Torque and angular momentum — when to use which in rotational problems

Question

A disc of mass $M$ and radius $R$ rotates freely at angular velocity $\omega_0$ . A small lump of clay of mass $m$ falls vertically and sticks to the rim. Find the new angular velocity. Should we use torque analysis or angular momentum conservation here?

(JEE Main & JEE Advanced pattern)

Solution — Step by Step

The key question: is there an external torque about the axis of rotation?

The clay falls vertically onto the rim. Gravity acts vertically, and the axis is also vertical — so gravity produces zero torque about this axis. The contact force during collision is internal.

No external torque $\implies$ angular momentum is conserved. We don’t need torque analysis.

The disc rotates with $\omega_0$ :

L_i = I_{\text{disc}} \cdot \omega_0 = \frac{1}{2}MR^2 \omega_0

The falling clay has zero angular momentum about the vertical axis (it falls along a vertical line through the rim — its velocity is parallel to the axis, so $\vec{r} \times \vec{p}$ about the axis is zero).

After sticking, the clay (at radius $R$ ) rotates with the disc:

L_f = \left(\frac{1}{2}MR^2 + mR^2\right)\omega_f

\frac{1}{2}MR^2 \omega_0 = \left(\frac{M}{2} + m\right)R^2 \omega_f

\omega_f = \frac{M\omega_0}{M + 2m}

The disc slows down because the moment of inertia increased while angular momentum stayed constant.

Why This Works

The decision between torque and angular momentum conservation comes down to one question: is the net external torque zero?

graph TD
    A["Rotational Problem"] --> B{"Net external torque<br/>about chosen axis?"}
    B -->|"Zero"| C["Conserve Angular Momentum<br/>L_i = L_f"]
    B -->|"Non-zero"| D["Use τ = Iα<br/>(Newton's 2nd for rotation)"]
    C --> E["Useful when: collision,<br/>explosion, person on disc"]
    D --> F["Useful when: force applied,<br/>friction torque, pulley problems"]
    D --> G{"Torque constant?"}
    G -->|"Yes"| H["α = τ/I, then use<br/>rotational kinematics"]
    G -->|"No"| I["Use angular impulse<br/>τΔt = ΔL"]

When external torque is zero, angular momentum conservation gives the answer in one equation — far simpler than tracking forces. When external torque is non-zero, we use $\tau = I\alpha$ (the rotational Newton’s second law) and integrate or use kinematics.

Alternative Method — Energy Approach (and Why It Fails Here)

You might think: use energy conservation. But this is an inelastic collision (clay sticks). Kinetic energy is NOT conserved — some energy is lost to deformation. So energy methods give the wrong answer here.

Energy conservation works for rotational problems where no collision occurs — like a spinning top slowing down due to friction (where we track energy lost to friction).

For JEE Advanced: whenever something “sticks” or “is placed gently,” think angular momentum conservation. The word “gently” is code for “no impulsive external torque.” If the problem says “suddenly,” also think angular momentum — the interaction happens too fast for external torques to produce significant angular impulse.

Common Mistake

Students conserve angular momentum about the wrong axis. In this problem, if you choose a horizontal axis instead of the vertical rotation axis, gravity DOES produce a torque, and the calculation breaks. Always conserve angular momentum about an axis where external torques vanish. For disc/turntable problems, the symmetry axis (vertical) is almost always the correct choice.

Question

Solution — Step by Step

Why This Works

Alternative Method — Energy Approach (and Why It Fails Here)

Common Mistake

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