“Acceleration towards the middle, size follows distance”
The test for SHM
If a = −ω²x, it is simple harmonic motion. Restoring, proportional, opposite in sign: all three must hold.
Class 11 Physics · Gujarat Board, CBSE, GUJCET and JEE
Oscillations formulas
Simple harmonic motion is one equation wearing many costumes: a spring, a pendulum, a floating block, a vibrating column. Once you can spot that the restoring force is proportional to displacement, the time period follows in one line.
15 formulas, 3 memory hooks, and the 5 mistakes that cost the most marks.
Describing SHM
| What | Formula | Symbols and notes |
|---|---|---|
| Displacement | x = A sin(ωt + φ) | A amplitude, ω angular frequency, φ phase constant |
| Velocity | v = Aω cos(ωt + φ) = ω√(A² − x²) | maximum at the mean position, zero at the ends |
| Acceleration | a = −ω²x | always directed towards the mean position; this defines SHM |
| Angular frequency | ω = 2π/T = 2πf | T is the time period |
| Maximum values | vmax = Aω, amax = Aω² | at x = 0 and x = A respectively |
Time periods
| What | Formula | Symbols and notes |
|---|---|---|
| Spring-mass system | T = 2π√(m/k) | independent of gravity, so it works the same on the Moon |
| Springs in series | 1/k = 1/k₁ + 1/k₂ | softer than either alone |
| Springs in parallel | k = k₁ + k₂ | stiffer than either alone |
| Simple pendulum | T = 2π√(l/g) | independent of the mass of the bob |
| Pendulum in a lift | T = 2π√(l/(g ± a)) | g + a going up, g − a going down |
| Floating cylinder | T = 2π√(hρ/ρlg) | h is the submerged height |
Energy in SHM
| What | Formula | Symbols and notes |
|---|---|---|
| Kinetic energy | KE = ½mω²(A² − x²) | greatest at the mean position |
| Potential energy | PE = ½mω²x² | greatest at the extremes |
| Total energy | E = ½mω²A² = ½kA² | constant, and proportional to the square of the amplitude |
| Frequency of energy | energy varies at twice the frequency of displacement | because both KE and PE peak twice per cycle |
Taught by Darshit Bhatt, founder and lead faculty at Darshit Sir Institute. Instrumentation and Control engineer, with a master's degree and a chartered degree, teaching in Paldi, Ahmedabad since 2006.
Know the formula but not when to use it?
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Memory hooks
How to remember this chapter.
“Spring ignores gravity, pendulum lives on it”
Which time period contains g
T = 2π√(m/k) has no g at all, so a spring keeps its period on the Moon. T = 2π√(l/g) does, so a pendulum runs slower there.
“Fast in the middle, still at the ends”
Where velocity and acceleration peak
Velocity is largest at the mean position where acceleration is zero; acceleration is largest at the extremes where velocity is zero.
Where marks go
The mistakes that cost the most.
- Thinking the time period depends on the amplitude. For SHM it does not, which is what makes pendulum clocks possible.
- Adding springs in series like resistors in parallel. Series springs add their reciprocals; parallel springs add directly.
- Forgetting the minus sign in a = −ω²x, which is the part that makes the motion oscillate at all.
- Using the length of the string alone for a physical pendulum. Only a simple pendulum uses T = 2π√(l/g).
- Saying total energy is proportional to the amplitude. It goes as the amplitude squared.
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