Torque and Rotational Equilibrium

🔧 Torque — The Rotational Cousin of Force

Part 1 of 7 — Torque: $\tau = rF\sin\theta$

You know that forces cause objects to accelerate in a straight line. But what causes objects to rotate? The answer is torque — a quantity that measures how effectively a force can cause rotation about a pivot point.

Think about opening a door: pushing near the hinge barely moves it, but pushing at the handle swings it easily. Same force, different torque.

Defining Torque

Torque ($\tau$, the Greek letter "tau") is defined as:

$\tau = rF\sin\theta$

Where:

• $r$ = distance from the axis of rotation (pivot) to where the force is applied
• $F$ = magnitude of the applied force

Sign Convention for Torque

By convention:

This follows the right-hand rule: curl the fingers of your right hand in the direction of rotation — your thumb points in the direction of the torque vector (out of the page for CCW, into the page for CW).

Example

A 50 N force is applied perpendicularly at a distance of 0.3 m from the pivot, causing clockwise rotation.

$\tau = -(0.3)(50)\sin 90° = -15 \text{ N·m}$

Torque Concept Check 🎯

Torque Calculation Drill 🧮

1. A 40 N force is applied perpendicularly ($\theta = 90°$) at a distance of 0.25 m from the pivot. What is the magnitude of the torque? (in N·m)

2. A 100 N force is applied at $\theta = 30°$ at a distance of 0.5 m from the pivot. What is the magnitude of the torque? (in N·m)

3. You want to produce a torque of 60 N·m by applying a perpendicular force at a distance of 0.4 m from the pivot. What force is needed? (in N)

Torque Fundamentals Review 🔍

Exit Quiz — Torque Basics ✅

📏 Lever Arm and Moment Arm

Part 2 of 7 — The Lever Arm Approach

In Part 1 we computed torque as $\tau = rF\sin\theta$. There is an equivalent — and often more intuitive — way to think about torque using the concept of the lever arm (also called the moment arm).

What Is the Lever Arm?

The lever arm (moment arm) $r_{\perp}$ is the from the axis of rotation to the of the force.

⚖️ Rotational Equilibrium

Part 3 of 7 — $\sum \tau = 0$

An object is in rotational equilibrium when it has no net torque acting on it. This means it is either not rotating at all, or rotating at a constant angular velocity (no angular acceleration).

The Condition for Rotational Equilibrium

$\sum \tau = 0$

This means the total clockwise torque equals the total counterclockwise torque:

$\sum {\tau }_{\text{CCW}}=$

🏗️ Static Equilibrium

Part 4 of 7 — $\sum F = 0$ AND $\sum \tau = 0$

An object in static equilibrium is completely at rest — not translating and not rotating. This requires TWO conditions to be satisfied simultaneously.

The Two Conditions for Static Equilibrium

Condition 1: Translational Equilibrium

$\sum F_x=0\text{and}$

🎢 Balancing Beams and See-Saw Problems

Part 5 of 7 — Classic Equilibrium Scenarios

See-saws, balance beams, and supported structures are the bread and butter of torque problems on the AP exam. Let's master the patterns!

The See-Saw Principle

For a see-saw balanced on a fulcrum:

$m_1 g \cdot d_1 = m_2 g \cdot d_2$

🛠️ Problem-Solving Workshop

Part 6 of 7 — Torque & Equilibrium Practice

Time to put everything together! This workshop focuses on systematic problem-solving for torque and equilibrium scenarios.

Master Problem-Solving Strategy

Step-by-Step Approach

1. Identify the system — What object(s) are in equilibrium?
2. Draw a free-body diagram — Show ALL forces at their actual points of application
3. Choose a pivot — Pick a smart point (where unknown forces act)
4. Set up equations:
   • $\sum F_x = 0$
   • $$

🎓 Synthesis & AP Review

Part 7 of 7 — Torque and Rotational Equilibrium

Let's bring together everything from this topic and practice the types of questions you'll see on the AP Physics 1 exam.

Key Concepts Summary

Torque

$\tau = rF\sin\theta = r_{\perp}F = rF_{\perp}$