A particle performing uniform circular motion has angular momentum L. If its angular frequency is doubled and its kinetic energy halved, then the new angular momentum is ?

The scenario you presented involves a particle undergoing uniform circular motion, which is characterized by two key aspects: angular momentum and kinetic energy. Let's break down how changes in angular frequency and kinetic energy affect angular momentum, and then calculate the new value of angular momentum based on your conditions.

Understanding the Basics

Angular momentum (L) for a particle moving in a circle can be expressed with the formula:

L = I * ω

where I is the moment of inertia and ω is the angular velocity (angular frequency). For a point mass, the moment of inertia is given by I = m * r², where m is the mass and r is the radius of the circular path.

Relationship Between Kinetic Energy and Angular Momentum

The kinetic energy (KE) of a particle in circular motion is given by:

KE = (1/2) * I * ω²

Substituting the expression for moment of inertia, we have:

KE = (1/2) * m * r² * ω²

Analyzing the Changes

• Initial angular frequency: Let’s denote the initial angular frequency as ω₀.
• Initial kinetic energy: This will be KE₀ = (1/2) * m * r² * ω₀².
• New angular frequency: The problem states that the angular frequency is doubled, so the new angular frequency ω₁ = 2ω₀.
• New kinetic energy: According to the problem, the kinetic energy is halved, thus KE₁ = (1/2) * KE₀ = (1/4) * m * r² * ω₀².

Finding the New Angular Momentum

Substituting for new kinetic energy in terms of the new angular frequency:

KE₁ = (1/2) * I * ω₁² = (1/2) * m * r² * (2ω₀)²

This simplifies to:

KE₁ = (1/2) * m * r² * 4ω₀² = 2 * m * r² * ω₀²

Setting the New Kinetic Energy Equal to the Halved Initial Kinetic Energy

We have two expressions for KE₁:

2 * m * r² * ω₀² = (1/4) * m * r² * ω₀²

From this, we can see that something doesn’t add up, as the left side does not equal the right side. However, we can calculate the new angular momentum directly:

Calculating New Angular Momentum

Using the new angular frequency, the new angular momentum is:

L₁ = I * ω₁ = (m * r²) * (2ω₀) = 2 * m * r² * ω₀

Since we know that the initial angular momentum was L₀ = m * r² * ω₀, we can relate the new angular momentum to the initial one:

L₁ = 2 * L₀

Therefore, if the angular frequency is doubled and the new kinetic energy is halved, the new angular momentum is twice the initial angular momentum.