Torque in Physics – Definition, Formula, Examples, and Applications

Torque is a fundamental concept in physics and engineering that describes how a force causes an object to rotate. Just as force causes linear acceleration, torque causes angular acceleration. Torque plays a vital role in simple machines, engines, robotics, biomechanics, and any system involving rotational motion.

Key Takeaways: Torque

• Torque is the rotational equivalent of force.
• It measures the tendency of a force to rotate an object about an axis or pivot point.
• The symbol for torque is τ (Greek letter tau).
• The SI unit of torque is the newton-meter (N·m).
• Torque depends on the magnitude of the force, the distance from the axis of rotation, and the angle at which the force is applied.
• Torque is sometimes called a moment, especially in engineering contexts.
• There are static (not causing angular acceleration) and dynamic (causing angular acceleration) torques.
• Torque multipliers amplify torque using levers, gears, or hydraulic systems.
• In engines, torque relates to rotational power output and affects acceleration.
• In robotics, torque determines how much load a motor can rotate or lift.

What Is Torque?

Torque is a measure of the force that causes an object to rotate about an axis. It is a vector quantity with both magnitude and direction. Torque is not just about how hard you push or pull, but where and how you apply the force.

Definition:

Torque is the rotational effect of a force applied at a distance from an axis of rotation.

Symbol:

The standard symbol for torque is τ (tau).

SI Unit:

The SI unit of torque is the newton-meter (N·m).
1 N·m = 1 kg·m²/s²

Torque vs Moment: Are They the Same?

In many contexts, torque and moment mean the same thing, which is the turning effect of a force. However, distinctions exist:

• In physics, “torque” usually refers to rotational force causing angular acceleration.
• In engineering and mechanics, the term moment is often used, especially for static systems.
• “Moment of force” is a broader term and may refer to different types (e.g., bending moment, moment of inertia).

Etymology and History:

• The word torque comes from the Latin torquere, meaning “to twist.”
• The term moment comes from momentum, reflecting its historical association with leverage and rotational influence.

Torque and Related Physical Quantities

There are several physical quantities that interact with torque:

Torque Formula and Explanation

The general equation for torque is:

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

Where:

• τ is the torque
• r is the lever arm (distance from the axis of rotation to the point of force application)
• F is the applied force
• θ is the angle between the force vector and lever arm

Special Cases

• When the force is perpendicular to the lever arm, $\sin(90°) = 1$, so:
  
  $$\tau = r \cdot F$$
• If the force is applied through the axis, there is no torque:
  
  $$\theta = 0° \Rightarrow \tau = 0$$

Example Physics Problem

Problem: A person uses a wrench to loosen a bolt by applying a force of 50 N at the end of a 0.3 m handle. The force is applied perpendicular to the handle. What is the torque?

Solution:
$$\tau = r \cdot F \cdot \sin(\theta)$$$$\tau = 0.3\ \text{m} \times 50\ \text{N} \times \sin(90^\circ)$$

τ = 15 N·m

Answer: 15 newton-meters of torque.

Static vs Dynamic Torque

Torque is either static or dynamic, depending on whether the object is rotating or held in place. Understanding the difference is important in engineering, construction, robotics, and physics experiments, where either maintaining position or creating motion is the goal.

Static torque refers to torque that does not result in angular motion. It occurs when forces are applied in such a way that the object stays in equilibrium. Even though the object isn’t turning, torque is still present. For example, holding a door open without moving it requires static torque to resist the door’s weight and hinge forces.

Dynamic torque, in contrast, causes an object to rotate or accelerate angularly. This is the torque you see when an engine powers a crankshaft, a gymnast spins, or a windmill blade turns in the wind. It’s associated with energy transfer, motion, and change in rotational velocity.

What Is a Torque Multiplier?

A torque multiplier is a mechanical device or arrangement that increases torque output from a given input. It is useful when high torque is needed but input force is limited.

Methods of Torque Multiplication

• Levers: Increasing the lever arm increases torque (e.g., using a longer wrench).
• Gear systems: A gear reduction system increases output torque while decreasing speed.
• Hydraulic multipliers: Use pressurized fluid to deliver amplified rotational force.
• Planetary gear sets: Common in tools and vehicles to adjust torque and speed.

Torque in Engines

Engine torque refers to the rotational force the engine produces at the crankshaft. It directly affects a vehicle’s acceleration and towing capability.

Key Concepts

• Higher torque = more rotational force = stronger push to move the car.
• Horsepower depends on both torque and engine speed (RPM):

$$\text{Horsepower} = \frac{\text{Torque (lb-ft)} \times \text{RPM}}{5252}$$

• Torque is highest at low to mid RPMs in most combustion engines.
• Performance vehicles often aim for a wide torque curve for flexibility.

Torque in Robotics

In robotics, torque is critical for determining a motor’s ability to rotate joints or move limbs.

Applications

• Servo motors are chosen based on required torque to lift loads.
• Joint torque affects movement smoothness, stability, and payload handling.
• Robots often require precise torque control to avoid overloading parts or causing mechanical failure.

Torque also plays a role in inverse kinematics, force feedback, and grip strength in robotic manipulators.

Torque and Rotational Equilibrium

An object is in rotational equilibrium when the net torque acting on it is zero. In this state, the object is either not rotating at all or is rotating at a constant angular velocity.
$$\sum \tau = 0$$

This principle is crucial in statics, where stability is the goal, such as in bridges, beams, or seesaws.

Vector Nature of Torque

Torque is a vector quantity, which means it has both magnitude and direction. The direction of the torque vector depends on the orientation of the force and the position vector (lever arm).
$$\vec{\tau} = \vec{r} \times \vec{F}$$

Use the right-hand rule to find the torque direction: point your fingers along $\vec{r}$, curl toward $\vec{F}$, and your thumb shows the torque vector direction (into or out of the plane).

Torque and Angular Momentum

Torque is the time rate of change of angular momentum:
$$\vec{\tau} = \frac{d\vec{L}}{dt}$$

Where $\vec{L}$ is angular momentum. This forms the rotational analog of Newton’s second law. In the absence of external torque, angular momentum remains conserved.

Biological Applications of Torque

Torque is fundamental to human movement. Muscles apply forces to bones at a distance from joints, creating rotational motion.

Examples:

• Biceps create torque around the elbow joint.
• Quadriceps generate torque to extend the knee.
• Calculating joint torques helps in sports training and injury prevention.

Measuring Torque

Several common instruments measure torque:

• Torque wrench: Measures applied torque in bolts and mechanical fasteners.
• Dynamometer: Measures torque and power output from engines.
• Strain gauges: Detect small deformations in rotating parts.
• Rotary torque sensors: Used in robotics and machinery to monitor torque in real time.

Torque in Simple Machines

Many simple machines rely on torque to increase force or mechanical advantage.

Examples:

• Levers use torque around a pivot.
• Gears transfer torque with adjustable speed ratios.
• Pulleys redirect force and distribute torque in lifting systems.
• Wheel and axle systems convert torque to motion in tools and vehicles.

Unit Conversions and Non-SI Units

Use correct units to prevent over- or under-tightening, especially in safety-critical applications.

Common Misconceptions About Torque

• Torque and force are the same: Torque causes rotation; force causes linear motion.
• Longer levers always give more torque: Only if the force is applied perpendicular to the lever arm.
• High speed means high torque: Torque causes angular acceleration, not speed.
• Only engines generate torque: Any applied force at a distance from a pivot causes torque.
• Torque is always positive: Torque can be positive or negative, depending on direction.

FAQs About Torque

Q: Is torque a vector or scalar?
A: Torque is a vector with both magnitude and direction.

Q: Can torque be negative?
A: Yes. Clockwise torque is typically negative, counterclockwise is positive.

Q: What is net torque?
A: The total of all torques acting on a system. If net torque = 0, there’s no angular acceleration.

Q: How is torque different from work?
A: Torque causes rotation; work is energy transfer. Torque × angle = work in rotational systems.

Q: How is torque measured?
A: With torque wrenches, dynamometers, strain gauges, or sensors, depending on the application.

References and Further Reading

• Kleppner, Daniel; Kolenkow, Robert (1973). An Introduction to Mechanics. McGraw-Hill. ISBN 9780070350489.
• Knight, Randall; Jones, Brian; Field, Stuart (2016). College Physics: A Strategic Approach (3rd technology update ed.). Boston: Pearson. ISBN 9780134143323.
• Kumar, Shitij; Savur, Celal; Sahin, Ferat (2021). “Survey of Human–Robot Collaboration in Industrial Settings: Awareness, Intelligence, and Compliance”. IEEE Transactions on Systems, Man, and Cybernetics: Systems. 51: 280–297. doi:10.1109/TSMC.2020.3041231
• Serway, R. A.; Jewett, J. W. Jr. (2003). Physics for Scientists and Engineers (6th ed.). Brooks Cole. ISBN 0-534-40842-7.
• Tipler, Paul (2004). Physics for Scientists and Engineers: Mechanics, Oscillations and Waves, Thermodynamics (5th ed.). W. H. Freeman. ISBN 0-7167-0809-4.