JEE Advanced 2024: Two discs joining by friction

Why angular momentum is the only tool

Two coaxial discs, one spinning and one at rest, are brought into contact and friction between them brings them to a common angular speed. Energy is lost to heat at the interface, so energy conservation is useless here. The conserved quantity is angular momentum about the common axis, because the frictional torques are internal to the two-disc system and there is no external torque on it.

Writing the balance

Initial angular momentum is I₁ω₁, since only the first disc spins. Final angular momentum is (I₁ + I₂)ω, because the discs lock to a common speed. Setting them equal gives ω = I₁ω₁ / (I₁ + I₂). The result is always smaller than the initial speed, which is the sanity check: the combined moment of inertia is larger, so the speed must drop.

Where the energy goes

The ratio of the final kinetic energy to the initial is I₁/(I₁ + I₂), so the fraction lost is I₂/(I₁ + I₂). The energy is not destroyed — it is converted to heat in the interface, and the rate of that conversion is proportional to the slip velocity, which is why the clutch takes time to lock rather than locking instantly.

The general two-body version

The clutch is the special case where one body starts at rest. When both discs already spin, the same conservation law applies with two terms on each side: I₁ω₁ + I₂ω₂ = (I₁ + I₂)ω, and the final speed is the angular-momentum weighted average of the two initial speeds. That is a useful thing to internalise, because the exam sometimes states the question as "a second disc of moment of inertia I₂ is gently lowered onto a spinning disc" and the candidate has to recognise the lowered disc as the second body rather than assume it starts from rest. The moment of inertia of the second disc about the common axis is the one that matters, and for a disc lowered onto a shaft that is the standard mR²/2.

The general two-body version

The clutch is the special case where one body starts at rest. When both discs already spin, the same conservation law applies with two terms on each side: I1 w1 + I2 w2 = (I1 + I2) w, and the final speed is the angular-momentum weighted average of the two initial speeds. That is a useful thing to internalise, because the exam sometimes states the question as "a second disc of moment of inertia I2 is gently lowered onto a spinning disc", and the candidate has to recognise the lowered disc as the second body rather than assume it starts from rest. The moment of inertia of the second disc about the common axis is the one that matters, and for a disc lowered onto a shaft that is the standard mR squared over two.

The two seconds that decide the question

A clutch problem in JEE is almost never asking about the clutch. It is asking about the two seconds in which the mechanism is active, and they behave differently. During the slipping phase the discs are not locked, so they rotate at different speeds and the friction torque acts on the slower disc to speed it up while acting on the faster disc to slow it down. The frictional torque on each disc has the same magnitude and the same sense of relative rotation, which is what lets the two angular accelerations be written in one equation.

Two facts follow, and the exam tests them separately. First, the torque decreases as the speeds approach each other, because the slip velocity shrinks — which is why the locking takes a finite time rather than being instantaneous. Second, the friction reverses direction if the discs ever overtake, so the process is self-limiting. A candidate who assumes the friction is constant while the discs slip will get the final speed right and the time wrong, and the time is usually the second sub-question.

The energy argument gives the same final state from the other side, and is the better way to check a slip in the algebra. The final kinetic energy can never exceed the initial, and the final speed can never exceed the faster disc's initial speed. If either of those bounds is violated, the conservation law was applied to the wrong quantity.

Why the wrong method fails so visibly

Candidates who conserved kinetic energy obtained a final speed larger than the initial one, which is impossible when two bodies lock together. The reason is structural, not arithmetic: kinetic energy is only conserved in elastic processes, and a slipping contact is the definition of an inelastic one. When an answer comes out larger than the starting value, the method is wrong, not the numbers.