How do you derive the formula for work done by torque in rotational motion?
Understanding the work done by torque is essential in rotational mechanics for JEE Main. This idea links applied force in a circle to changes in rotational energy. Many real-world mechanisms, such as wheels and engines, use this principle. Mastery of its definitions, formulas, and typical mistakes will strengthen your command of this classic topic.
In rotational motion, when a force causes an object to rotate about an axis, it creates torque. The work done by torque measures how much energy is transferred as the object undergoes angular displacement. It relates directly to mechanical concepts like rotational work-energy theorem and is foundational for solving JEE Main physics problems.
Definition and Physical Concept of Work Done by Torque
The work done by torque is the energy imparted to or by a rotating body due to torque acting over an angular displacement. When constant torque is applied, the object rotates by a certain angle, and energy changes occur, typically seen as changes in rotational kinetic energy. For linear systems, force and displacement matter; for rotational ones, it’s torque and angular displacement.
In JEE-level problems, you’ll often be asked to find work done when a torque acts through a given angle. This scenario is common in machines, flywheels, electric motors, and magnetic dipoles in a field. The distinction between torque’s role in rotation and force’s role in translation is a classic comparison point for JEE questions.
Formula for Work Done by Torque and Key Variables
The fundamental formula for work done by torque is:
| Physical Quantity | Symbol | SI Unit |
|---|---|---|
| Torque | τ | N·m |
| Angular Displacement | θ | radian |
| Work Done by Torque | W | joule (J) |
W = τ × θ, if torque is constant and θ is in radians. If torque varies, use W = ∫ τ · dθ.
Here, τ (torque) is the turning effect of force, θ is angular displacement, and W is work done by torque. These definitions often appear in quick-reference tables and problems throughout rotational motion of a rigid body and torque formula pages.
Step-by-Step JEE Derivation of Work Done by Torque
- Recall the work done by a force: W = F · s, where s is linear displacement.
- Analogously, work in rotation is dW = τ dθ for a small angle dθ.
- Integrate both sides for finite rotation: W = ∫ τ · dθ.
- If τ is constant, this becomes W = τ θ.
- By the rotational work-energy theorem, W = ΔK.E. Here, ΔK.E. = ½ I (ω2 - ω02), where I is moment of inertia and ω is angular speed.
This derivation is a staple in practice questions and appears in nearly every rotational mechanics chapter in work, energy, and power and rotational motion resources on Vedantu.
Worked Example: Applying Work Done by Torque in a JEE Problem
A constant torque of 12 N·m acts on a flywheel, causing an angular displacement of π/4 radians. Calculate the work done by torque on the flywheel.
- Given: τ = 12 N·m; θ = π/4 radians
- Formula: W = τ × θ
- Substitute: W = 12 × (π/4) = 3π J
- Final Answer: 3.14 × 3 = 9.42 J
In a JEE Main context, answers should always use proper SI units and report the final value: work done is 9.42 J.
- Torque in magnetic field questions may use W = pE(cosθi - cosθf) for dipoles.
- Moment of inertia problems often relate torque-work to changes in rotational kinetic energy.
- Numerical problems sometimes require angular displacement to be found from angular acceleration data, connecting this principle to moment of inertia.
- Conceptual traps include mixing up force (linear work) and torque (rotational work).
Exam Tips, Mistakes, and Applications of Work Done by Torque
- The formula W = τ θ applies only for constant torque scenarios.
- Always use radians (not degrees) for θ in JEE calculations.
- Work done by torque is zero if there’s no angular displacement.
- Negative work indicates torque opposes motion, e.g., braking systems.
- Applications include DC motors, magnetic dipoles, and rotational pulleys.
- Compare carefully: force–displacement (linear work) vs torque–angular displacement (rotational work).
- Errors often arise from using degrees or from neglecting direction of torque/displacement.
- Refer to difference between work and power to clarify these distinctions.
Summary Table: Core Points on Work Done by Torque
| Aspect | Rotational Motion | Linear Motion |
|---|---|---|
| Quantity Doing Work | Torque (τ) | Force (F) |
| Displacement Type | Angular Displacement (θ) | Linear (s) |
| Basic Formula | W = τ θ | W = F s |
| SI Unit | Joule (J) | Joule (J) |
For last-minute revision, focus on the formula, conditions for its use, and units. Practising questions from rotational motion practice paper and work, energy, and power important questions can enhance grasp and speed.
JEE aspirants often make errors in the direction (sign) or mismatching units in these problems. Review of solved examples in work, energy, and power revision notes is strongly recommended. The Vedantu Physics team regularly updates these with exam-style problems and error-proofed solutions.
FAQs on Work Done by Torque – Formula, Derivation, and Applications
1. How do you calculate work done by torque?
Work done by torque is calculated by multiplying the constant torque (τ) applied to a body by the angular displacement (θ) it produces (in radians).
The formula is:
Work (W) = Torque (τ) × Angular displacement (θ)
- τ: Torque applied (in Nm)
- θ: Angular displacement (in radians)
2. How is work done and torque related?
Work done and torque are directly related in rotational motion, similar to how force and displacement are related in linear motion.
Specifically:
- Work done (W) is the product of torque (τ) and angular displacement (θ).
- If a varying torque is applied, work is found by integrating torque with respect to angle.
- This relationship forms the basis of the rotational work-energy theorem.
3. What is the formula for work done by torque?
The formula for work done by torque is: W = τ × θ, where W is work, τ is torque in newton-metre (Nm), and θ is angular displacement in radians.
For non-constant torque:
- Use W = ∫ τ dθ.
4. What is the rotational work-energy theorem?
The rotational work-energy theorem states that the work done by torque on a rigid body equals the change in its rotational kinetic energy.
- W = ΔK = (1/2)I(ω_2^2 - ω_1^2)
- I is moment of inertia
- ω is angular velocity
5. What are the SI units of rotational work and torque?
The SI unit of rotational work is the joule (J), and the SI unit of torque is newton-metre (Nm).
- Both rotational and linear work use joules
- Torque has the same dimensional formula as work, but represents rotation
6. What is the work done by a constant torque acting on a body rotating through an angle?
When a constant torque (τ) acts on a body and it rotates through an angle (θ) in radians, work done is:
- W = τ × θ
7. Derive the equation for work done by torque in rotational motion.
The work done by torque is derived using the definition of work in rotational systems.
- For a small angular displacement dθ, small work done, dW = τ dθ.
- Integrating over total angle gives: W = ∫ τ dθ = τ θ for constant torque.
8. How is the formula for torque in terms of work written?
Torque (τ) can be expressed in terms of work (W) and angular displacement (θ):
- τ = W / θ
9. Give an example problem: How much work is done by a torque of 5 Nm rotating a shaft through 90 degrees?
First, convert angular displacement to radians:
90° = (π/2) radians.
Work done, W = τ × θ = 5 Nm × (π/2) = (5π/2) Joules ≈ 7.85 J.
This is a standard Class 11–12 numerical type question.
10. What is the work done by torque in a magnetic field on a magnetic dipole?
The work done by torque on a magnetic dipole of moment M in a magnetic field B:
- W = MB (cos θ_1 – cos θ_2)
- M is magnetic dipole moment
- B is magnetic field strength
- θ_1 and θ_2 are initial and final angles
11. What is the dimensional formula of work done by torque?
The dimensional formula for work done by torque is [ML2T-2], the same as energy and linear work.
- M = mass
- L = length
- T = time
