Rolling Motion
a = g sinθ/(1 + k²) — 4 shapes race down an incline; sphere wins.
Key Notes
Rolling without slipping: a rotating body translates AND rotates, with v_CM = R·ω (rolling condition).
The point of contact with the ground is INSTANTANEOUSLY AT REST.
Total KE of a rolling body: K = ½Mv² + ½Iω². For rolling condition: K = ½Mv²(1 + I/MR²).
Rolling friction acts at contact point — small (compared to sliding friction) because the contact area doesn't slip.
Static friction at contact provides the torque for rolling — it does NO WORK (point of contact at rest).
On an incline: a = g·sin θ / (1 + I/MR²). Different bodies have different accelerations.
Solid sphere rolls fastest on an incline, hollow ball slower, then ring slowest — depends on I/MR² ratio.
Pure rolling: no energy lost to friction. Slipping: kinetic friction acts and energy is lost.
Formulas
Rolling condition
No slipping between contact and ground.
Total KE (rolling)
Combined translation + rotation.
Acceleration on incline (rolling)
Smaller for larger I/MR² (rotational inertia hogs energy).
Min coefficient of friction (rolling)
Below this μ, body slips instead of rolling.
Important Points
The CONTACT POINT in pure rolling is INSTANTANEOUSLY AT REST — counterintuitive but true.
Static friction at contact point does NO WORK because the point doesn't move during instantaneous contact.
Solid sphere (I = 2MR²/5) is fastest: a = (5/7)g·sin θ. Hollow sphere (I = 2MR²/3) slower. Ring (I = MR²) slowest: a = ½g·sin θ.
Sliding (frictionless) block on incline reaches bottom first — no rotational KE to share.
If you START a solid disk rolling with too little static friction, it SLIPS — and kinetic friction takes over (acts and does work).
Rolling resistance is real but small — comes from deformation at contact, not slipping.
Rolling Motion notes from sciphylab (also known as SciPhy, SciPhy Lab, SciPhy Labs, Physics Lab). Class 11 physics revision for JEE Mains, JEE Advanced, NEET UG, AP Physics 1/2/C, SAT, and CUET-UG.