CLASS 11 · CHAPTER 6 · MECHANICS

Rotational Motion

Every spin, a figure skater, a spinning wheel, a planet, obeys the same rule: nothing changes angular momentum except an outside twist. Pull mass toward the axis, and the system finds its own way to spin faster.

01 · See it

Watch it happen

Distance of each mass from the axis: 0.50 m

0.500 kg·m²

Moment of inertia

2.40 rad/s

Angular velocity

1.20 kg·m²/s

Angular momentum

1.00×

Rotational KE vs. start

Pull the masses inward: angular momentum stays fixed at 1.20 kg·m²/s by construction, but the spin rate, and the kinetic energy, both climb sharply.

Nothing external is twisting this system as you drag the slider, the pull on the masses acts straight along the rod, toward the centre, so it creates zero torque about the axis. With no external torque, angular momentum has nowhere to go.

02 · Derive it

Where the formula comes from

For two point masses $m$ at distance $r$ from the axis, moment of inertia is just $I = \sum m_i r_i^2$ :

I = 2mr^2

Angular momentum and the rotational form of Newton’s second law:

\begin{gathered} L = I\omega \\[6px] \tau_{\text{net}} = \dfrac{dL}{dt} \end{gathered}

When $\tau_{\text{net}} = 0$ , $L$ stays fixed, even as $I$ itself changes:

Conservation of angular momentum

\textcolor{#e08a1e}{I_i\omega_i} = \textcolor{#e08a1e}{I_f\omega_f}

03 · Break it

Where the shortcut stops working

“Angular momentum is conserved” gets misheard as “nothing about the spin changes.” Substitute $\omega = L/I$ into the kinetic energy formula and a different story appears:

KE_{\text{rot}} = \tfrac{1}{2}I\omega^2 = \dfrac{L^2}{2I}

With $L$ pinned constant, $KE_{\text{rot}}$ is proportional to $1/I$ . Shrink the moment of inertia and the kinetic energy climbs, exactly what the simulation above shows in the “KE vs. start” readout. That extra energy is not a loophole: it’s the real work your muscles do pulling the masses in against their own outward-flung momentum. Conservation of $L$ was never a promise about energy.

04 · Master it

Apply it under exam conditions

Q1. A skater spinning at 2 rad/s with arms out (I = 2.5 kg·m²) pulls their arms in, reducing I to 1.0 kg·m². Find the new angular velocity and how the kinetic energy has changed.

\omega_f = \dfrac{I_i\omega_i}{I_f} = \dfrac{2.5\times2}{1.0} = \textcolor{#e08a1e}{5\text{ rad/s}}

\dfrac{KE_f}{KE_i} = \dfrac{I_i}{I_f} = \textcolor{#e08a1e}{2.5\times}

Q2. A wheel with I = 0.4 kg·m² starts from rest under a constant torque of 2 N·m. Find its angular velocity after 5 s, and the number of revolutions completed.

\alpha = \dfrac{\tau}{I} = \dfrac{2}{0.4} = 5\text{ rad/s}^2 \;\Rightarrow\; \omega = \alpha t = \textcolor{#e08a1e}{25\text{ rad/s}}

\theta = \tfrac{1}{2}\alpha t^2 = 62.5\text{ rad} = \dfrac{62.5}{2\pi} \approx \textcolor{#e08a1e}{9.95\text{ revolutions}}

05 · FAQs

Quick answers

What's the difference between torque and force?+

Force changes how fast something moves in a straight line; torque changes how fast it spins. The same force applied further from the axis produces more torque, which is exactly why a longer wrench needs less push to loosen a stubborn bolt.

Is angular momentum always conserved?+

No, only when there's no net external torque on the system. An ice skater pulling their own arms in is an internal force, it exerts zero torque about the spin axis, so L stays fixed even though the skater is clearly doing something.

Why does a figure skater spin faster with arms pulled in?+

Because angular momentum L = Iω stays constant. Pulling the arms in shrinks the moment of inertia I, and since L can't change, ω has to rise to compensate.

Does pulling your arms in take energy?+

Yes. Rotational kinetic energy actually increases, since KE = L²/2I and I has gone down. That extra energy comes from real muscular work done pulling the masses inward, not from nowhere.

What's the rotational analogue of Newton's second law?+

τ_net = dL/dt, directly mirroring F_net = dp/dt from linear mechanics. Torque plays the role force plays, and angular momentum plays the role linear momentum plays.

Related concepts

Physics doesn’t stay inside chapter boundaries. Neither should you.

Mechanics

Laws of Motion

Push a block and watch friction switch from static to kinetic the instant it slips.

Mechanics

Gravitation

Launch a satellite and watch its orbit turn elliptical, or escape entirely.