Simple Harmonic Motion (SHM) Class 11: All Formulas & Formula Sheet with Free PDF Download (Physics, JEE & NEET)
Simple Harmonic Motion Formula Sheet — Competishun
Simple Harmonic Motion (SHM) Class 11: All Formulas & Formula Sheet with Free PDF Download (Physics, JEE & NEET)
Simple Harmonic Motion is one of those chapters that shows up everywhere. The same ideas that describe a spring bouncing on a table also explain how a pendulum swings, how atoms vibrate in a crystal, how AC circuits behave, and even how sound waves travel. If you understand SHM properly once, you will find echoes of it in almost every advanced physics topic you study later.
This page gives you the complete SHM formula sheet for Class 11 in one place. It covers the definition and conditions for SHM, displacement, velocity and acceleration equations, energy in SHM, spring-mass systems, the simple pendulum, and damped oscillations, with worked examples. Download the free PDF below and keep it with you for revision.
Download the SHM Formula Sheet PDF
Get all SHM formulas including displacement, velocity, acceleration, energy, spring systems, pendulum and damped oscillations in one clean PDF, free. Perfect for Class 11, JEE and NEET revision.
Download Free PDFWhat is Simple Harmonic Motion?
Simple Harmonic Motion is a special type of periodic oscillatory motion where the restoring force on the object is always directed toward the mean (equilibrium) position and is directly proportional to the displacement from that position.
The negative sign is essential. It tells you the force always points back toward the centre. The larger the displacement, the larger the restoring force pulling it back. This is what makes the motion oscillate smoothly instead of going off in one direction.
Key Terms and Definitions
| Term | Symbol | Meaning |
|---|---|---|
| Amplitude | A | Maximum displacement from mean position |
| Time period | T | Time for one complete oscillation (seconds) |
| Frequency | f | Number of oscillations per second (Hz) |
| Angular frequency | ω | Rate of change of phase (rad/s) |
| Phase | φ | Initial position and direction of motion at t = 0 |
| Mean position | x = 0 | Equilibrium point; net force is zero here |
| Extreme position | x = ±A | Turning points; velocity is zero here |
| Relation: T = 1/f = 2π/ω — these three are different ways to express the same idea. | ||
1. Displacement, Velocity and Acceleration in SHM
These three equations describe where the particle is, how fast it is moving and how quickly its velocity is changing at any instant.
| Quantity | Formula | Note |
|---|---|---|
| Displacement | x = A sin(ωt + φ) | Or A cos(ωt + φ) depending on initial condition |
| Velocity | v = Aω cos(ωt + φ) | v = dx/dt |
| Velocity at position x | v = ω√(A² − x²) | Most useful formula in numericals |
| Acceleration | a = −Aω² sin(ωt + φ) | a = dv/dt = −ω²x |
| Maximum velocity | vmax = Aω | At mean position, x = 0 |
| Maximum acceleration | amax = Aω² | At extreme position, x = ±A |
| Velocity is maximum at mean position and zero at extremes. Acceleration is maximum at extremes and zero at mean position. | ||
2. Time Period and Frequency
| Relation | Formula |
|---|---|
| Angular frequency | ω = 2πf = 2π/T |
| Time period from ω | T = 2π/ω |
| Frequency from T | f = 1/T |
| Differential equation of SHM | d²x/dt² + ω²x = 0 |
3. Spring-Mass System
A mass attached to a spring and displaced from its natural length is the most classic example of SHM. The spring force F = negative kx provides the restoring force.
| Quantity | Formula | Note |
|---|---|---|
| Restoring force | F = −kx | k is the spring constant (N/m) |
| Angular frequency | ω = √(k/m) | m is mass in kg |
| Time period | T = 2π√(m/k) | Independent of amplitude |
| Springs in series | 1/keff = 1/k1 + 1/k2 | Effective spring constant decreases |
| Springs in parallel | keff = k1 + k2 | Effective spring constant increases |
| Spring cut in ratio n:(1-n) | k' = k/n (for length fraction n) | Shorter piece has larger k |
| Time period of spring-mass system does NOT depend on amplitude or gravity. It depends only on m and k. | ||
4. Simple Pendulum
A simple pendulum is a point mass suspended by a massless inextensible string. For small angles (less than about 5 degrees), the motion is approximately SHM.
| Quantity | Formula | Note |
|---|---|---|
| Restoring force | F = −mg sinθ ≈ −mgθ | Small angle approximation: sinθ ≈ θ |
| Angular frequency | ω = √(g/L) | L is length of pendulum in metres |
| Time period | T = 2π√(L/g) | Independent of mass and amplitude |
| Seconds pendulum | T = 2 s, L ≈ 1 m | Standard reference pendulum |
| Effect of gravity | T ∝ 1/√g | Pendulum beats slower at higher altitude |
| Simple pendulum time period depends on length L and gravity g only. Mass of the bob does not matter. | ||
5. Energy in SHM
In SHM, energy continuously shifts between kinetic and potential form. The total mechanical energy stays constant throughout the motion as long as there is no damping.
| Type of Energy | Formula | When Maximum |
|---|---|---|
| Kinetic energy (KE) | KE = ½mω²(A² − x²) | At mean position, x = 0 |
| Potential energy (PE) | PE = ½mω²x² = ½kx² | At extreme positions, x = ±A |
| Total energy (E) | E = ½mω²A² = ½kA² | Constant; independent of x |
| KE = PE (equal energy) | x = ±A/√2 | At this displacement, KE and PE are equal |
| Total energy in SHM is proportional to the square of the amplitude. Double the amplitude means four times the energy. | ||
6. Damped and Forced Oscillations
In real situations, oscillations die down over time because of resistive forces like air resistance or friction. This is called damping.
| Concept | Formula / Condition |
|---|---|
| Damping force | Fd = −bv (b = damping coefficient) |
| Amplitude with damping | A(t) = A0e−bt/2m |
| Under-damped | Oscillates with decreasing amplitude (b² < 4mk) |
| Critically damped | Returns to mean position fastest, no oscillation (b² = 4mk) |
| Over-damped | Returns slowly, no oscillation (b² > 4mk) |
| Resonance (forced oscillations) | Maximum amplitude when driving frequency equals natural frequency |
| For JEE, focus on understanding resonance and the condition b squared versus 4mk. | |
All SHM Formulas at a Glance
| Formula | What It Means |
|---|---|
| a = −ω²x | Defining condition of SHM |
| x = A sin(ωt + φ) | Displacement as a function of time |
| v = ω√(A² − x²) | Velocity at displacement x |
| vmax = Aω | Maximum velocity at mean position |
| amax = Aω² | Maximum acceleration at extreme position |
| T = 2π/ω | Time period from angular frequency |
| T = 2π√(m/k) | Spring-mass system time period |
| T = 2π√(L/g) | Simple pendulum time period |
| E = ½kA² | Total energy in SHM |
| KE = ½mω²(A²−x²) | Kinetic energy at displacement x |
| PE = ½mω²x² | Potential energy at displacement x |
| keff(series) = k1k2/(k1+k2) | Springs in series |
| keff(parallel) = k1 + k2 | Springs in parallel |
Worked Examples
| Problem | Solution |
|---|---|
| Time period of spring: k = 100 N/m, m = 1 kg | T = 2π√(1/100) = 2π/10 = 0.628 s |
| Max velocity: A = 0.05 m, ω = 10 rad/s | v_max = Aω = 0.05 × 10 = 0.5 m/s |
| Velocity at x = 3 cm: A = 5 cm, ω = 10 | v = 10√(25−9) = 10√16 = 40 cm/s |
| Total energy: k = 200 N/m, A = 0.1 m | E = ½ × 200 × 0.01 = 1 J |
| Pendulum length for T = 2 s (g = 10) | L = g(T/2π)² = 10 × (1/π)² ≈ 1.01 m |
| Spring cut into 3 equal parts, original k = 60 | Each piece: k' = 3 × 60 = 180 N/m |
Common Mistakes to Avoid
- Forgetting the negative sign in a = negative omega squared x: The negative sign is the defining feature of SHM. Without it, there is no restoring force.
- Thinking amplitude affects time period: For a spring-mass system or simple pendulum, time period is independent of amplitude. This is a very common wrong assumption.
- Using v = A omega everywhere: That is only the maximum velocity. At any other position, use v = omega times root of (A squared minus x squared).
- Getting spring combination formulas wrong: Series springs follow the reciprocal rule, just like resistors in parallel. Parallel springs simply add up, like resistors in series.
- Applying pendulum formula without small angle condition: T = 2 pi root (L/g) only holds for small angles (less than about 5 degrees). For large angles, the actual time period is longer.
- Confusing where KE and PE are maximum: KE is maximum at mean position (x = 0). PE is maximum at extreme positions (x = plus or minus A). Not the other way around.
Why SHM Matters for JEE and NEET
- Consistent weightage: SHM and Oscillations appear in every JEE Main and NEET paper, typically contributing two to four questions per exam.
- Foundation for waves: Wave motion, sound and light are all extensions of oscillatory behaviour. SHM is the building block.
- Connects to other chapters: Spring systems link to Hooke's Law and elasticity. Energy in SHM connects to Work, Energy and Power. Pendulum connects to gravity.
- Scoring chapter: The concept variety is moderate and the formula set is compact. Students who master this chapter reliably score full marks on SHM questions in JEE Main.
How to Use This Formula Sheet
- Understand the condition a = negative omega squared x first: Every SHM formula flows from this one condition. Understand it physically before memorising the rest.
- Memorise v = omega root (A squared minus x squared): This is the single most used formula in SHM numericals. You will apply it in almost every calculation.
- Know both spring and pendulum time period formulas: These are direct one-mark questions in JEE Main and NEET. There is no excuse for not having them memorised.
- Practise energy problems: Questions about at which position KE equals PE, or what fraction of total energy is kinetic at a given x, are very common. Solve at least ten of these.
- Skim the PDF before every test: A quick scan of this sheet before a mock exam refreshes all the key formulas and prevents silly mistakes under pressure.
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Frequently Asked Questions — Simple Harmonic Motion
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