Torque and Rotational Motion
Torque and Rotational Motion
Definition: Torque is the rotational equivalent of force, causing an object to rotate about an axis; rotational motion describes how objects spin or revolve under such twisting forces.
How It Works
- Torque depends on three things: the magnitude of the applied force, the distance from the pivot point to where the force is applied (the lever arm), and the angle between the force and the lever arm.
- Maximum torque occurs when force is applied perpendicular to the lever arm; torque drops to zero if the force is applied directly along the lever arm (e.g., pushing straight into a door hinge line does nothing to open it).
- An object’s resistance to changes in rotation is its moment of inertia, which plays the same role for rotation that mass plays for linear motion in Newton’s second law.
- Moment of inertia depends not just on an object’s mass, but on how that mass is distributed relative to the axis of rotation, mass farther from the axis contributes disproportionately more.
- Angular momentum is the rotational analog of linear momentum, and it’s conserved in a system with no net external torque, exactly as linear momentum is conserved with no net external force.
- Rotational kinetic energy depends on both moment of inertia and angular velocity, letting a spinning object store and release energy, as in a flywheel.
- The right-hand rule gives torque and angular momentum a direction: curl your fingers in the direction of rotation, and your thumb points along the rotation axis.
Illustration
Under the Hood
τ = r × F = rF·sin(θ) (torque, cross product form)τ_net = I·α (rotational form of Newton's second law)L = I·ω (angular momentum)KE_rot = ½I·ω² (rotational kinetic energy)I = Σm·r² (moment of inertia, point masses)Worked Example 1: Torque from a wrench Given: a 0.30 m wrench applies 150 N perpendicular to a bolt. Step 1: apply τ = rF·sin(θ), with θ = 90°, so sin(90°) = 1. Step 2: τ = 0.30 × 150 × 1. Answer: τ = 45 N·m.
Worked Example 2: Torque at an angle Given: the same 0.30 m wrench and 150 N force, but applied at 40° from perpendicular to the handle (so 50° from the handle’s length). Step 1: apply τ = rF·sin(θ), where θ is measured from the lever arm, θ = 50°. Step 2: sin(50°) ≈ 0.766. Answer: τ = 0.30 × 150 × 0.766 ≈ 34.5 N·m, less than the perpendicular case.
Worked Example 3: Angular acceleration of a disk Given: a solid disk (I = 0.50 kg·m²) experiences a net torque of 4.0 N·m. Step 1: apply τ = I·α, so α = τ/I. Step 2: α = 4.0 / 0.50. Answer: α = 8.0 rad/s².
Worked Example 4: Conservation of angular momentum (figure skater) Given: a skater spins at 2.0 rad/s with arms out (I₁ = 4.0 kg·m²); she pulls her arms in, reducing I to 1.0 kg·m². Step 1: apply conservation of angular momentum, I₁ω₁ = I₂ω₂. Step 2: ω₂ = (I₁ω₁) / I₂ = (4.0 × 2.0) / 1.0. Answer: ω₂ = 8.0 rad/s, she spins four times faster after pulling her arms in.
Worked Example 5: Rotational kinetic energy of a flywheel Given: a flywheel with I = 2.0 kg·m² spins at ω = 100 rad/s. Step 1: apply KE_rot = ½Iω² = ½ × 2.0 × (100)². Step 2: KE_rot = 1.0 × 10,000. Answer: KE_rot = 10,000 J, stored energy the flywheel can release to smooth out power delivery.
Moment of Inertia by Shape
- Solid disk/cylinder about its center axis: I = ½MR², mass distributed close to the axis on average.
- Solid sphere about its center: I = (2/5)MR², even more mass concentrated near the axis.
- Thin hoop/ring about its center: I = MR², all mass at maximum radius, giving the largest moment of inertia for a given mass and radius.
- Thin rod about its center: I = (1/12)ML², compared to (1/3)ML² about one end, the same rod resists rotation less when spun about its middle.
- This is why a figure skater or diver curls into a tighter shape to spin faster: reducing the average radius of their mass reduces I, and conserved angular momentum forces ω to increase.
Why It Matters
- Torque governs everything from tightening a bolt with a wrench to how car and motorcycle engines deliver rotational power to the wheels.
- Gyroscopes use conservation of angular momentum to maintain orientation, critical in navigation systems, drones, and spacecraft attitude control.
- Wind turbine and engine designers must calculate torque curves to match power delivery to real-world load requirements.
- Understanding moment of inertia lets engineers design flywheels that store rotational energy efficiently for hybrid vehicles and grid energy storage.
- Sports biomechanics, from a golf swing to a diver’s tuck, is largely the applied physics of torque and angular momentum.
- Torque wrenches are calibrated tools used in manufacturing and auto repair to tighten bolts to precise specifications, preventing both under- and over-tightening.
- Robotics engineers size motor torque carefully so joints can lift and move rated loads without stalling.
Common Pitfalls
- Confusing torque with force: torque depends on where and at what angle a force is applied, not just its magnitude.
- Forgetting the angle term in τ = rF·sin(θ); applying force parallel to the lever arm produces zero torque, no matter how large the force.
- Assuming moment of inertia depends only on mass. It depends heavily on how that mass is distributed relative to the rotation axis, identical masses in different shapes have very different I.
- Mixing up angular velocity (ω, rad/s) with linear velocity (v, m/s); they’re related by v = ωr but are not interchangeable.
- Believing torque always causes rotation to start from rest. Torque causes angular acceleration; an object can have zero net torque while still spinning at constant angular velocity.
- Forgetting that angular momentum conservation requires zero net external torque, internal torques (like a skater’s own muscles) don’t violate the conservation law.
Comparison
| Linear Quantity | Rotational Analog | Relationship |
|---|---|---|
| Force (F) | Torque (τ) | τ = rF·sin(θ) |
| Mass (m) | Moment of inertia (I) | I = Σmr² |
| Velocity (v) | Angular velocity (ω) | v = ωr |
| Momentum (p = mv) | Angular momentum (L = Iω) | Analogous conservation law |
| Kinetic energy (½mv²) | Rotational KE (½Iω²) | Same energy form, rotational variables |
| Newton’s second law (F=ma) | Rotational second law (τ=Iα) | Direct rotational analog |
Example
Using a longer wrench to loosen a stuck bolt works because increasing the lever arm distance increases torque for the same applied force, making it far easier to overcome the bolt’s resistance. Figure skaters exploit conservation of angular momentum directly: pulling their arms and legs close to their body reduces their moment of inertia, and since angular momentum stays constant, their spin rate increases dramatically without any extra torque applied.
History
- Archimedes described the principle of the lever, an early form of torque reasoning, over 2,000 years ago, famously claiming he could move the Earth given a long enough lever and a place to stand.
- The formal vector treatment of torque as r × F developed alongside classical mechanics in the 18th and 19th centuries, building on Newton’s laws.
- Leonhard Euler extended Newtonian mechanics to rigid body rotation in the 1700s, formalizing moment of inertia and the rotational equations of motion still used today.
- Gyroscopes, which rely on angular momentum conservation, were rigorously studied by Léon Foucault in the 1850s and became essential to navigation within decades.
FAQ
Why is it easier to loosen a bolt with a longer wrench? Torque equals force times lever arm distance, so increasing the lever arm produces more torque for the same applied force, requiring less muscular effort to reach the torque needed to turn the bolt.
Does a spinning object need a constant torque to keep spinning? No. Once spinning, an object with no net external torque (like friction) continues rotating at constant angular velocity, exactly analogous to Newton’s first law for linear motion.
Why do tightrope walkers carry long poles? A long pole increases the moment of inertia about the walker’s forward-backward tipping axis, making it harder to start rotating (falling), and giving them more time and torque control to correct their balance.
Is torque the same thing as work or energy? No, though they share units that can look similar (N·m for torque, J for work/energy). Torque is a rotational force; it only does work when it acts through an angular displacement, W = τ·θ.
Why does a car engine’s peak torque matter more than horsepower for towing? Torque directly determines how much twisting force is available at the wheels to accelerate a heavy load, especially at low RPM, while horsepower reflects the rate of doing work and matters more for sustained high-speed performance.
Related Terms
Referenced by