JEE Main 2021
LEVELJEE Advanced
Visualized Solution
The Sigma Insight: Conservation of Angular Momentum
The Setup
A Tale of Two Discs
Imagine you are in a physics laboratory, observing a fascinating experiment. Before you are two solid discs, mounted on the same vertical frictionless axle. However, they are currently separated, spinning completely independently of one another.
The top disc has a moment of inertia and is spinning with an initial angular velocity . The bottom disc is a bit different; it has a moment of inertia and is spinning with its own initial angular velocity . They are like two dancers spinning to their own rhythm, completely unaware of each other.
Suddenly, the support holding the top disc is removed! It drops down, landing squarely on the bottom disc. What happens next is a beautiful display of rotational dynamics.
The Collision
When Worlds Collide
The moment the two discs touch, they don't instantly spin together. Because they are initially rotating at different speeds, their surfaces grind against each other. This grinding is kinetic friction in action. The faster disc tries to speed up the slower one, while the slower disc tries to drag down the faster one.
This internal tug-of-war continues until the friction finally does its job, locking the two discs together. From that moment on, they spin as a single, unified object with a brand new, common angular velocity, which we will call .
But how do we find this new speed? And more importantly, what was the cost of this violent, grinding collision?
The Master Equation
Conservation of Angular Momentum
To find the final speed, we need to look for a conserved quantity. During the collision, the frictional forces acting between the discs are purely internal to our two-disc system. By Newton's Third Law, the torque exerted by the top disc on the bottom disc is exactly equal and opposite to the torque exerted by the bottom disc on the top disc.
Because there is no external twisting force (no outside hand spinning the axle), the net external torque on the system is exactly zero:
When the net external torque is zero, the total angular momentum of the system must remain perfectly constant. This is the Law of Conservation of Angular Momentum.
Let's write down the initial angular momentum. It is simply the sum of the individual momenta of the two discs:
After the collision, the discs are locked together. Their combined moment of inertia is , and they spin at the common speed . So, the final angular momentum is:
Equating the initial and final states, we get our master equation:
Solving for the final common angular velocity , we find:
This formula is incredibly elegant. It is the exact rotational equivalent of finding the final velocity in a perfectly inelastic linear collision!
The Energy Toll
Calculating the Loss
Now we arrive at the core of the problem. While angular momentum was perfectly conserved, kinetic energy was not. The kinetic friction that locked the discs together did negative work, dissipating some of the system's rotational kinetic energy as heat and sound.
To find out exactly how much energy was lost, we need to calculate the difference between the initial kinetic energy and the final kinetic energy:
Let's set up the raw equations. The initial kinetic energy is the sum of the kinetic energies of the two freely spinning discs:
The final kinetic energy is the energy of the combined, locked system spinning at the new speed :
Now, we substitute our expression for into the final kinetic energy equation:
The Algebraic Dance
Simplifying the Mess
At first glance, this equation looks like an algebraic nightmare. But if we take a deep breath and proceed carefully, it will simplify beautifully.
First, notice that the term in the numerator cancels out one of the terms in the denominator of the squared fraction:
Next, we need to combine these terms into a single fraction. We do this by taking a common denominator of :
Now, we carefully expand the terms in the numerator. Watch out for the negative sign!
The Grand Finale
A Beautiful Symmetry
Look closely at the expanded numerator. The magic of algebra is about to happen. The terms cancel each other out perfectly. The terms also cancel each other out perfectly!
We are left with only the cross terms:
Notice that every single remaining term contains the product . Let's factor that out:
Do you recognize the expression inside the parentheses? It is the perfect square expansion of !
Substituting this beautifully simplified numerator back into our full equation, we arrive at our final answer:
This final expression is not just a random jumble of variables; it is a profound physical truth. It perfectly mirrors the energy loss formula for a perfectly inelastic linear collision (), where acts as the "reduced moment of inertia" and is the relative angular velocity. Understanding this deep symmetry will make you a true master of rotational mechanics!
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