Simple Pendulum – Definition, Formulae, Derivation, Examples

A device in which generally a point mass is fixed to a string which is light and inextensible and is suspended from constant support is known as a simple pendulum. The mean position in a simple pendulum is usually the point from where the vertical line passes. Length of the pendulum is denoted by L and is the vertical distance from the suspension point to the center of mass of the body suspended provided it is in its mean position. This type of pendulum tends to have a single resonant frequency.  

Basically, a simple pendulum is a mechanical device demonstrating periodic motion. This has a small circular bob suspended by a thin inextensible string from any fixed end and has length L. It performs oscillatory motion, which is driven by gravitational pull and occurs in the vertical plane. Generally, the suspended bob at the extreme end of the string is massless. The time period of the pendulum can be increased by increasing the length of the string. The time period of the pendulum is independent of the mass of the suspended bob. The time period of a pendulum generally depends on the position of the bob and acceleration due to gravity as it is not uniform in every place on earth.

Simple Pendulum formulae

We can see the pendulum is used for various things such as clocks or if you observe keenly even a swing in the garden is like a pendulum. Let’s look further into this. Important Formulae,

Time Period = 2π/ω₀ = 2π × √(L/g)

Potential energy = mgL(1-cosθ)

Kinetic energy = (½)mv²

Total Energy = Kinetic Energy + Potential Energy =  (½)mv² + mgL(1 – cosθ) = constant.

Some essential terms

• Oscillatory motion – Any to and fro motion performed by the pendulum in a periodic movement and when the pendulum is at its central position then the position is called equilibrium position.
• Time period – It is generally the total time taken by a pendulum to complete one full oscillation, it is denoted by ‘T’.
• Amplitude – The distance between the equilibrium position and the extreme position of the pendulum.
• Length – Length of the string is generally the distance between the fixed end of a string to the center of mass.

Derivation of the time period of a Simple Pendulum

Provided condition,

1. It is a frictionless surrounding.
2. The arms of the pendulum are stiff and massless.
3. Acceleration due to gravity is constant.
4. The motion of the pendulum is in an accurate plane.

We can use the motion equation,

T – mg cosθ = mv²L

Here, torque brings mass to the equilibrium position,

Τ = mgL × sinθ = mgsinθ × L = I × α

Sin θ ≈ θ, for small angles of oscillations.

So,  Iα = -mgLθ

Α = -(mgLθ)/I

-ω₀² θ = -(mgLθ)/I

ω₀² θ = (mgL)/I

ω₀ = √(mgL/I)

By using, I = ML²

Here, I = moment of inertia of the bob

ω₀ = √(g/L)

Hence, the time period of the simple pendulum is given by,

T = 2π/ω₀ = 2π × √(L/g)

Derivation of the potential energy of the simple pendulum

As we know the basic equation of potential energy is,

Potential energy = mgh

Here, m = mass of the object, g = acceleration due to gravity, and h = height of the object.

In simple pendulum movement, the pendulum is constrained by the string. The height here is written in terms of the angle and length of the string.

Therefore, height = L(1 – cosθ)

If θ = 90°, then the pendulum is considered to be at its highest point.

Cos90° = 0, and h = L, then potential energy = mgL

If θ = 0°, then the pendulum is considered to be at its lowest point.

Cos0° = 1, and h = L(1 – 1) = 0, then potential energy = 0

Therefore, potential energy at all points is given as,

Potential energy = mgL(1-cosθ)

Derivation for the Total Energy

As we know the basic equation of kinetic energy is,

Kinetic energy = (½)mv²

Here, m is the mass of the pendulum and v is the velocity of the pendulum.

Kinetic Energy = 0, at maximum displacement, and is maximum at zero displacements. Though the total energy is a constant being a function of time.

The mechanical energy of the pendulum.

In a simple pendulum, the mechanical energy of a simple pendulum remains to be conserved.

Total Energy = Kinetic Energy + Potential Energy =  (½)mv²+ mgL(1-cosθ) = constant.

Points to remember

• Change in temperature of the system may affect the time period of the pendulum as the time period depends on the length of the pendulum.
• A simple pendulum tends to be placed in a non-inertial frame of reference.
• If the mean position of the pendulum changes the value of ‘g’ would be replaced by ‘g_effective’, to determine the time period.

Example – A lift moving with acceleration ‘a’ upwards, then T = 2π × √(L/g_eff) =  2π √[L/(g + a)]

Sample Questions  

Question 1: What is a Simple Pendulum?

Answer:

An idealized body comprising of a bob or particle suspended to one end of the string and the other end fixed to a rigid support. Pendulum when pulled from one side tends to move in to and fro periodic motion and swings in vertical plane due to gravitational pull. This motion is oscillatory and periodic and is so termed as Simple Harmonic motion.

Question 2: In a simple pendulum, what is the effective length?

Answer:

Effective length in a simple pendulum is the length of the string from rigid support to the center of mass of the pendulum. The Center of mass of the pendulum is generally the center point of the bob. Simply, effective length is the distance between the suspension point to the center of the bob of the pendulum.

Question 3: The time period of a Simple Pendulum is 2.4seconds. What is the length of the pendulum? (g=10m/s²).

Solution:

Given,

T = 2.4seconds

To find,

Length of pendulum =?

Formula for time period,

T = 2π/ω₀ = 2π × √(L/g)  

2.4 =  2π × √(L/10)

(2.4)² = (2π)² (L/10)

L = 1.46m.

Question 4: The time period of a Simple Pendulum is 1.2seconds. What is the length of the pendulum? (g = 10m/s²).

Solution:  

Given,

T = 1.2seconds

To find,

Length of pendulum = ?

Formula for time period,

T = 2π/ω₀ = 2π × √(L/g)  

1.2 =  2π × √(L/10)

(1.2)² = (2π)² (L/10)

L = 0.36m.

Question 5: Find the length of a pendulum that has a period of 3.6 seconds then find its frequency.

Solution:  

Given,

T = 3.6seconds

To find,

Length of pendulum =?

Frequency =?

Formula for time period,

T = 2π/ω₀ = 2π × √(L/g)  

3.6 =  2π × √(L/10)

(3.6)² = (2π)² (L/10)

L = 3.2m.

Frequency = 1/T = 1/(3.6) = 0.27

Question 6: The acceleration due to gravity on the surface of the moon is 1.8 m/s². What is the time period of a simple pendulum on the moon if its time period on the earth is 3.6s?

Solution:

Given,

T = 3.6seconds

g = 1.8m/s²

To find,

Length of pendulum = ?

Formula for time period,

On earth,

T = 2π/ω₀ = 2π × √(3.6/10)  

3.6 =  2π × √(L/10)

(3.6)² = (2π)² (L/10)

L = 3.2m

On the moon,

T = 2π/ω₀ = 2π × √(3.2/1.8)  

T =  2π × √(3.2/1.8)

T =  2π × 1.3

T = 8.164seconds.

Therefore, the time period of the pendulum on the moon is 8.164 seconds.

Question 7: The length of the pendulum is 2m, what is the time period? (g = 10m/s²).

Solution:

Given,

L = 2m

g=10m/s²

To find,

Time period = ?

Formula for time period,

T = 2π/ω₀ = 2π × √(L/g)

T = 2π × √(2/10)

T = 2.80seconds.