Rigid Bodies and Constrained Spring Systems
For a small generalized displacement q, a constrained elastic system satisfies I_eff q double-dot + k_q q = 0, where k_q follows from U = k_q q^2/2.
Why this shows up in the exam
Rods attached to springs · Rolling discs or rings with springs · Amplitude of an internal point in a spring assembly
Learn the idea
For rotational or constrained motion, convert each spring extension into the selected generalized coordinate. A rod or rolling body makes different attachment points move by different amounts, so geometry weights each spring's restoring effect.
🧠 Memory hook: Geometry first, spring algebra second.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- omega = sqrt(k_q/I_eff) — small angular frequency in a rotational coordinate
- U = sum_i k_i (Delta l_i)²/2 — total spring energy under the constraint
- Delta l_i approximately equals c_i q — small-displacement geometric relation for each attachment
How to approach it
- 1Choose the independent coordinate
- 2Relate all point motions to it
- 3Form total kinetic and potential energies and compare coefficients
Common slip-ups that cost marks
- •Using translational mass instead of moment of inertia
- •Giving every attachment the same extension
- •Omitting rolling kinetic energy
🌟 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.
A mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
More from Oscillations and Waves
Superposition, Beats, and Resonance
Explore the principles of superposition, formation of beats, beat frequency, resonance, and phase difference in wave phenomena.
Standing Waves and Harmonics
Understand the formation of standing waves, harmonics in strings and organ pipes (open and closed), and the relationship between displacement and pressure in stationary waves.
Velocity and Acceleration in SHM
Explore how velocity and acceleration vary with displacement and time in SHM, including maximum values and their physical significance.
Simple Pendulum and Spring Systems
Study the time period formulas, energy considerations, and behavior of simple pendulums and spring-mass systems, including effects of non-inertial frames and spring combinations.
Basics of Simple Harmonic Motion (SHM)
Understand the defining features of SHM, including its standard equations, conditions, and the relationship between displacement, velocity, and acceleration.
Wave Motion and Wave Equation
Learn the fundamentals of wave motion, including the wave equation, its parameters, direction, and the distinction between particle and wave velocities.