum, represented by the symbol \( L \), is calculated using the formula \( L = I \omega \), where \( I \) is the moment of inertia and \( \omega \) is the angular velocity. Due to this interrelationship, if the moment of inertia changes, the angular velocity must adjust to keep the angular momentum constant, given no external torque is applied.

Imagine an ice skater spinning. As they pull their arms in, their moment of inertia decreases, which in turn increases their spinning speed. This phenomenon vividly illustrates the conservation of angular momentum.

Moment of Inertia

Moment of inertia is a measure of an object's resistance to changes in its rotational motion. It serves as the rotational analogue to mass in linear motion. The greater an object's moment of inertia, the harder it is to change its rotational speed.

The formula for moment of inertia depends on how mass is distributed relative to the axis of rotation. In a simple scenario, such as a point mass rotating at a distance, the moment of inertia \( I \) is calculated as \( I = mr^2 \), where \( m \) is the mass and \( r \) is the distance from the axis. However, for a rigid body, a more commonly used formula is \( I = \frac{1}{2}m r^2 \).

In practice, when dealing with complex systems, each component's contribution to the total moment of inertia is considered. This helps in accurately predicting how the body's rotation will respond to forces. Understanding moment of inertia is crucial as it affects how an object's angular velocity changes during rotational movements.

Angular Velocity

Angular velocity is a vector quantity that represents the rate of rotation of an object around an axis. It specifies how many radians an object rotates per second and is measured in radians per second (rad/s).

When you look at a rotating system where angular momentum is conserved, angular velocity plays a key role alongside the moment of inertia. Any alteration in the system's configuration, like changes in the distribution of mass, can lead to variations in angular velocity as it strives to maintain constant angular momentum.

An effective way to visualize this is through a spinning dancer or figure skater. They adjust their speed to spin faster or slower by simply changing the positioning of their arms and legs, which modifies their moment of inertia, subsequently changing their angular velocity accordingly.

Understanding angular velocity is vital in many areas, from designing efficient mechanical systems to understanding natural phenomena in the universe, as it provides insights into how objects rotate and interact within various systems.