DPST1021 Chap.7 Rotation, Torque and Angular Momentum
Rotation, Torque and Angular Momentum
Rotation closes the Newtonian mechanics block of Physics 1A, just before gravitation. Lectures 14 and 15, given in week 5, set every translational quantity beside its rotational partner: angle with position, angular velocity with velocity, moment of inertia with mass, torque with force and angular momentum with linear momentum.
The course outline notes that parts of this topic are taught in laboratory and problem-solving classes rather than lectures, yet all of it can appear in Test 1.
The chapter is built as a comparison. A dictionary table lets you translate a familiar linear result into its rotational form, including the constant-angular-acceleration equations.
Careful attention goes to moment of inertia because, unlike mass, it depends on the chosen axis; you compute it from the particle sum and from the integral given on the formula sheet, and see why mass far from the axis dominates.
Torque is introduced as a cross product, then used in Newton's second law for rotation and in the two equilibrium conditions explored in the rigid-body laboratory.
The final part combines translational and rotational kinetic energy for objects that roll without slipping, explaining why a sphere beats a hoop down the same slope.
The topic finishes with angular momentum and its conservation, using spinning-stool and turntable situations where shrinking the moment of inertia raises the spin rate. Lab 4 on rotational inertia offers a direct measurement to set against these predictions.
What this chapter covers
- 01
Rotational analogues: pair angle, angular velocity, angular acceleration, inertia, torque and angular momentum with their translational partners and convert results between them.
- 02
Constant angular acceleration: apply the rotational versions of the kinematics equations, working in radians, and convert to revolutions only at the end.
- 03
Axis-dependent inertia: compute I from the particle sum or the integral of r squared dm, and explain why it changes when the axis moves.
- 04
Torque and rotational dynamics: evaluate torque as a cross product, find its direction with the right-hand rule, and solve problems with net torque equal to I alpha.
- 05
Rolling and rotational energy: split kinetic energy into translational and rotational shares using v equal to R omega, and compare speeds of different shapes.
- 06
Angular momentum: use L equal to I omega for rigid bodies and conserve it whenever the net external torque about the axis is zero.
Worked example · free
Spinning up a flywheel
- 1Newton's second law for rotation: α = τ/I = 1.5/0.25 = 6.0 rad/s², constant because the torque is constant.
- 1Angular speed: ω = ω0 + αt = 0 + 6.0(4.0) = 24 rad/s.
- 1Angle: θ = ½αt² = ½(6.0)(4.0)² = 48 rad, about 7.6 revolutions.
- 1Kinetic energy: K = ½Iω² = ½(0.25)(24)² = 72 J.
- 1Check with work: W = τθ = 1.5 × 48 = 72 J, matching the kinetic energy gained from rest.
Key terms
- Angular velocity
- The rate of change of angular position, measured in rad/s, with its direction along the rotation axis by the right-hand rule.
- Moment of inertia
- The rotational counterpart of mass, the sum of each mass element times the square of its distance from a stated axis.
- Torque
- The turning effect of a force about an axis, equal to the cross product of the position vector and the force.
- Lever arm
- The perpendicular distance from the axis to the line of action of a force, so that torque equals force times lever arm.
- Rolling without slipping
- Motion in which the contact point is momentarily at rest, so the centre speed equals radius times angular speed.
- Angular momentum
- A vector equal to the cross product of position and linear momentum, reducing to I times omega for a rigid body about a fixed axis.
Rotation, Torque and Angular Momentum FAQ
Why must angles be in radians for rotational formulas?
The link between arc length and angle, s equals r theta, only holds when theta is measured in radians, because the radian is defined as arc length divided by radius. Relations such as v equals r omega and tangential acceleration equals r alpha inherit that definition, so degrees or revolutions give answers off by constant factors.
Is a body in equilibrium if the net force on it is zero?
Not necessarily. Two equal and opposite forces acting along different lines form a couple: the net force is zero, yet the pair produces a torque and starts the body turning. Full static equilibrium also requires the net torque about every point to vanish, which is why balance problems need both a force equation and a torque equation.
How can a skater speed up without anyone pushing her?
Ice exerts almost no torque about her vertical axis, so her angular momentum stays fixed. Drawing her arms in reduces her moment of inertia, and the angular speed rises to compensate. Her kinetic energy actually increases, with the extra energy supplied by the muscular work she does drawing her arms inward.
Exam move
Make your own two-column dictionary of translational and rotational quantities and test yourself by converting a linear result into its rotational partner, then back again. Practise moment-of-inertia sums for a few point-mass arrangements with the axis in different places, and carry out at least one integral for a uniform rod so the formula sheet definition feels concrete.
For torque problems, write the position and force vectors in components and compute the cross product, then confirm the size with rF sin theta. Balance problems reward choosing the pivot at an unknown force. For rolling, express the total kinetic energy in terms of the centre speed before using energy conservation.
Finish with angular momentum questions where the moment of inertia changes, checking each time whether an external torque is present, and use the laboratory work on rigid bodies and rotational inertia to link the algebra to measurements.
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