How Do You Calculate Work Done by Torque?
For JEE Main Physics, a thorough grasp of the concept of work done by torque is essential when analysing rotational systems. This topic links rotational motion to energy, letting you solve a wide range of problems involving pulleys, wheels, flywheels, and even electric motors. Mistaking torque-based work for linear work is a common source of error, so it’s critical to understand the definitions, mathematical formulation, and the stepwise derivation as presented in NCERT texts and Vedantu’s expert content.
Physical Meaning of Work Done by Torque
When a force acts tangentially on the edge of a rotating object, it generates torque, which tends to produce angular acceleration. If the object rotates through an angular displacement, the torque performs work. This idea is the rotational counterpart to linear work done, but with different variables—torque replaces force, and angular displacement substitutes for linear displacement.
Stepwise JEE Derivation: Work Done by Torque Formula
Let us derive the quantitative relation for work done by a constant torque acting on a rigid body over a certain angular displacement. This is a key part of JEE Main Physics theory, so all steps are shown:
(1) Work done by a force over a displacement is:
dW = **F** · **dr**
In circular (rotational) motion, consider a small angular displacement dθ at a distance r from the axis.
(2) The arc length ds corresponding to dθ is:
ds = r dθ
(3) The force acts tangentially, so work done for dθ is:
dW = F ds = F r dθ
(4) By definition, torque τ = F r
Substitute for F r:
dW = τ dθ
(5) For a finite rotation from θ_i to θ_f:
W = ∫θ_iθ_f τ dθ
If torque τ is constant:
W = τ (θ_f − θ_i) = τ θ, where θ is the net angular displacement in radians.
Thus, the work done by torque is W = τ θ.
Variables and SI Units in Work Done by Torque
In the equation for the work done by torque, each variable has a specific physical meaning and SI unit, which is vital for correct problem-solving at JEE Main level.
| Physical Quantity | Symbol | SI Unit |
|---|---|---|
| Torque | τ | Newton-metre (N·m) |
| Angular Displacement | θ | radian (rad) |
| Work Done by Torque | W | Joule (J) |
Example: Calculating Work Done by Torque in a JEE Scenario
Suppose a wheel experiences a constant torque of 12 N·m and rotates through an angle of π/4 radians. Inserting the values into W = τ θ:
W = 12 × (π/4) = 3π joules.
The work done by torque is 3π J.
Common Traps and Applications in JEE Main
JEE Main examiners often set questions that require you to distinguish between linear and rotational work concepts. Mixing up units or using degrees instead of radians for θ is a typical error. Negative work occurs when torque opposes the direction of angular displacement, as seen in braking mechanisms or frictional effects on a rotating shaft.
- Always express angular displacement in radians for calculation
- The formula W = τ θ applies only when torque is constant
- Work is zero if there is no rotation, even if torque is present
- Negative work signals opposing motion or retarding torques
- Understand energy transfer in pulleys, flywheels, and motors
Applying the rotational work formula is essential for systems like DC motors, rotational pulleys, and for analysing turning effects in competitive exams. For a deeper look at the principle itself, see Understanding Torque.
Comparing Work Done by Torque and Force: Tabular Overview
To master rotational mechanics, it is important to distinguish how work is calculated for rotational versus linear systems. The following table highlights the key differences:
| Aspect | Rotational Motion | Linear Motion |
|---|---|---|
| Quantity doing Work | Torque (τ) | Force (F) |
| Displacement Type | Angular Displacement (θ) | Linear Displacement (s) |
| Formula | W = τ θ | W = F s |
| SI Unit | Joule (J) | Joule (J) |
For exam revision on related topics, Vedantu’s Work Energy And Power Revision Notes offers reliable, JEE-aligned practice and concise concept summaries that help strengthen your problem-solving skills.
Making Work Done by Torque Second Nature for JEE
Consistent problem practice, careful attention to units and directions, and a strong grasp of the meaning behind each variable are crucial for mastering questions on work done by torque. As always, check the provided torque and angular displacement for constancy, and interpret the sign of your answer in the context of energy transfer. Revisiting anchor concepts like Work Energy And Power ensures that rotational mechanics feels logical and approachable under exam pressure, putting you ahead in JEE Main Physics.
FAQs on Understanding Work Done by Torque in Physics
1. What is work done by torque?
Work done by torque refers to the energy transferred when a force causes an object to rotate about an axis. It is calculated as the product of torque and the angular displacement:
- Work (W) = Torque (𝜏) × Angular displacement (θ)
- Units: Joules (J)
- Relevant for systems involving circular or rotational motion
2. How is the formula for work done by torque derived?
The formula for work done by torque is derived from the definition of work in rotational systems.
- Work done (W) = Torque (𝜏) × Angular displacement (θ), where θ is in radians
- Derived from the relationship: W = Force × Distance, considering the tangential component of force in circular motion
- Used when a rigid body rotates about a fixed axis under applied torque
3. What are the SI units for work done by torque?
The SI unit of work done by torque is the joule (J).
- Torque (𝜏) has units of newton metre (N·m)
- Angular displacement (θ) is measured in radians
- Thus, Work (W) = N·m × rad = Joule (J) since radians are dimensionless
4. Is torque a scalar or vector quantity?
Torque is a vector quantity because it has both magnitude and direction.
- The direction is determined using the right-hand rule
- Shown along the axis of rotation
- It's crucial in defining the direction of rotational effect
5. What is the relation between torque and work in rotational motion?
In rotational motion, work done is directly proportional to torque and angular displacement.
- Work (W) = Torque (𝜏) × Angular displacement (θ)
- If torque and angular displacement are in the same direction, work is positive
- Relevant in examples like engines, machines, and levers
6. Can work be done if the axis of rotation does not move?
Yes, work can be done by torque even if the axis of rotation is fixed, as long as there is angular displacement.
- For example, opening a door about its hinge does work through rotational movement
- If torque is applied but no movement occurs, then no work is done
7. What happens if the torque is perpendicular to the angular displacement?
If torque is perpendicular to the angular displacement, no work is done.
- Work (W) = τ × θ × cos(𝛼), where 𝛼 = angle between torque and displacement vectors
- If 𝛼 = 90°, cos(90°) = 0, so W = 0
- It highlights that only the component of torque in the direction of rotation does work
8. What is the physical significance of work done by torque?
The physical significance of work done by torque is that it represents the energy transferred to rotate an object.
- It helps calculate kinetic energy in rotational systems
- Essential for analyzing engines, motors, and rotating machinery
- Links rotational motion to energy concepts
9. What factors affect the work done by torque?
Three main factors affect work done by torque:
- Magnitude of torque (τ) applied
- Angular displacement (θ) covered
- Direction of torque relative to rotation
10. How is work done by torque applied in daily life?
Work done by torque is seen in many everyday activities requiring rotation.
- Turning a screwdriver or spanner
- Pedaling a bicycle
- Opening or closing a door
- Using a manual winch or steering wheel
11. What is the difference between torque and force?
Torque measures the tendency to rotate an object, while force causes linear motion.
- Force is measured in newtons (N) and acts in a straight line
- Torque is measured in newton-metres (N·m) and causes angular acceleration
- Torque = Force × Perpendicular distance from axis
12. How do you calculate the work done by variable torque?
For variable torque, work done is calculated using integration.
- Work (W) = ∫ τ dθ
- This is used when torque changes with respect to angle
- Often applied in complex machinery and engineering systems
