Work and Torque

Work and Torque: Vectors in Motion

Overview

In physics, vectors represent the mechanics of how the world moves. This lesson explores two critical applications: Work (the scalar product) and Torque (the vector product).

Understanding these allows you to predict how much energy is transferred and how effectively a force can cause rotation.

Welcome! Today we bridge the gap between abstract math and physical reality. On one side, we have linear motion and energy, known as Work. On the other, we have rotational influence, known as Torque. Both rely on vector multiplication, but they produce very different results. Work uses the dot product to measure energy transfer. It tells us how much force actually contributes to moving an object. Torque uses the cross product to measure rotational tendency. It tells us how effectively a force can turn a bolt or a door.

Work is the dot product of force and displacement.
Torque is the cross product of lever arm and force.
Work is a scalar; Torque is a vector.

Work: The Power of Projection

The Dot Product in Action

Mechanical work occurs when a force causes an object to move. It only counts the component of force that acts along the direction of motion.

Formula: $W = \vec{F} \cdot \vec{d} = |\vec{F}||\vec{d}|\cos(\theta)$
Units: Joules (J)

Let's look at Work. Imagine pulling this crate. The displacement vector is horizontal. As you change the angle of your pull, notice how the horizontal component changes. Because we use the dot product, only the force parallel to the movement counts. If you pull straight up at 90 degrees, the cosine is zero, and you do no work on the horizontal movement!

Work is a scalar quantity.
The cosine of the angle determines the effective force.
Work is zero when force is perpendicular to displacement.

Torque: The Physics of Rotation

The Cross Product in Action

Torque measures the tendency of a force to rotate an object about a pivot. It is the cross product of the lever arm (r) and the force (F).

Formula: $\vec{\tau} = \vec{r} \times \vec{F}$
Magnitude: $|\vec{\tau}| = |\vec{r}||\vec{F}|\sin(\theta)$

Now, let's turn to Torque. Here we have a wrench. The lever arm, vector r, goes from the bolt to your hand. When you pull the handle, you create torque. Unlike work, torque uses the sine of the angle. If you pull directly away from the bolt at 0 degrees, the sine is zero, and the bolt won't budge. You get the most 'swing' at a perpendicular 90-degree angle.

Torque is a vector quantity.
Maximum torque occurs at 90 degrees.
The direction is determined by the Right-Hand Rule.

The Right-Hand Rule

Finding the Direction

Since torque is a vector, it has a direction. We use the Right-Hand Rule (RHR) to find it.

Point fingers along r.
Curl fingers toward F.
Your thumb points toward torque.

Torque isn't just a number; it's a vector pointing in space. Let's practice the Right-Hand Rule. Click the steps to see how the torque vector emerges from the interaction of the lever and the force. First, align your hand with the lever arm, vector r. Next, curl your fingers in the direction the force is pushing. Your thumb now points in the direction of the torque vector!

Torque is perpendicular to both r and F.
Order matters: r cross F is not the same as F cross r.

Avoid the 'Zero' Trap

Test your intuition. Drag the force vector to find the angles that result in zero work or zero torque.

Engineers must know when forces are wasted. Drag the force arrows on both the crate and the wrench to reach the 'Zero' target. Watch how the math changes as you move. Correct! At 90 degrees, the force is perpendicular to the motion. No work is done. Exactly! When you pull parallel to the lever arm, there is no rotational component. Torque is zero.

Work = 0 when Force ⊥ Displacement.
Torque = 0 when Force ∥ Lever Arm.

The Stubborn Bolt Challenge

A technician is pulling a 0.5m wrench with a force of 100N at an angle of 30° to the handle. Diagnose the issue: Is this efficient? How can they maximize the torque?

Analyze the scenario described. Write a short explanation of how much torque is being produced and what the technician should change to be more efficient.

Identify r and F.
Apply the sine component.
Suggest optimization (90 degrees).