Rotational Motion and Gravitation
What this domain tests
Moment of inertia and its theorems, torque and angular momentum, rolling motion, and gravitation: variation of g, orbital and escape speeds, Kepler's laws.
Core rules
- I = Σmr²; ring about its axis MR², disc MR²/2, solid sphere 2MR²/5, rod about its centre ML²/12.
- Parallel axis I = I_cm + Md²; perpendicular axis I_z = I_x + I_y for a plane lamina.
- τ = Iα and L = Iω, conserved when external torque is zero (a skater pulling the arms in spins faster).
- Rolling without slipping: v = ωR; total KE = ½Mv²(1 + I/MR²). Rolling down an incline, a solid sphere arrives first, then a disc, then a ring.
- g_h = g(R/(R+h))²; g_d = g(1 − d/R); orbital v = √(GM/r); escape v_e = √(2GM/R) = √2 times orbital speed at the surface, independent of mass and direction.
Worked example
Compare the speeds at the bottom of an incline of height h for a solid sphere and a ring, both rolling from rest. mgh = ½mv²(1 + I/mR²). Sphere: I/mR² = 2/5, so v = √(10gh/7). Ring: I/mR² = 1, so v = √(gh). The sphere is faster because a larger share of its energy goes into translation.
Traps that cost marks
- Using a moment of inertia taken about the wrong axis.
- Writing v = ωR for a body that is slipping.
- Thinking escape speed depends on the projectile's mass or launch angle.
- Forgetting that g is maximum at the surface and zero at the centre.