Acceleration Down an Incline
For a symmetric rigid body rolling without slipping down a fixed incline, its centre acceleration is g sin theta divided by one plus I_cm/(MR squared), provided static friction can enforce the constraint.
Why this shows up in the exam
Race of rings, discs, and spheres · Travel-time calculations on inclines · Inferring inertia from rolling acceleration
Learn the idea
A rolling body's incline acceleration is reduced because gravity must build both translation and rotation. A rolling body's incline acceleration is reduced because gravity must build both translation and rotation. 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: More rotational inertia means less downhill acceleration.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- a_cm = g sin(theta)/(1 + I_cm/(M R²)) — Down-slope acceleration for pure rolling on a fixed incline.
- alpha = a_cm/R — Angular acceleration from the no-slip constraint.
How to approach it
- 1Write force and torque equations
- 2Use a=R alpha with I_cm
- 3Check friction direction and limiting value
Common slip-ups that cost marks
- •Assuming acceleration is g sin theta
- •Choosing the body with larger mass as faster
- •Ignoring whether static friction is sufficient
🌟 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.