Torque and Rotational Motion
This unit is Unit 2's Newton's-second-law playbook translated into rotational language: force becomes torque, mass becomes moment of inertia, and linear kinematics becomes angular kinematics, lesson for lesson. The payoff is that free-body-diagram intuition from dynamics transfers almost directly -- the new work is learning to compute torque (including sign and lever arm) and moment of inertia, which is exactly what Unit 8 then needs to handle rotational energy and angular momentum.
What you'll learn
- Calculate torque produced by a force applied at a given position and angle.
- Apply the rotational analog of Newton's second law (Στ = Iα).
- Determine moment of inertia for point masses and common shapes.
- Analyze static equilibrium problems involving torque balance.
- Relate angular kinematic quantities using rotational equations analogous to linear motion.
- Distinguish center of mass from center of gravity in torque problems.
- Solve problems with multiple torques acting on an extended object.
1. Torque
Torque is the rotational analog of force — it measures a force's tendency to cause rotation about an axis: \(\tau = rF\sin\theta\), where \(r\) is the distance from the axis to the point where the force is applied, and \(\theta\) is the angle between the force and the line from the axis to that point. Equivalently, \(\tau = F \cdot (\text{lever arm})\), where the lever arm is the perpendicular distance from the axis to the force's line of action.
Torque depends on *where* a force is applied, not just how large it is — a small force far from the axis, applied perpendicular to the lever, can produce more torque than a large force applied close to the axis or at a shallow angle.
2. Rotational Equilibrium
An object is in complete static equilibrium only when both \(\Sigma F = 0\) *and* \(\Sigma \tau = 0\) — zero net force alone doesn't prevent an object from spinning if the torques are unbalanced.
A powerful exam technique: you may choose *any* point as the pivot when writing the torque equation, even if the object doesn't actually rotate about that point. Choosing the pivot at the location of an unknown force makes that force's lever arm zero, eliminating it from the equation entirely — often the fastest way to solve for a different unknown without needing to solve a system of equations.
3. Moment of Inertia
Moment of inertia, \(I\), is the rotational analog of mass — it measures an object's resistance to changes in rotational motion. For point masses, \(I = \sum m_i r_i^2\), where \(r_i\) is each mass's distance from the axis. Because \(r\) is squared, mass far from the axis contributes disproportionately more than the same mass placed close to the axis — moment of inertia depends on *how* mass is distributed, not just how much there is.
Common shapes have standard formulas about their center: a solid disk or cylinder, \(I = \tfrac{1}{2}MR^2\); a thin hoop or ring, \(I = MR^2\); a solid sphere, \(I = \tfrac{2}{5}MR^2\); a thin rod about its center, \(I = \tfrac{1}{12}ML^2\).
4. Newton's Second Law for Rotation
The rotational analog of \(F = ma\) is \(\Sigma\tau = I\alpha\): net torque equals moment of inertia times angular acceleration. This is used whenever a pulley, disk, or rod has mass and cannot be treated as an idealized massless object.
5. Rotational Kinematics
Rotational kinematics mirrors linear kinematics exactly, with angular quantities replacing linear ones: \(\omega = \omega_0 + \alpha t\), \(\theta = \theta_0 + \omega_0 t + \tfrac{1}{2}\alpha t^2\), and \(\omega^2 = \omega_0^2 + 2\alpha\,\Delta\theta\). Every technique used for linear motion (choosing the right equation, using graphs) carries over directly.
6. Relating Linear and Angular Quantities
For a point on a rotating rigid body at distance \(r\) from the axis: linear speed \(v = \omega r\), tangential acceleration \(a_t = \alpha r\), and centripetal acceleration \(a_c = \omega^2 r = v^2/r\). These relationships connect the rotational description of a spinning object (in \(\omega\), \(\alpha\)) to the actual linear motion of any specific point on it — the rim of a wheel, the tip of a fan blade, and so on.
Key equations
- τ = rF sinθ — Any torque calculation
- Στ = Iα — Rotational analog of F=ma
- I = Σmr^2 (point masses); shape formulas (e.g. I = 1/2 MR^2 for a disk) — Calculating moment of inertia
- ω = ω0 + αt; θ = θ0 + ω0t + 1/2αt^2 — Rotational kinematics
- v = ωr; a_tangential = αr — Relating linear and rotational quantities