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Simple Harmonic Motion Calculator

Calculate the period, frequency, and motion characteristics of a spring-mass system with interactive visualization

Parameters

mⓘ
N/mⓘ
kgⓘ
ⓘ
Show Trail

Controls

xⓘ

Calculated Values

Angular Frequency:
3.16;rad/s3.16;rad/s
Period:
1.99;s1.99;s
Frequency:
0.50;Hz0.50;Hz
Maximum Velocity:
3.16;m/s3.16;m/s
Maximum Acceleration:
10.00;m/s210.00;m/s²
Total Energy:
5.00;J5.00;J

Examples

Example 1: Light Spring System

A spring with constant 5 N/m and mass 0.5 kg with amplitude 0.3 m.

  • Angular Frequency: 3.163.16
  • Period: 1.991.99
  • Frequency: 0.500.50

Example 2: Heavy Mass System

A heavier mass (2 kg) with a stiffer spring (20 N/m) and amplitude 0.5 m.

  • Angular Frequency: 3.163.16
  • Period: 1.991.99
  • Frequency: 0.500.50

Example 3: Damped Oscillator

Same system as Example 1 but with damping ratio 0.1.

  • Angular Frequency: 3.163.16
  • Period: 1.991.99
  • Frequency: 0.500.50

Visualization

Simple Harmonic Motion

Simple Harmonic Motion (SHM) is a fundamental type of periodic motion that occurs when a restoring force is proportional to the displacement from equilibrium and acts in the opposite direction. This type of motion is ubiquitous in nature and appears in many physical systems.

The key characteristic of SHM is that the acceleration is always directed toward the equilibrium position and is proportional to the displacement. This creates a sinusoidal motion pattern that repeats with a constant period, regardless of the amplitude.

In a spring-mass system, Hooke's Law provides the restoring force: F = -kx, where k is the spring constant and x is the displacement. This force creates SHM with an angular frequency ω = √(k/m), where m is the mass.

The period of SHM is independent of the amplitude, a property known as isochronism. This means that whether you pull the spring a little or a lot, it will complete one oscillation in the same time (assuming no damping).

Damping introduces energy loss to the system, causing the amplitude to decrease over time. The damping ratio ζ determines the type of motion: underdamped (ζ < 1) shows oscillatory decay, critically damped (ζ = 1) returns to equilibrium fastest without oscillation, and overdamped (ζ > 1) returns slowly without oscillation.

Key Concepts

  • Restoring Force: A force that always acts toward the equilibrium position
  • Amplitude: The maximum displacement from equilibrium
  • Period: The time for one complete oscillation
  • Frequency: The number of oscillations per unit time
  • Angular Frequency: The rate of change of the phase angle
  • Damping: The gradual reduction in amplitude due to energy loss

Real-World Applications

  • Mechanical systems: Springs, pendulums, and vibrating machinery
  • Electrical circuits: LC oscillators and AC circuits
  • Musical instruments: String vibrations and sound waves
  • Seismology: Earthquake detection and building design
  • Quantum mechanics: Harmonic oscillator model for molecular vibrations

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Physics Equations

Angular Frequency:
ω=km\omega = \sqrt{\frac{k}{m}}
Period:
T=2πmkT = 2\pi\sqrt{\frac{m}{k}}
Frequency:
f=1T=12πkmf = \frac{1}{T} = \frac{1}{2\pi}\sqrt{\frac{k}{m}}
Position:
x(t)=Ae−ζωtcos⁡(ωt)x(t) = A e^{-\zeta\omega t} \cos(\omega t)
Velocity:
v(t)=−Aωe−ζωt[sin⁡(ωt)+ζcos⁡(ωt)]v(t) = -A\omega e^{-\zeta\omega t} [\sin(\omega t) + \zeta\cos(\omega t)]
Acceleration:
a(t)=−kmx(t)a(t) = -\frac{k}{m}x(t)

Step-by-Step Solution

See how the main results are calculated.

1

Step 1: Calculate Angular Frequency

First, we calculate the angular frequency using the formula:

Equation:

ω=km\omega = \sqrt{\frac{k}{m}}

Calculation:

ω=10.001.00=3.16 rad/s\omega = \sqrt{\frac{10.00}{1.00}} = 3.16 \text{ rad/s}

Explanation:

The angular frequency determines how fast the system oscillates.

2

Step 2: Calculate Period

The period is the time for one complete oscillation:

Equation:

T=2πω=2πmkT = \frac{2\pi}{\omega} = 2\pi\sqrt{\frac{m}{k}}

Calculation:

T=2π3.16=1.99 sT = \frac{2\pi}{3.16} = 1.99 \text{ s}

Explanation:

The period is independent of the amplitude (for undamped motion).

3

Step 3: Calculate Frequency

Frequency is the reciprocal of the period:

Equation:

f=1Tf = \frac{1}{T}

Calculation:

f=11.99=0.50 Hzf = \frac{1}{1.99} = 0.50 \text{ Hz}

Explanation:

Frequency tells us how many oscillations occur per second.

4

Step 4: Calculate Maximum Velocity

The maximum velocity occurs when the object passes through the equilibrium position:

Equation:

vmax=Aωv_{max} = A\omega

Calculation:

vmax=1.00×3.16=3.16 m/sv_{max} = 1.00 \times 3.16 = 3.16 \text{ m/s}

Explanation:

This is the fastest speed the object reaches during its motion.

5

Step 5: Calculate Total Energy

The total mechanical energy is conserved in undamped motion:

Equation:

E=12kA2E = \frac{1}{2}kA^2

Calculation:

E=12×10.00×(1.00)2=5.00 JE = \frac{1}{2} \times 10.00 \times (1.00)^2 = 5.00 \text{ J}

Explanation:

This energy oscillates between kinetic and potential forms.

Frequently Asked Questions (FAQ)

What is Simple Harmonic Motion (SHM)?

Simple Harmonic Motion 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.

What are the key characteristics of SHM?

SHM has constant amplitude, constant period, and the motion follows a sinusoidal pattern. The acceleration is always directed toward the equilibrium position and is proportional to the displacement.

How does damping affect the motion?

Damping reduces the amplitude over time. Critical damping (ζ=1) brings the system to rest in the shortest time without oscillation. Overdamping (ζ>1) brings it to rest slowly without oscillation. Underdamping (ζ<1) causes oscillatory decay.

What is the relationship between period and mass?

The period is proportional to the square root of mass: T ∝ √m. This means heavier masses oscillate more slowly, while lighter masses oscillate faster.

How does spring constant affect the motion?

The period is inversely proportional to the square root of spring constant: T ∝ 1/√k. Stiffer springs (higher k) result in faster oscillations, while softer springs result in slower oscillations.

Practice MCQs

  1. What is the period of a spring-mass system with k = 16 N/m and m = 1 kg?
  2. If the mass is doubled while keeping the spring constant the same, the period becomes:
  3. The frequency of a spring-mass system is:
  4. In SHM, the acceleration is maximum when:
  5. What happens to the energy in a damped oscillator?

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