SHM Criterion and Restoring Dynamics
For one-dimensional ideal SHM about x = 0, the net force obeys F = -kx, equivalently d2x/dt2 = -omega^2 x, with constant positive k and omega.
Why this shows up in the exam
Testing whether a force law gives SHM · Finding frequency from force per displacement · Linearising motion near stable equilibrium
Learn the idea
Motion is simple harmonic when acceleration is proportional and opposite to displacement from stable equilibrium. A displaced system is pulled back more strongly the farther it moves from equilibrium, so inertia carries it through and the restoring process repeats.
🧠 Memory hook: SHM needs a minus sign and one power of displacement.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- F = -kx — linear restoring force measured from stable equilibrium
- a = -omega² x — kinematic test for ideal simple harmonic motion
- omega = sqrt(k/m) — angular frequency for an effective mass m and stiffness k
How to approach it
- 1Locate the stable equilibrium
- 2Write the net force for a small signed displacement
- 3Match the linear term to -m omega squared x
Common slip-ups that cost marks
- •Measuring x from an unstretched position instead of equilibrium
- •Calling every periodic motion SHM
- •Using a displacement-dependent stiffness as a 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.
A mass of 1 kg is attached to a spring of force constant 100 N/m. Find the angular frequency of small oscillations.
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