Rolling Energy on Slopes and Tracks
For a rigid body rolling without slipping on a fixed surface with negligible dissipative resistance, gravitational potential energy converts into both translational and rotational kinetic energy.
Why this shows up in the exam
Rolling down or up inclines · Loop and track speed calculations · Comparing maximum heights of rolling bodies
Learn the idea
Static friction in ideal pure rolling does no work at a stationary contact, so mechanical energy can be conserved. Static friction in ideal pure rolling does no work at a stationary contact, so mechanical energy can be conserved. 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: Height pays for both glide and spin.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- M g Delta h = (1/2) M v² + (1/2) I_cm v²/R² — Energy conversion from rest through vertical drop Delta h.
- h_max = K_initial/(M g) — Rise height when rolling energy converts back to gravity and no loss occurs.
How to approach it
- 1Choose initial and final heights
- 2Write total rolling kinetic energy
- 3Check whether slipping or rolling resistance invalidates conservation
Common slip-ups that cost marks
- •Treating static friction as dissipative work
- •Omitting rotational energy
- •Using path length instead of vertical height
🌟 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.