Physics Formula for Torque

Physics Formula for Torque: A Complete Guide

Introduction

Torque is a fundamental concept in physics, particularly in the study of rotational dynamics. Just as force causes linear motion, torque causes rotational motion. Torque plays a crucial role in mechanical systems, from simple door hinges to complex machinery in automotive and aerospace engineering.

What is Torque?

Torque, also known as the moment of force, is a measure of the tendency of a force to rotate an object about an axis. It depends on two main factors: the magnitude of the force applied and the distance from the axis of rotation (also called the moment arm or lever arm).

Definition

Torque is defined as the product of the force applied and the perpendicular distance from the axis of rotation to the line of action of the force.

Symbol and Unit

Torque is commonly denoted by the Greek letter τ (tau). Its SI unit is the Newton-meter (N·m).

The Torque Formula

The basic formula for torque is:

$$ \tau = r \times F $$

Where:

τ = Torque (in Newton-meters, N·m)
r = Distance from the axis of rotation to the point where the force is applied (in meters)
F = Force applied (in Newtons)

If the force is not perpendicular to the lever arm, the formula becomes:

$$ \tau = r \cdot F \cdot \sin(\theta) $$

Where:

θ = The angle between the force vector and the lever arm

Understanding the Components

Lever Arm (r)

The lever arm is the distance from the axis of rotation to the point where the force is applied. The longer the lever arm, the greater the torque for the same amount of force.

Force (F)

This is the external effort applied to make the object rotate. It must be perpendicular to the lever arm for maximum torque.

Angle (θ)

The sine component, sin(θ), accounts for the angle between the force vector and the lever arm. When the angle is 90°, sin(θ) is 1, giving maximum torque.

Positive and Negative Torque

Torque has direction, just like force. In physics, torque can be either clockwise or counterclockwise:

Clockwise torque is generally considered negative.
Counterclockwise torque is generally considered positive.

This sign convention helps in analyzing complex systems involving multiple torques acting simultaneously.

Net Torque

When more than one torque acts on a body, the net torque is the algebraic sum of all the torques. The object’s rotational motion depends on this net torque.

$$ \tau_{\text{net}} = \sum \tau $$

If the net torque is zero, the object remains in rotational equilibrium — it either remains at rest or continues to rotate at constant angular velocity.

Torque and Rotational Motion

Torque is directly related to angular acceleration, similar to how force is related to linear acceleration. Newton’s Second Law for rotational motion is given by:

$$ \tau = I \cdot \alpha $$

Where:

I = Moment of inertia (rotational equivalent of mass)
α = Angular acceleration (in radians per second squared)

Example Problems

Example 1: Torque with Perpendicular Force

A wrench is used to loosen a bolt. The length of the wrench is 0.3 m, and a force of 50 N is applied perpendicular to the wrench. What is the torque?

Using the formula:

$$ \tau = r \cdot F = 0.3 \cdot 50 = 15 \, \text{N·m} $$

The torque applied is 15 N·m.

Example 2: Torque with an Angle

A person applies a force of 40 N at an angle of 60° to a lever arm that is 0.5 m long. Find the torque.

Using the angle formula:

$$ \tau = r \cdot F \cdot \sin(\theta) = 0.5 \cdot 40 \cdot \sin(60^\circ) $$

$$ \tau = 0.5 \cdot 40 \cdot 0.866 = 17.32 \, \text{N·m} $$

The torque is approximately 17.32 N·m.

Applications of Torque

Engineering

Torque is essential in engineering applications, including designing engines, turbines, and structural components that must withstand rotational forces.

Automobiles

In vehicles, torque is a measure of the engine's ability to do work. It determines the pulling power and acceleration capability.

Tools

Wrenches and screwdrivers utilize torque to turn bolts and screws. Torque wrenches are calibrated tools that apply a specific torque level.

Sports

In sports like golf or baseball, players generate torque through twisting motions to deliver powerful swings.

Rotational Equilibrium

An object is in rotational equilibrium when the sum of the torques acting on it is zero. This is a key principle in statics and is often used to analyze bridges, buildings, and cranes.

Torque vs. Work

Although torque and work both use the unit Newton-meter, they are not the same. Torque causes rotation, while work involves moving an object in the direction of a force.

Comparison Table

Property	Torque	Work
Cause	Rotation	Linear motion
Formula	τ = r × F	W = F × d
SI Unit	Newton-meter (N·m)	Joule (J)

Factors Affecting Torque

Force magnitude: Greater force results in greater torque.
Lever arm length: Longer lever arms amplify torque.
Angle of force: Torque is maximum when the force is perpendicular to the lever arm.

Detailed Look at the Moment of Inertia

To further understand torque in rotational systems, we need to explore the concept of moment of inertia (I). It is the rotational equivalent of mass and describes how much torque is needed to rotate an object about an axis. Moment of inertia depends on both the mass of the object and how that mass is distributed in relation to the axis of rotation.

Mathematically, the moment of inertia is given by:

$$ I = \sum m_i r_i^2 $$

Where:

m_i is the mass of the i-th particle in the object
r_i is the distance of the i-th particle from the axis of rotation

For continuous rigid bodies, integration is used instead of summation:

$$ I = \int r^2 \, dm $$

Different shapes have different standard moments of inertia. For example:

Solid disk about its center: $ I = \frac{1}{2}MR^2 $
Thin rod about center: $ I = \frac{1}{12}ML^2 $
Solid sphere about diameter: $ I = \frac{2}{5}MR^2 $

Conservation of Angular Momentum

In the absence of external torque, the angular momentum of a rotating system remains constant. This principle is known as the law of conservation of angular momentum, and it is a direct consequence of Newton’s laws in rotational form.

Angular momentum $ L $ is defined as:

$$ L = I \cdot \omega $$

Where:

I is the moment of inertia
ω is the angular velocity

If no external torque acts on the system:

$$ \frac{dL}{dt} = \tau = 0 \Rightarrow L = \text{constant} $$

Example: Figure Skater

A figure skater spinning with arms extended can pull in her arms to spin faster. When she reduces her moment of inertia by bringing in her arms, her angular velocity increases to conserve angular momentum.

Torque in Equilibrium Analysis

In static equilibrium problems, such as analyzing bridges or support beams, torque helps determine unknown forces or tensions. For a body to be in full equilibrium:

Net force must be zero: $ \sum F = 0 $
Net torque must also be zero: $ \sum \tau = 0 $

Example: See-Saw Problem

Consider a see-saw balanced at its midpoint. If a 30 kg child sits 2 meters from the pivot, how far from the pivot must a 20 kg child sit on the opposite side to balance it?

Let distance be d:

$$ (30)(9.8)(2) = (20)(9.8)(d) \Rightarrow 588 = 196d \Rightarrow d = 3 \, \text{m} $$

The 20 kg child must sit 3 meters away from the pivot to maintain balance.

Torque in Machinery and Rotational Power

Torque is closely tied to rotational power, especially in engines, motors, and turbines. Rotational or angular power is given by:

$$ P = \tau \cdot \omega $$

Where:

P is the power (in Watts)
τ is the torque (in Newton-meters)
ω is the angular velocity (in radians per second)

Practical Example: Car Engine

If a car engine delivers 200 N·m of torque at 3000 rpm, we can calculate the power:

Convert rpm to rad/s:

$$ \omega = \frac{2\pi \cdot 3000}{60} = 314.16 \, \text{rad/s} $$

Now calculate power:

$$ P = 200 \cdot 314.16 = 62832 \, \text{W} = 62.8 \, \text{kW} $$

So the engine delivers 62.8 kilowatts of rotational power.

Torque in Biomechanics

Torque also plays a major role in human movement and biomechanics. Muscles generate torque at joints to produce motion. For example, the torque exerted by the biceps muscle at the elbow allows lifting objects.

Example: Human Arm

Suppose a person holds a 5 kg weight in their hand with the arm extended 0.4 m from the elbow. The torque about the elbow joint is:

$$ \tau = r \cdot F = 0.4 \cdot (5 \cdot 9.8) = 0.4 \cdot 49 = 19.6 \, \text{N·m} $$

Measuring Torque

Torque can be measured using a torque wrench or a torque sensor. In industry and automotive settings, digital torque sensors are used for precision.

Torque Wrench

A torque wrench is calibrated to apply a specific torque, ensuring that bolts and nuts are not under- or over-tightened.

Digital Torque Sensors

These are used in machines to monitor torque in real time, improving safety and efficiency in operations like robotic arms, CNC machines, and electric motors.

Common Mistakes with Torque

Forgetting to use perpendicular force in torque calculation
Using incorrect units (e.g., forgetting to convert cm to meters)
Assuming torque equals force (they are related but not the same)
Mixing up clockwise and counterclockwise signs in net torque

Tips for Solving Torque Problems

Always draw a free-body diagram (FBD)
Choose a pivot point that simplifies calculations
Use consistent units
Apply trigonometry when forces are at an angle
Double-check direction of rotation (sign convention)

Conclusion (Extended)

Torque is a central concept in both theoretical and applied physics. From balancing beams and analyzing levers to powering car engines and understanding muscle movement, torque affects a vast array of systems. Its correct application allows engineers to design safer structures, develop more efficient machines, and optimize performance across many industries.

Whether in classical mechanics or advanced engineering, mastering torque equips students, researchers, and professionals with the tools to tackle real-world mechanical challenges. Understanding its relationship with moment of inertia, angular momentum, and equilibrium leads to deeper insights into how the physical world operates.

As you encounter problems involving rotation, remember that torque is the key that unlocks the motion. Grasping this concept empowers you to analyze and innovate across fields where rotational forces govern motion.