Impact and Capture about a Pivot
For a short collision involving a pivoted rigid body, angular momentum about the pivot is conserved when other external angular impulses are negligible, while mechanical energy is generally not conserved for sticking or capture.
Why this shows up in the exam
Bullets embedding in rods · Particles sticking to rotating discs · Impulsive strikes on hinged bodies
Learn the idea
During a brief impact at a pivot, pivot impulse has zero torque about the pivot, often conserving angular momentum there. During a brief impact at a pivot, pivot impulse has zero torque about the pivot, often conserving angular momentum there. 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: At impact, choose the pivot to erase its unknown impulse torque.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- L_about_pivot,before = L_about_pivot,after — Impact relation when pivot angular impulse is zero.
- (r cross m v)_i = I_total omega_f — Particle capture by a body rotating about the pivot.
How to approach it
- 1Choose the pivot as angular-momentum origin
- 2Write pre-impact particle and body angular momenta
- 3Use final combined inertia, then compute any energy loss separately
Common slip-ups that cost marks
- •Conserving linear momentum despite a pivot impulse
- •Conserving kinetic energy for sticking
- •Using inertia that excludes the captured mass
🌟 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.