## 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 = 2π√(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**

**/**

**2π**or

**f = (1/**)

**2π****√(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 = 2π/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 = ω ^{2}r** or

**α = ω**A

^{2}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√(A ^{2} – x^{2})**

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 360^{0} in Simple Harmonic Motion

In a simple harmonic motion, **360 ^{0} = 2π rad**

### 10. Restoring Force Exerted by a Spring

F = – kx

Where

k = force constant

x = displacement

### 11. Energy in Simple Harmonic Motion

The formula for energy in simple harmonic motion is

E = (1/2)mv^{2} + (1/2)kx^{2}

## Note:

If θ = Angle

r = A = radius of the circle

s = distance

Then

**θ = s/r**

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