Angular Position, Velocity, and Acceleration
For rotation about a fixed axis, angular velocity is the time derivative of angular displacement and angular acceleration is the derivative of angular velocity, with signs set by one chosen rotational sense.
Why this shows up in the exam
Describing spinning wheels and shafts · Reading angular motion graphs · Connecting periodic rotation to revolutions
Learn the idea
Angular velocity and acceleration are time derivatives of angular position with a signed axis convention. Angular velocity and acceleration are time derivatives of angular position with a signed axis convention. 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: Theta differentiates to omega, then alpha.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- omega = dtheta/dt — Instantaneous angular velocity about the fixed axis.
- alpha = domega/dt = d2theta/dt2 — Instantaneous angular acceleration in the same sign convention.
How to approach it
- 1Set the axis and positive sense
- 2Differentiate or integrate with initial conditions
- 3Convert revolutions or rpm only after tracking units
Common slip-ups that cost marks
- •Mixing degrees and radians in calculus
- •Treating angular speed as signed angular velocity
- •Changing the positive rotation direction
🌟 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.