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Rotational Motion & Torque cheat sheet - grade 11-12

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Physics Grade 11-12

Rotational Motion & Torque Cheat Sheet

A printable reference covering angular kinematics, torque, moment of inertia, rotational dynamics, angular momentum, and rolling motion for grades 11-12.

TopicsRotationTorqueForcesMotionNewton's LawsMomentumPhysicsStudy Guide
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Rotational motion describes how objects spin, rotate, roll, and turn about an axis. This cheat sheet helps students connect linear motion ideas to angular quantities such as angular velocity, angular acceleration, torque, and moment of inertia. It is useful for solving problems involving wheels, pulleys, disks, rods, doors, gears, and rotating systems in equilibrium or acceleration.

The most important idea is that rotation depends not only on force, but also on where and how the force is applied. Torque follows τ=rFsin⁡θ\tau = rF\sin\theta, rotational dynamics follows ∑τ=Iα\sum \tau = I\alpha, and angular momentum follows L=IωL = I\omega. Rolling motion combines translation and rotation using relationships such as v=rωv = r\omega and a=rαa = r\alpha.

Key Facts

  • Angular displacement, angular velocity, and angular acceleration are related by ω=ΔθΔt\omega = \frac{\Delta \theta}{\Delta t} and α=ΔωΔt\alpha = \frac{\Delta \omega}{\Delta t}.
  • For constant angular acceleration, the rotational kinematics equations include ωf=ωi+αt\omega_f = \omega_i + \alpha t and θ=ωit+12αt2\theta = \omega_i t + \frac{1}{2}\alpha t^2.
  • Tangential speed and angular speed are related by v=rωv = r\omega, where rr is the distance from the rotation axis.
  • Tangential acceleration and angular acceleration are related by at=rαa_t = r\alpha, while centripetal acceleration is ac=rω2=v2ra_c = r\omega^2 = \frac{v^2}{r}.
  • Torque is given by τ=rFsin⁡θ\tau = rF\sin\theta, where θ\theta is the angle between the lever arm and the applied force.
  • Newton's second law for rotation is ∑τ=Iα\sum \tau = I\alpha, where II is the moment of inertia and α\alpha is angular acceleration.
  • Rotational kinetic energy is Krot=12Iω2K_{rot} = \frac{1}{2}I\omega^2, and total kinetic energy for rolling without slipping is K=12mv2+12Iω2K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2.
  • Angular momentum is L=IωL = I\omega for a rigid body, and it is conserved when the net external torque is zero, so Li=LfL_i = L_f.

Vocabulary

Angular Displacement
Angular displacement is the change in rotational position, usually measured in radians as Δθ\Delta \theta.
Angular Velocity
Angular velocity is the rate of change of angular displacement, given by ω=ΔθΔt\omega = \frac{\Delta \theta}{\Delta t}.
Torque
Torque is the rotational effect of a force and is calculated with τ=rFsin⁡θ\tau = rF\sin\theta.
Moment of Inertia
Moment of inertia is a measure of an object's resistance to angular acceleration, written as I=∑mr2I = \sum mr^2 for point masses.
Angular Momentum
Angular momentum is rotational momentum, given by L=IωL = I\omega for a rigid rotating body.
Rolling Without Slipping
Rolling without slipping occurs when the contact point is instantaneously at rest and the motion satisfies v=rωv = r\omega.

Common Mistakes to Avoid

  • Using force instead of torque: This is wrong because rotation depends on both the force and the lever arm, so the correct quantity is τ=rFsin⁡θ\tau = rF\sin\theta.
  • Forgetting the angle in torque: This is wrong because only the perpendicular component of force causes rotation, so τ\tau is not always equal to rFrF.
  • Using mass instead of moment of inertia: This is wrong because rotational acceleration depends on mass distribution, so ∑τ=Iα\sum \tau = I\alpha uses II, not just mm.
  • Mixing degrees and radians in angular equations: This is wrong because formulas such as v=rωv = r\omega and at=rαa_t = r\alpha require angles in radians.
  • Ignoring rotational kinetic energy in rolling problems: This is wrong because a rolling object has both translational energy and rotational energy, so K=12mv2+12Iω2K = \frac{1}{2}mv^2 + \frac{1}{2}I\omega^2.

Practice Questions

  1. 1 A wheel starts from rest and has angular acceleration α=4.0 rad/s2\alpha = 4.0\,\text{rad/s}^2 for t=3.0 st = 3.0\,\text{s}. Find its final angular velocity ωf\omega_f.
  2. 2 A force of 25 N25\,\text{N} is applied perpendicular to a wrench at a distance r=0.30 mr = 0.30\,\text{m} from the pivot. Find the torque τ\tau.
  3. 3 A solid disk has I=0.50 kg⋅m2I = 0.50\,\text{kg}\cdot\text{m}^2 and angular speed ω=12 rad/s\omega = 12\,\text{rad/s}. Find its rotational kinetic energy KrotK_{rot}.
  4. 4 Two equal forces are applied to a door, one near the hinge and one near the handle. Explain which force creates the larger torque and why.

Understanding Rotational Motion & Torque

A useful first step is choosing the axis and keeping one sign convention throughout a problem. Rotation can be called positive in the counterclockwise direction and negative in the clockwise direction, as long as the choice stays fixed. Angular quantities describe the motion of the whole object, but each point on that object can have a different linear speed.

A point near the rim of a bicycle wheel moves much faster than a point near the hub. Radians are important because they connect arc length directly to radius.

One full turn contains two pi radians. Students often lose marks by mixing revolutions, degrees, and radians, or by treating the distance traveled by a rim point as the displacement of the wheel's center.

Torque is about turning effectiveness, not simply force size. A small force applied far from a hinge can turn a door more easily than a large force applied close to the hinge. Only the part of a force that acts sideways to the lever arm produces turning.

A force aimed directly toward the pivot has no turning effect, even if it is large. This explains why a wrench works best when pushed at its end and at right angles to its handle. In equilibrium problems, every force needs a clear line of action.

Choosing the pivot cleverly can remove unknown forces from the calculation because forces through the pivot produce zero torque. Balance requires no net turning tendency, even when several individual torques are present.

Moment of inertia measures how strongly an object resists a change in its spin. It depends on mass, shape, and the chosen axis. Mass placed farther from the axis has a much larger effect than mass close to it.

A hoop and a solid disk with equal mass and radius do not respond the same way to the same torque. The hoop is harder to speed up because more of its mass lies near the rim. The axis matters too.

A ruler is easier to rotate about its center than about one end. When an object is made from several parts, find the contribution of each part about the same axis before combining them. This idea appears in ceiling fans, flywheels, playground roundabouts, and vehicle wheels.

Rolling without slipping has a subtle contact point. At any instant, the point touching the ground is temporarily at rest relative to the ground, while the center of the object keeps moving. Static friction can provide the torque needed for rolling, especially on a slope or when a wheel starts moving.

In ideal rolling on a fixed surface, static friction does no energy transfer because the contact point does not slide. If a wheel skids, kinetic friction converts some mechanical energy into thermal energy. Objects with different shapes race down an incline differently because their energy is divided differently between forward motion and spin.

Angular momentum gives another powerful viewpoint. A spinning skater speeds up by pulling mass inward, while a diver rotates faster during a tuck. These changes occur with little external torque, so the total angular momentum remains unchanged.