Constant and Variable Angular Kinematics
For fixed-axis rotation with constant angular acceleration, angular displacement, angular velocity, and time satisfy the standard uniformly accelerated equations; for variable acceleration they must be integrated instead.
Why this shows up in the exam
Flywheel run-up and braking · Counting revolutions during speed changes · Integrating prescribed angular acceleration
Learn the idea
Angular motion follows kinematic equations analogous to linear motion when angular acceleration is constant. Angular motion follows kinematic equations analogous to linear motion when angular acceleration is constant. Start from a clear axis, origin, body, and reference frame; the geometry and constraints then decide which rotational law is safe to use.
🧠 Memory hook: Use SUVAT only when alpha truly stays constant.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- omega = omega₀ + alpha t — Angular velocity for constant alpha.
- theta-theta₀ = omega₀ t + (1/2) alpha t² — Angular displacement for constant alpha.
- omega² = omega₀² + 2 alpha (theta-theta₀) — Time-eliminated relation for constant alpha.
- omega = omega₀ + integral alpha(t) dt — Required form when alpha varies with time.
How to approach it
- 1Identify whether alpha is constant
- 2Convert to radians and seconds
- 3Apply initial conditions and reject inconsistent signs
Common slip-ups that cost marks
- •Applying constant-alpha equations to alpha(t)
- •Forgetting initial angular position
- •Using rpm directly in SI equations
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
Masses 1 kg and 3 kg lie at x = 0 and x = 4 m. Find the x-coordinate of their centre of mass.
More from Motion of System of Particles and Rigid Body
Conservation of momentum and angular momentum
The total linear and angular momentum of a system remains constant in the absence of external forces or torques, including during collisions and rotational motion.
Moment of inertia and radius of gyration
Moment of inertia quantifies how mass is distributed with respect to an axis of rotation, and the radius of gyration is a measure related to this distribution.
Torque and rotational equilibrium
Torque is the rotational analogue of force, causing angular acceleration, and equilibrium occurs when the net torque on a body is zero.
Center of mass: definition and calculation
The center of mass is the point representing the mean position of the mass in a system, and can be calculated for discrete particles or continuous bodies.
Rotational kinematics and dynamics
Rotational kinematics describes the motion of rotating bodies, while dynamics relates torque, angular acceleration, and rotational kinetic energy.
Centre of Mass of Discrete Particles
For discrete particles, the centre-of-mass position is the vector sum of each mass times its position divided by total mass; this point governs translation even when the particles move relative to one another.