JEE Main 2020, 05 Sep Shift-I
LEVELJEE Main
Visualized Solution
The Sigma Insight: Conservation of Angular Momentum
The problem of coupling two rotating discs is a classic scenario in rotational mechanics, often referred to as a "perfectly inelastic rotational collision." Let's dive into the physics of why the wheels slow down and where the energy goes.
Analyzing the Setup
Imagine a wheel spinning freely on a shaft. It has a moment of inertia and an angular speed . This wheel possesses both angular momentum and rotational kinetic energy.
Suddenly, a second wheel, three times as massive in terms of inertia (), is dropped onto the same shaft. Initially, this second wheel is at rest. Because they are now forced to share the same shaft, they will rub against each other until they lock together and spin at a new, common angular speed.
The Master Equation
Conservation of Angular Momentum
When the wheels couple, there is a lot of friction between them. However, this friction is an internal force within the two-wheel system. The torque exerted by the first wheel on the second is exactly equal and opposite to the torque exerted by the second wheel on the first.
Since there is no external torque acting on the system from the outside world, the total angular momentum must remain strictly conserved.
Initially, only the first wheel is spinning:
Finally, both wheels spin together as a single unit with a combined inertia of , at a new common speed :
Equating the two gives us the final speed:
The system is now spinning at one-fourth of its original speed.
The Energy Crisis
While angular momentum is conserved, kinetic energy is absolutely not. The friction that brought the two wheels to the same speed did negative work, dissipating a significant amount of kinetic energy as heat. Let's calculate exactly how much was lost.
The initial kinetic energy is just the energy of the first wheel:
The final kinetic energy is the energy of the combined system:
Final Calculation
The Fractional Loss
The question asks for the fractional loss in kinetic energy, which is the ratio of the energy lost to the initial energy.
First, find the absolute loss:
Now, divide by the initial kinetic energy to find the fraction:
A staggering of the initial kinetic energy was lost to heat and sound during the coupling process! This perfectly mirrors a perfectly inelastic collision in linear mechanics, where objects stick together and maximize their kinetic energy loss.
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