Torque and Angular Momentum

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Torque and Angular Momentum

TL;DR

Torque is the rotational version of force—it measures how hard you twist something around a pivot. Angular momentum is the rotational version of momentum—it measures how much "rotational motion" a system has. They're related by τ = dL/dt, just like force and linear momentum.

1. The Mental Model

Think of torque as leverage in rotation. When you push a door near its handle, you get lots of rotational effect; push near the hinge, you get almost nothing. That's torque. Angular momentum is the rotational stubbornness of a spinning object—a spinning ice skater has high angular momentum and resists being pushed into a different spin. The core insight: torque is what changes angular momentum, exactly as force changes linear momentum.

2. The Core Material

2.1 Defining Torque

Torque τ measures the twisting effect of a force around a pivot. It depends on three things:

• The magnitude of the force F
• How far from the pivot you apply it (the lever arm r)
• The angle between the force and the lever arm

The formula is:

$$\boldsymbol{\tau} = \mathbf{r} \times \mathbf{F}$$

If you apply the force perpendicular to the lever arm, then:

$$\tau = rF$$

Units: Newton·meters (N·m).

Direction: Use the right-hand rule. Curl your fingers in the direction of rotation; your thumb points in the direction of τ.

Real example: loosening a bolt. A 0.3 m wrench with 100 N perpendicular force gives τ = 30 N·m. That same 100 N applied along the wrench (parallel) gives zero torque—no twist at all.

2.2 Defining Angular Momentum

Angular momentum L measures how much rotational motion a system has. For a point mass at distance r moving with velocity v:

$$\mathbf{L} = \mathbf{r} \times \mathbf{p} = \mathbf{r} \times m\mathbf{v}$$

For a rigid body rotating around a fixed axis with angular velocity ω:

$$L = I\omega$$

where I is the moment of inertia—the rotational analog of mass. It tells you how hard it is to spin the object. A disk and a hoop of the same mass rotating at the same ω have different I (the hoop has more—more mass is far from the axis), so they have different L.

Units: kg·m²/s.

2.3 The Connection: τ = dL/dt

This is the Newton's second law of rotation:

$$\boldsymbol{\tau}_{\text{net}} = \frac{d\mathbf{L}}{dt}$$

This is the key. Just as F_net = dp/dt in linear motion, torque is the rate of change of angular momentum. No net torque? Angular momentum stays constant. This is conservation of angular momentum.

graph TD
    A["Net Torque τ_net"] -->|applies to| B["System"]
    B -->|changes| C["Angular Momentum L"]
    D["No torque?"] -->|means| E["L is constant"]
    C -->|if constant| F["Object keeps same spin"]
    G["Friction/external forces"] -->|create| A
    H["Isolated system"] -->|has| D

2.4 Angular Momentum Conservation

If the net external torque on a system is zero, angular momentum is conserved:

$$L_{\text{initial}} = L_{\text{final}}$$

Classic example: an ice skater spins with arms out (I = high, ω = low, so L = some value). They pull their arms in (I decreases). Since no external torque acts, L stays the same, so ω must increase—they spin faster.

Another real case: a planet orbiting the Sun. Far from the Sun, it moves slowly; near the Sun (perihelion), it moves fast. Angular momentum about the Sun's center is conserved.

3. Worked Example

Problem: A uniform disk of mass M = 2 kg and radius R = 0.5 m rotates at ω = 4 rad/s on a frictionless axle. You apply a tangential force F = 10 N at the rim perpendicular to the radius. How long until the disk reaches ω = 8 rad/s?

Step 1: Find the moment of inertia of the disk about its center.
$$I = \frac{1}{2}MR^2 = \frac{1}{2}(2)(0.5)^2 = 0.25 \text{ kg·m}^2$$

Step 2: Calculate the torque applied.
$$\tau = rF = (0.5)(10) = 5 \text{ N·m}$$

Step 3: Use τ = dL/dt. Since L = Iω and I is constant:
$$\tau = I\frac{d\omega}{dt}$$
$$\frac{d\omega}{dt} = \frac{\tau}{I} = \frac{5}{0.25} = 20 \text{ rad/s}^2$$

Step 4: Use kinematics. From ω₀ = 4 rad/s to ω = 8 rad/s:
$$\omega = \omega_0 + \alpha t$$
$$8 = 4 + 20t$$
$$t = 0.2 \text{ s}$$

The disk reaches 8 rad/s in 0.2 seconds.

4. Production Pitfalls & Best Practices

4.1 Real-World Best Practices

1. Always define your axis. Angular momentum and torque both depend on which point you're calculating them about. For a wheel rolling, pick the axle, not some random point. Different axis = different L.

2. Use components consistently. If you decompose torque into x, y, z components, keep the same coordinate system throughout. Mixing conventions (sometimes radial, sometimes Cartesian) is where signs flip and answers turn wrong.

3. Watch out for "no friction" vs "no slipping." Frictionless motion means torque from external forces only. A rolling wheel without slipping has static friction (does no work on the contact point, but does create torque). Don't ignore it.

4. For changing moment of inertia, use L directly. If I changes (like a skater pulling their arms in), use L = Iω and check that τ_net = 0 implies L_initial = L_final. Don't try to use α = τ/I—that only works for constant I.

5. Label "about which point" in every calculation. Write "L about the center of mass" or "τ about the pivot." Examiners and colleagues will know exactly what you mean.

4.2 Common Bugs & Anti-Patterns

Bug 1: Forgetting that torque is a vector (or at least has a sign).

# WRONG
torque = r * F  # scalar multiplication, no direction info

# RIGHT
import numpy as np
r_vec = np.array([0, 0, 0.5])  # position vector (meters)
F_vec = np.array([10, 0, 0])   # force vector (Newtons)
tau = np.cross(r_vec, F_vec)   # cross product gives direction
print(tau)  # [0, 5, 0] — 5 N·m around the y-axis

Bug 2: Using τ = Iα without checking that I is constant.

# WRONG (if I changes)
L_final = L_initial + tau * delta_t  # only works if I is constant!

# RIGHT (always valid)
tau = dL_dt  # directly use definition
L_final = L_initial + tau * delta_t  # works even if I changes
omega_final = L_final / I_final      # then divide by the new I

Bug 3: Confusing "no friction" with "conserved angular momentum."

# WRONG reasoning:
# A wheel slides on a frictionless surface; therefore L is conserved.
# FALSE! Friction at the axle or an external force can still create torque.

# RIGHT reasoning:
# Check: are there ANY external torques about the center of mass?
# If yes, L changes. If no, L is constant.

Bug 4: Applying τ = Iα to a multi-object system.

# WRONG
# Two gears mesh. You find I_total and α_total and think you can use τ = Iα.
# The internal torques matter; you can't just add moments of inertia.

# RIGHT
# Apply τ = Iα to EACH gear separately, accounting for the force
# (and thus torque) one exerts on the other.

4.3 When to Use What (Decision Matrix)

5. Now Try It

Exercise: A student sits on a rotating stool holding two 2 kg weights at arm's length (r = 0.8 m from the axis). They're spinning at ω₀ = 2 rad/s. The student's moment of inertia (without weights) is I_s = 1.5 kg·m². You may treat each weight as a point mass.

1. Calculate the initial total angular momentum L₀.
2. The student pulls the weights in to r = 0.3 m. Assuming no external torque, calculate the final angular velocity ω_f.
3. Verify that angular momentum is conserved by computing L_f and checking that L_f = L₀.

Hint: I_weights = 2 × m × r², and this changes when r changes.

Success looks like: You get a specific numerical value for ω_f (should be around 6.5–7 rad/s), and you show that L_initial and L_final match when you recalculate I after the radius shrinks.

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