Torque Formulas: Comprehensive Reference for Students

Torque Formulas

Dimensional Formula of Torque

Dimensional Formula: [M¹ L² T⁻²]

Derivation:

• Torque (τ) = Force (F) × Distance (r)
• Force = Mass × Acceleration = [M] × [L T⁻²] = [M L T⁻²]
• Distance = [L]
• Therefore, τ = [M L T⁻²] × [L] = [M L² T⁻²]

Important Note:

The dimensional formula of torque is the same as that of work and energy, though they represent different physical quantities. This is why torque and energy have the same dimensions but different physical meanings.

Detailed Explanation of Key Formulas

1. Basic Torque Formula (τ = r × F × sin θ)

This is the most fundamental formula for calculating torque. The cross-product nature means:

• Maximum torque occurs when θ = 90° (sin 90° = 1)
• Zero torque when force is parallel to the lever arm (θ = 0° or 180°)
• The perpendicular distance (lever arm) is crucial for efficiency

Example: Opening a door requires less force when you push farther from the hinges.

2. Torque and Moment of Inertia (τ = I × α)

This is the rotational equivalent of Newton’s Second Law (F = ma):

• Larger moment of inertia requires more torque for the same angular acceleration
• Fundamental for analyzing rotating machinery and systems

Example: A heavy flywheel needs more torque to accelerate than a lighter one.

3. Power-Torque Relationship (τ = P/ω)

Critical for understanding engines and motors:

• At constant power, higher speed means lower torque
• At constant power, lower speed means higher torque
• Used extensively in automotive and mechanical engineering

Example: Cars have maximum torque at lower RPMs and maximum power at higher RPMs.

4. Induction Motor Torque Formula

For three-phase induction motors, the torque depends on:

• Supply voltage (V)
• Rotor resistance (R₂)
• Slip (s) – the difference between synchronous and actual speed
• Stator and rotor reactances (X₁, X₂)

The torque-slip characteristic shows:

• Starting torque at s = 1
• Maximum torque at intermediate slip
• Running torque at low slip values

Torque Formula Summary for Class 12

For CBSE and state board students, the essential formulas are:

1. τ = r × F × sin θ (Vector form: τ = r × F)
2. τ = I × α
3. τ = dL/dt where L = I × ω
4. Relation with angular momentum: L = r × p
5. Work done by torque: W = τ × θ

Units and Conversions

Frequently Asked Questions about Torque

Q. What is the basic formula of torque?

The basic formula of torque is τ = r × F × sin θ, where τ is torque, r is the distance from the axis of rotation (lever arm), F is the applied force, and θ is the angle between the force vector and lever arm. When the force is perpendicular to the lever arm, the formula simplifies to τ = r × F.

Q. What is the dimensional formula of torque?

The dimensional formula of torque is [M¹ L² T⁻²]. This is derived from torque = force × distance, where force has dimensions [M L T⁻²] and distance has dimensions [L]. Interestingly, this dimensional formula is identical to that of work and energy, though they represent different physical quantities.

Q. Is the dimensional formula of torque the same as work?

Yes, the dimensional formula of torque [M L² T⁻²] is the same as that of work and energy. However, torque is a vector quantity (has direction), while work is a scalar quantity. Additionally, torque is measured in Newton-meters (N·m), while energy is measured in Joules (J), even though they have the same dimensions.

Q. What is the relationship between torque and angular acceleration?

The relationship is given by τ = I × α, where τ is torque, I is the moment of inertia, and α is angular acceleration. This is the rotational analog of Newton’s second law (F = ma). It shows that torque is directly proportional to angular acceleration when moment of inertia is constant.

Q. How do you calculate torque in an electric motor?

For electric motors, torque can be calculated using τ = (P × 60)/(2πN), where P is power in watts and N is speed in RPM. For induction motors specifically, the torque formula is τ = (3 × V² × R₂)/(ω_s × [(R₁ + R₂/s)² + (X₁ + X₂)²]), which accounts for electrical parameters like voltage, resistance, slip, and reactance.

Q. What is the formula for torque in terms of power?

The formula relating torque to power is τ = P/ω, where P is power (in watts) and ω is angular velocity (in rad/s). Alternatively, for rotational speed in RPM: P = (2πNτ)/60. This relationship is fundamental in understanding engine and motor performance.

Q. How is torque different from force?

Force causes linear acceleration (F = ma), while torque causes rotational acceleration (τ = Iα). Force is measured in Newtons (N), while torque is measured in Newton-meters (N·m). Torque depends on both the magnitude of force and the distance from the axis of rotation, whereas force is independent of position.

Q. What is the torque formula for Class 12 physics?

For Class 12, the key torque formulas are:

• Basic: τ = r × F × sin θ (or vector form: τ = r × F)
• Rotational dynamics: τ = I × α
• Angular momentum: τ = dL/dt
• Work-energy: W = τθ These formulas cover the CBSE and state board curriculum comprehensively.

Q. What is the maximum torque formula for an induction motor?

The maximum torque (also called pull-out torque or breakdown torque) of an induction motor is given by: τ_max = (3 × V²)/(2ω_s × [R₁ + √(R₁² + (X₁ + X₂)²)]). This maximum occurs at a specific slip value and represents the highest torque the motor can produce before stalling.

Q. Can torque be negative?

Yes, torque can be negative depending on the chosen sign convention. By convention, counterclockwise rotation is considered positive torque, and clockwise rotation is negative torque. A negative torque indicates that the force tends to rotate the object in the opposite direction to the chosen positive direction. Net torque is the algebraic sum of all individual torques acting on a body.

Concepts for Students

Direction of Torque (Right-Hand Rule)

• Point fingers in direction of r (position vector)
• Curl fingers toward F (force vector)
• Thumb points in direction of torque vector

Equilibrium Conditions

• Translational equilibrium: ΣF = 0
• Rotational equilibrium: Στ = 0

Applications of Torque

1. Wrenches and spanners (longer handles = more torque)
2. Door handles (placed far from hinges)
3. Electric motors and generators
4. Gear systems and transmissions
5. Seesaws and levers
6. Steering wheels
7. Gyroscopes and spinning tops