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"Show that the motion is simple harmonic and find its period" is one of the most common question types in oscillations. Many students jump to T = 2π√(m/k) or T = 2π√(L/g) and hope one fits. That works for the textbook spring and pendulum, and fails on anything new: a floating block, a ball in a bowl, a body in a tunnel. This post gives one method that works for all of them, with worked examples.
What makes a motion simple harmonic?
The NCERT Class 11 chapter Oscillations defines SHM through its restoring force: the force is proportional to the displacement from equilibrium and directed towards it, F = −kx. That equation gives x = A cos(ωt + φ) with ω = √(k/m) and period T = 2π/ω. The JEE (Main) 2026 syllabus lists SHM, phase, the spring's restoring force and force constant, energy in SHM and the simple pendulum. Check the syllabus for your exam year.
So "show it is SHM" means: write the net force for a small displacement x and show it has the form −(constant) × x. The constant plays the role of k, whatever its physical origin.
Where do students usually go wrong?
Somroob and Wattanakasiwich (2017) gave a 16-question conceptual test to university students taking physics courses. Most had misconceptions about restoring force and had trouble connecting the mathematical solution to the real motion, especially the phase angle. Many also struggled to read mechanical energy from graphs and diagrams. These are the same weak points that show up in exam answers: students memorise T but cannot say what force brings the body back.
A three-step method
Find equilibrium. Where is the net force zero?
Displace by a small x and write the net force (or net torque for rotation). Use only the extra force caused by the displacement.
Match to F = −k_eff x. Then ω = √(k_eff/m) and T = 2π√(m/k_eff). For rotation, τ = −c θ gives T = 2π√(I/c).
If step 3 gives a force that is not proportional to x, the motion is periodic but not SHM. Using a small-angle or small-displacement approximation is usually what makes it SHM.
Worked example 1: springs in parallel and in series
A 2.0 kg block is attached to two springs of force constants 100 N/m and 300 N/m.
Parallel (both stretch by x): F = −(100 + 300)x, so k = 400 N/m. ω = √(400/2.0) ≈ 14.1 rad/s and T = 2π/14.1 ≈ 0.44 s.
Series (same force in each, extensions add): 1/k = 1/100 + 1/300, so k = 75 N/m. ω = √(75/2.0) ≈ 6.12 rad/s and T ≈ 1.03 s.
A common error is to add series constants directly. Series springs are softer than either spring alone.
Worked example 2: a floating block
A wooden block of uniform cross-section floats upright in water with 10 cm of it below the surface. It is pushed down a little and released. Find the period, ignoring water resistance.
At equilibrium, buoyancy balances weight: ρ_w A h g = m g, where h = 0.10 m is the submerged depth.
Push down by x: extra buoyant force = ρ_w A g x, upward, so F = −(ρ_w A g)x. This is SHM with k = ρ_w A g.
T = 2π√(m/k) = 2π√(ρ_w A h / (ρ_w A g)) = 2π√(h/g) = 2π√(0.10/9.8) ≈ 0.63 s.
The block's density and area cancel. Only the submerged depth matters.
Worked example 3: a tunnel through the Earth
Imagine a straight tunnel through the centre of a uniform Earth of radius R. Inside, gravity at distance x from the centre is g(x/R), so the force on a mass m is F = −(mg/R)x. That is SHM with k = mg/R, so T = 2π√(R/g). With R = 6.4 × 10⁶ m and g = 9.8 m/s², T ≈ 5.1 × 10³ s, about 85 minutes. Our post on gravitation and satellite problems shows that a satellite skimming the surface has the same period.
Which other mistakes cost marks?
Using T = 2π√(L/g) at large angles. The pendulum formula needs the small-angle approximation sin θ ≈ θ.
Thinking a vertical spring has a different period. Gravity shifts the equilibrium point (by mg/k) but not the period. For m = 0.50 kg and k = 200 N/m, the static stretch is 2.45 cm and T = 2π/20 ≈ 0.31 s, the same as on a smooth table.
Confusing ω with frequency. ω = 2πf. Frequency is in hertz, ω in rad/s.
Reading energy graphs wrongly. Kinetic energy is greatest at equilibrium and zero at the extremes; total energy ½kA² stays constant.
If you set up SHM equations comfortably but freeze on unfamiliar systems, compare your force equation with a full solution. The Fermi Problem Library has a step-by-step solver that explains the reasoning, so you can check whether your restoring-force step matches. For the calculus behind x = A cos(ωt + φ), see the calculus you need for JEE physics.
A practice routine
Day 1: Five "show it is SHM" questions. Write only steps 1 to 3, no period.
Day 2: Find the periods for the same five.
Day 3: Two rotational SHM problems (a rod pivoted at one end, a disc on a torsion wire).
Day 4: Energy and phase: sketch x, v and a against time for one system, then a timed mixed set.
Choose one unfamiliar oscillating system from your book and, before looking at the solution, write its restoring force equation in one line.
