Animated Solution for Physics - Rotational Motion: 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 beauty of physics often lies in how different concepts interlock to form a cohesive understanding of the universe. In this problem, we are exploring the elegant dance between three fundamental rotational quantities: Angular Momentum ($L$), Kinetic Energy ($K$), and Angular Frequency ($\omega$).

Imagine you are spinning a stone tied to a string. The stone possesses kinetic energy because it is moving, and it possesses angular momentum because it is rotating around a central point. But how exactly are these two quantities related? Let's break down the DNA of rotational motion.

To begin, we must recall the two most critical equations in rotational mechanics. First, the angular momentum of a rotating body is defined as the product of its moment of inertia ($I$) and its angular velocity ($\omega$):

Second, the rotational kinetic energy ($K$) is given by half the product of the moment of inertia and the square of the angular velocity:

At first glance, these look like two separate equations. However, if you look closely at the kinetic energy formula, you will notice that the term $I\omega$ is hidden right inside it! We can rewrite the kinetic energy equation by splitting the ${\omega }^2$ term:

Now, we can perform a beautiful substitution. Since we know that $L=I\omega$, we can replace the $(I\omega )$ term in our kinetic energy equation with $L$. This yields a direct relationship between kinetic energy, angular momentum, and angular velocity:

This is a incredibly powerful formula because it completely bypasses the moment of inertia ($I$). For our specific problem, we want to find out what happens to the angular momentum. So, let's rearrange this equation to solve for $L$:

This is our Master Equation. It tells us exactly how angular momentum responds to changes in kinetic energy and angular frequency.

The problem presents us with a hypothetical scenario: What if the kinetic energy is halved, and the angular frequency is doubled? Let's define our new state variables:

New Kinetic Energy: $K^{\prime }=\frac{K}{2}$

New Angular Frequency: ${\omega }^{\prime }=2\omega$

We simply plug these new conditions into our Master Equation to find the new angular momentum ($L^{\prime }$):

Now, it's just a matter of careful algebra. In the numerator, the $2$ and the $\frac{1}{2}$ cancel each other out perfectly, leaving us with just $K$. The denominator remains $2\omega$.

To see how this relates to our original angular momentum, let's factor out a $\frac{1}{4}$ from the expression:

Since we already established that our original angular momentum is $L=\frac{2K}{\omega }$, we can substitute $L$ back into the equation:

And there we have it! By halving the kinetic energy and doubling the angular frequency, the new angular momentum becomes exactly one-fourth of its original value. This problem perfectly illustrates why finding direct mathematical relationships between variables is often much faster and more elegant than calculating each component individually.