Physical and Torsional Pendulums
For small angular displacement, a physical pendulum has period determined by pivot inertia and centre-of-mass offset, while a torsional pendulum is restored by torque proportional to twist angle.
Why this shows up in the exam
Compound pendulum timing · Measuring moment of inertia by oscillation · Torsion-wire experiments
Learn the idea
A rigid body's small oscillations depend on restoring torque and inertia about the suspension axis. A rigid body's small oscillations depend on restoring torque and inertia about the suspension axis. 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: Period compares rotational inertia with restoring-torque strength.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- T_physical = 2 pi sqrt(I_O/(M g d)) — Small-angle period with centre of mass distance d below pivot.
- T_torsion = 2 pi sqrt(I/kappa) — Period for restoring torque -kappa theta.
How to approach it
- 1Identify the restoring torque near equilibrium
- 2Linearize for small angle
- 3Use inertia about the actual suspension axis
Common slip-ups that cost marks
- •Using I_cm instead of pivot inertia
- •Applying the small-angle formula at large amplitude
- •Confusing torsion constant with spring constant
🌟 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.