Torque Formula

The Formula

\tau = rF\sin(\theta) (distance times force times sine of angle)

When to use: How hard you're twisting something. Depends on force AND distance from pivot.

Quick Example

Opening a door: push far from the hinge (more torque), push near the hinge (less torque).

Notation

\tau (tau) is torque in newton-metres (N·m), r is the distance from the axis of rotation to the point of force application, F is the applied force, and \theta is the angle between \vec{r} and \vec{F}.

What This Formula Means

The rotational equivalent of force; a measure of how much a force tends to cause an object to rotate about an axis.

How hard you're twisting something. Depends on force AND distance from pivot.

Formal View

Torque is defined as the cross product of the position vector and the force vector: \vec{\tau} = \vec{r} \times \vec{F}, with magnitude \tau = rF\sin\theta. The net torque on a rigid body equals I\alpha, where I is the moment of inertia and \alpha is the angular acceleration.

Worked Examples

Example 1

easy
A force of 20 \text{ N} is applied at the end of a 0.5 \text{ m} wrench, perpendicular to the wrench. What is the torque produced?

Solution

  1. 1
    Torque is the rotational equivalent of force: \tau = rF\sin\theta.
  2. 2
    Since the force is perpendicular to the lever arm, \theta = 90° and \sin 90° = 1.
  3. 3
    \tau = rF = 0.5 \times 20 = 10 \text{ N m}

Answer

\tau = 10 \text{ N m}
Torque measures the tendency of a force to cause rotation. It depends on the force magnitude, the distance from the pivot point, and the angle between the force and the lever arm.

Example 2

medium
A 3 \text{ m} seesaw has a pivot at its center. A 30 \text{ kg} child sits at one end. How far from the center must a 45 \text{ kg} child sit to balance the seesaw? Use g = 9.8 \text{ m/s}^2.

Example 3

medium
A 30 N force is applied at the end of a 0.4 m wrench at 60\u00b0 to the handle. Calculate the torque.

Common Mistakes

  • Using the total length of the object instead of the perpendicular distance (lever arm) from the pivot to the line of action of the force.
  • Forgetting to include the \sin\theta factor when the force is not perpendicular to the lever arm, which overestimates the torque.
  • Confusing torque with force — a large force applied at the pivot produces zero torque because the lever arm is zero.

Why This Formula Matters

Torque governs every rotational system — from opening doors and tightening bolts to engine design and robotic arms. Understanding torque is critical for mechanical engineering and structural analysis.

Frequently Asked Questions

What is the Torque formula?

The rotational equivalent of force; a measure of how much a force tends to cause an object to rotate about an axis.

How do you use the Torque formula?

How hard you're twisting something. Depends on force AND distance from pivot.

What do the symbols mean in the Torque formula?

\tau (tau) is torque in newton-metres (N·m), r is the distance from the axis of rotation to the point of force application, F is the applied force, and \theta is the angle between \vec{r} and \vec{F}.

Why is the Torque formula important in Physics?

Torque governs every rotational system — from opening doors and tightening bolts to engine design and robotic arms. Understanding torque is critical for mechanical engineering and structural analysis.

What do students get wrong about Torque?

Torque is not force—same force at different distances produces different torques.

What should I learn before the Torque formula?

Before studying the Torque formula, you should understand: force, circular motion.

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