🔄 Simple Harmonic Motion (Spring-Mass) Visualizer
OscillationsWhat's happening? ω = √(k/m). Bigger k = faster oscillation. Bigger mass = slower. Watch KE and PE trade places — total energy stays constant.
Simple Harmonic Motion is a high-weightage JEE topic — 2-3 questions every year.
SHM appears in block-spring systems, pendulums, torsional oscillators, and even in wave problems. JEE tests both equation-solving and physical interpretation of SHM.
Key formulas
SHM Formulas
Angular frequency
ω = √(k/m)
For spring: ω = √(k/m). For pendulum: ω = √(g/L).
Displacement
x = A·cos(ωt + φ)
General solution. φ is phase constant from initial conditions.
Velocity & Acceleration
v = -Aω·sin(ωt + φ)
a = -ω²x
Acceleration always points to center. Max v = Aω, max a = Aω².
Total Energy
E = ½kA²
Constant. KE = ½mv², PE = ½kx². They swap but E stays fixed.
Pendulum period
T = 2π√(L/g)
Only depends on L and g — not mass or amplitude (small angles).
Damped SHM
x = Ae^(-bt/2m)·cos(ω't + φ)
ω' = √(ω₀² - (b/2m)²). Amplitude decays exponentially.
Step-by-step problem solving
How to solve SHM problems in JEE
- Identify the system — block-spring, simple pendulum, or physical pendulum.
- Check if SHM is valid — force must be proportional to displacement and opposite (F = -kx).
- Find ω — ω = √(k/m) for spring, √(g/L) for pendulum.
- Use energy conservation — E = ½kA² = ½mv² + ½kx² to find v at any x.
- Apply initial conditions — use x(0) and v(0) to find amplitude A and phase φ.
Common JEE mistakes
What students get wrong
❌ Confusing ω with frequency
ω = 2πf, not f. JEE asks for angular frequency — use ω = √(k/m), not f = 1/T.
❌ Forgetting energy is constant
At extreme position all PE, at center all KE. The sum is always E = ½kA² — use this shortcut!
❌ Using large-angle pendulum formula
T = 2π√(L/g) only works for small angles (< 10°). For larger angles, use T = 2π√(L/g) · [1 + (1/16)θ₀² + ...]
❌ Mixing up x, v, a relationships
a = -ω²x (acceleration leads displacement by 180°). v = -Aω·sin(ωt+φ) leads x by 90°. Phase matters!
FAQs
SHM Questions
Is SHM always motion in a straight line?
Yes, SHM is always along a straight line (the x-axis). Even though the oscillator moves in 1D, the graph of x vs t is sinusoidal — not a wave on a string. The key signature: a = -ω²x.
Why is acceleration always toward the center?
In SHM, the restoring force always points toward equilibrium (F = -kx). Since F = ma, acceleration a = -(k/m)x = -ω²x. The minus sign means acceleration and displacement are opposite — hence "toward center."
What's the difference between SHM and periodic motion?
ALL SHM is periodic, but NOT all periodic motion is SHM. SHM requires the special condition a = -ω²x. A bouncing ball is periodic but its acceleration is g (constant), not proportional to displacement.
Can energy ever equal zero in SHM?
No — E = ½kA² always. At the extremes, PE = E and KE = 0. At the center, KE = E and PE = 0. Energy shifts between KE and PE but the total never changes.