Simple Harmonic Motion Calculator

Calculate and visualize oscillations in spring-mass systems and pendulums

m
N/m
kg
m
m/s²
s
Time at which to calculate position, velocity, acceleration
°
Initial phase angle in degrees
Angular Frequency (ω)
14.14 rad/s
ω = √(k/m)
Frequency (f)
2.25 Hz
f = ω/(2π)
Period (T)
0.444 s
T = 1/f
Max Velocity (vmax)
1.414 m/s
vmax = ωA
Kinetic Energy (KE)
0.25 J
KE = ½mv²
Potential Energy (PE)
0.25 J
PE = ½kx²
Total Energy (E)
0.50 J
E = KE + PE
Motion at Time t
Parameter Value Description
Position (x) 0.0707 m x = A cos(ωt + φ)
Velocity (v) -1.0 m/s v = -Aω sin(ωt + φ)
Acceleration (a) -14.14 m/s² a = -Aω² cos(ωt + φ)
Phase 45° Phase angle at time t

Understanding Simple Harmonic Motion

1

What is Simple Harmonic Motion? SHM is a type of periodic motion where the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement.

2

Key Equations:

x(t) = A cos(ωt + φ) v(t) = -Aω sin(ωt + φ) a(t) = -Aω² cos(ωt + φ)

Where:
A = Amplitude
ω = Angular frequency
φ = Phase angle
t = Time
3

Spring-Mass System:

  • Angular frequency: ω = √(k/m)
  • Period: T = 2π√(m/k)
  • Frequency: f = 1/T = (1/2π)√(k/m)
4

Simple Pendulum:

  • Angular frequency: ω = √(g/L)
  • Period: T = 2π√(L/g)
  • Frequency: f = 1/T = (1/2π)√(g/L)

Energy Conservation: In SHM, total mechanical energy is conserved and alternates between kinetic and potential energy.

E = ½kA² = ½mv² + ½kx²
Energy in SHM
  • Potential Energy: PE = ½kx²
  • Kinetic Energy: KE = ½mv²
  • Total Energy: E = ½kA² (constant)
  • At equilibrium (x=0): PE=0, KE=max
  • At extremes (x=±A): PE=max, KE=0

Applications of SHM

1

Clocks and Timekeeping: Pendulum clocks use SHM principles for accurate time measurement.

2

Musical Instruments: Strings in instruments vibrate with SHM to produce musical tones.

3

Seismology: Seismometers use pendulum systems to detect and measure earthquakes.

4

Automotive: Car suspension systems use springs and dampers based on SHM principles.

SHM Examples

System Typical Period
Grandfather clock pendulum 2 seconds
Guitar string (A4) 0.00227 seconds (440 Hz)
Car suspension 0.5-1.5 seconds
Atom in crystal lattice 10-13 seconds
Earthquake aftershock 10-30 seconds
Human walking 1 second (approx)

Frequently Asked Questions

Simple Harmonic Motion (SHM) is a special type of periodic motion where the restoring force is directly proportional to the displacement and acts in the opposite direction. Key characteristics include:

  • Sinusoidal motion pattern (sine or cosine wave)
  • Acceleration proportional to displacement but opposite in direction
  • Conservation of total mechanical energy
  • Constant period independent of amplitude

Common SHM systems include spring-mass systems, pendulums (small angle approximation), and LC circuits.

Angular frequency (ω) and frequency (f) both describe oscillation rates but have distinct differences:

Parameter Definition Units Relationship
Angular frequency (ω) Rate of phase angle change radians/second (rad/s) ω = 2πf
Frequency (f) Number of oscillations per second Hertz (Hz) f = 1/T

For spring-mass systems: ω = √(k/m)

For pendulum systems: ω = √(g/L)

Total mechanical energy is conserved in ideal SHM systems because:

  • No non-conservative forces do work
  • Energy continuously converts between kinetic and potential forms
  • Total energy E = ½kA² (spring systems) or E = mgL(1-cosθ) (pendulums)

Energy transformation process:

  1. At equilibrium (x=0): Maximum kinetic energy, zero potential energy
  2. At maximum displacement (x=±A): Zero kinetic energy, maximum potential energy
  3. At other positions: Sum of kinetic and potential energy remains constant
Energy equation: E = ½mv² + ½kx² = constant

The phase angle (φ) determines the initial state of the oscillating system:

  • Specifies starting position and direction of motion
  • Measured in radians or degrees (0° to 360°)
  • Affects the instantaneous values of position, velocity, and acceleration

Common phase angle values:

Phase Angle Initial Condition
φ = 0° Starts at maximum displacement (x = A)
φ = 90° (π/2 rad) Starts at equilibrium moving negative
φ = 180° (π rad) Starts at minimum displacement (x = -A)
φ = 270° (3π/2 rad) Starts at equilibrium moving positive

Phase difference between two oscillators determines their synchronization.

Amplitude (A) is the maximum displacement from equilibrium and influences SHM in several ways:

  • Energy: Total energy proportional to A² (E ∝ A²)
  • Velocity: Maximum velocity proportional to A (vₘₐₓ ∝ A)
  • Acceleration: Maximum acceleration proportional to A (aₘₐₓ ∝ A)
  • Period/Frequency: Independent of amplitude (isochronism)

Key points:

  1. Doubling amplitude quadruples total energy
  2. Doubling amplitude doubles maximum velocity
  3. Period remains unchanged regardless of amplitude
  4. Large amplitudes may violate SHM assumptions (e.g., pendulum at large angles)

Isochronism: The period of SHM is independent of amplitude, a property discovered by Galileo in pendulum experiments.

Simple Harmonic Motion and Uniform Circular Motion are mathematically related:

Aspect Simple Harmonic Motion Uniform Circular Motion
Motion Type Linear oscillation Circular path
Projection SHM is the projection of UCM UCM can generate SHM
Equations x = A cos(ωt + φ) x = R cos(ωt), y = R sin(ωt)
Phase Difference Position and velocity 90° out of phase Position components 90° out of phase
Applications Springs, pendulums Rotating systems, orbits

The connection:

  • SHM can be considered as the projection of uniform circular motion onto a diameter
  • The angular frequency ω is identical in both systems
  • The reference circle helps visualize SHM phase relationships