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Simple Harmonic Motion Formula

Simple Harmonic Motion Formulae

Simple Harmonic Motion (SHM) is a fundamental concept in physics that describes the repetitive, back-and-forth motion exhibited by various systems. The key formula related to simple harmonic motion include:


Video Explanation

Here is a video explanation of simple harmonic motion and how to apply its equations to solve a problem:

1. Generalized Equation of Simple Harmonic Motion

The formula for a generalized simple harmonic motion is

x = A cos ( ω t + ϕ )

Where x is the displacement

A = Amplitude

ω = Angular velocity

t = time

ϕ = Phase angle

We use a generalized equation for the simple harmonic motion of a body that undergoes oscillation around a stable equilibrium position. Furthermore, for us to apply this formula [x = A cos ( ω t + ϕ )], the system should exhibit a restoring force proportional to its displacement from the equilibrium position. Additionally, the motion should be periodic with a constant frequency and amplitude. The acceleration needs to be directly proportional to the displacement in the opposite direction.

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2. Amplitude (A)

The formula for calculating the amplitude of a simple harmonic motion is:

A = v / ω

Where A is the amplitude

v = linear speed or velocity

ω = angular velocity

3. Period (T)

The period of a simple harmonic motion is the time taken to complete one cycle, oscillation or vibration. The formula to calculate the period of an oscillating body is

Period (T) = Time taken (t) / number of oscillations (n)

Which implies that T = t / n

We can also use the following formulae for period:

a. T = 1 / f

b. T = 2π / ω

c. T = √(k/m)

Where f is the frequency of oscillations.

k = force constant

Period is measured in seconds (s).

4. Frequency (f)

Frequency of a simple harmonic motion is the number of complete oscillations per second. The formula for calculating frequency is

Frequency (f) = 1 / period (T)

which shows that

f = 1 / T

The other formulae for frequency is f = ω / or f = (1/)√(k/m)

ω = angular frequency

k = force constant

m = mass of the body

The unit is a cycle per second (s-1) or Heartz(Hz).

5. Angular Velocity or Frequency (ω)

The formula for calculating angular velocity is

ω = θ / t, or ω = 2πf

Where ω = Angular velocity

θ = Angle

t = time

f = frequency

We can also use the formula below to calculate angular velocity

ω = √α/r, ω = /T or ω = √α/A

The unit of angular velocity is radians per second (rads-1)

Another formula we can use for angular frequency is

ω = √(k/m)

where k = force constant

m = mass of the body

6. Angular Acceleration (α)

For angular acceleration, we use

α = ω2r or α = ω2A

The unit of angular acceleration is also radians per second (rads-1)

7. Linear Speed or Velocity (v)

The formula for linear speed is

v = ωr or v = ωA

We can also use the formula below to calculate linear speed

v = ω√(A2 – x2)

The unit for linear speed is meters per second (m/s)

8. Linear Acceleration (a)

The formula for calculating linear acceleration of an oscillating body is

a = αr

r = radius (measured in meters, m)

9. Meaning of 3600 in Simple Harmonic Motion

In a simple harmonic motion, 3600 = 2π rad

10. Restoring Force Exerted by a Spring

F = – kx


k = force constant

x = displacement

11. Energy in Simple Harmonic Motion

The formula for energy in simple harmonic motion is

E = (1/2)mv2 + (1/2)kx2


If θ = Angle

r = A = radius of the circle

s = distance


θ = s/r

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