Simple Harmonic Motion Calculator
Physics Class 12
Equation: 10 * cos(12.57t + 0)
At t=1s: 10.0000 m
Key Formulas
y = A cos(ωt + φ)
v = -Aω sin(ωt + φ)
a = -Aω² cos(ωt + φ)
ω = 2πf
Period T
0.5000 s
ω
12.5664 rad/s
Max Velocity
125.66 m/s
Simple Harmonic Motion - Class 12 Physics Complete Guide
Simple Harmonic Motion (SHM) is one of the most fundamental and important topics in Class 12 Physics. It describes a special type of oscillatory motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. SHM forms the foundation for understanding more complex wave phenomena, including sound waves, light waves, and quantum mechanics. From the swinging of a pendulum to the vibration of guitar strings, SHM is everywhere in nature.
What is Simple Harmonic Motion?
Simple Harmonic Motion is defined as the motion in which the acceleration of a particle is always directed toward a fixed point (called the mean position or equilibrium) and is directly proportional to its displacement from that point. Mathematically, a particle executing SHM satisfies:
where a is acceleration, x is displacement from equilibrium, and ω (omega) is the angular frequency. The negative sign indicates that acceleration is always opposite to displacement - it is a restoring force trying to bring the particle back to equilibrium.
Key Quantities and Formulas
x = A cos(ωt + φ)
Displacement equation
v = -Aω sin(ωt + φ)
Velocity equation
a = -Aω² cos(ωt + φ)
Acceleration equation
ω = 2πf = 2π/T
Angular frequency
Understanding Each Term
- A (Amplitude) = The maximum displacement from the mean position. It determines the maximum value of displacement, velocity, and acceleration.
- ω (Angular Frequency) = Rate of change of phase, measured in radians per second. Higher ω means faster oscillation.
- T (Period) = Time taken to complete one full oscillation or cycle. T = 2π/ω
- f (Frequency) = Number of oscillations per second. f = 1/T = ω/2π
- φ (Phase) = Determines the initial state of the motion at t = 0. Phase difference explains why two identical pendulums might be at different positions at the same time.
Maximum and Minimum Values
| Quantity | Maximum Value | Minimum Value | Occurs When |
|---|---|---|---|
| Displacement (x) | A | -A | cos = ±1 |
| Velocity (v) | Aω | -Aω | sin = 0 |
| Acceleration (a) | Aω² | -Aω² | cos = ±1 |
Solved Example
Problem: A particle executes SHM with amplitude 10 cm and frequency 2 Hz. At t = 0, it is at its maximum displacement. Find (a) the period, (b) angular frequency, (c) displacement at t = 0.25 s, (d) maximum velocity.
Solution:
(a) Period: T = 1/f = 1/2 = 0.5 s
(b) Angular frequency: ω = 2πf = 2π × 2 = 4π rad/s ≈ 12.57 rad/s
(c) Displacement: At t = 0, x = A (maximum), so φ = 0
x = A cos(ωt) = 10 × cos(4π × 0.25) = 10 × cos(π) = 10 × (-1) = -10 cm
(d) Maximum velocity: v_max = Aω = 10 × 4π = 40π cm/s ≈ 125.7 cm/s
Energy in Simple Harmonic Motion
In SHM, energy continuously transforms between kinetic and potential forms while total energy remains constant:
Total Energy (E) = Kinetic Energy (K) + Potential Energy (U)
K = (1/2)mv² = (1/2)mω²A²sin²(ωt + φ)
U = (1/2)kx² = (1/2)mω²A²cos²(ωt + φ)
E = (1/2)mω²A² = constant
Real-World Examples of SHM
- Simple Pendulum: For small oscillations, a pendulum exhibits SHM with period T = 2π√(L/g)
- Mass-Spring System: A mass attached to a spring oscillates with SHM with T = 2π√(m/k)
- Tuning Forks: The prongs vibrate with SHM producing a pure tone
- Guitar Strings: Vibrate in SHM when plucked, producing musical notes
- LC Circuits: Electrical oscillations in inductors and capacitors behave like SHM
Differences Between SHM and General Oscillatory Motion
| Feature | Simple Harmonic Motion | General Oscillatory Motion |
|---|---|---|
| Force | F = -kx (linear) | Any restoring force |
| Shape | Sinusoidal | Any periodic shape |
| Period | Independent of amplitude | May depend on amplitude |
Important Note: In true SHM, the period is completely independent of amplitude (this is called isochronism). However, real-world oscillators like pendulums only approximate SHM for small angles. At larger amplitudes, the motion becomes anharmonic.