Simple Pendulum

A simple pendulum is a device composed of a point mass attached to a light, inextensible string and suspended from a fixed support. The vertical line passing through the fixed support is the mean position of the pendulum. The vertical distance between the point of suspension and the center of mass of the suspended body, when it is in the mean position, is called the length of the pendulum and is denoted by $L$. This form of the pendulum is based on a resonant system with a single resonant frequency.

The simple pendulum is a mechanical system that sways or moves in an oscillatory motion. This motion occurs in a vertical plane and is mainly driven by the gravitational force. Interestingly, the bob that is suspended at the end of a thread is very light, or even massless. The period of a simple pendulum can be extended by increasing the length of the string while taking the measurements from the point of suspension to the middle of the bob. However, it should be noted that if the mass of the bob is changed, the period remains unchanged. The period is mainly influenced by the position of the pendulum in relation to Earth, as the strength of the gravitational field is not uniform everywhere.

We will explore the simple pendulum on this page and learn about the conditions under which it performs simple harmonic motion. Additionally, we will derive an interesting expression for its period. Pendulums are commonly used in many different instances, such as in clocks to keep track of time, as a child’s swing, and even as a sinker on a fishing line.

Important Terms

The Oscillatory Motion of a Simple Pendulum: Oscillatory motion is defined as a back-and-forth motion of the pendulum in a regular pattern, with the centre point of oscillation known as the equilibrium position.

The Time Period of a Simple Pendulum: It is defined as the time taken by the pendulum to complete one full oscillation and is represented by the letter ‘T’.

The Amplitude of a Simple Pendulum: It is defined as the distance travelled by the pendulum from its equilibrium position to one side.

The length of a simple pendulum is defined as the distance between the point of suspension and the centre of the bob, denoted by l.

Time Period of Simple Pendulum

A point mass M is suspended from the end of a light, inextensible string whose upper end is fixed to a rigid support. The mass is displaced from its mean position.

Assumptions:

The air provides almost no friction to the system.

The arm of the pendulum does not bend, compress, or have mass.

The pendulum swings in a perfect plane

Gravity remains constant!

Simple Pendulum image 2

Derivation of the Time Period of a Simple Pendulum

Using the equation of motion, $T - m g \cos θ = m v^{2} L$

The torque tending to restore the mass to its equilibrium position

$τ = m g L \times \sin θ = m g \sin θ \times L = I \times α$

For small angles of oscillations, $\sin θ \approx θ$

Therefore, $I_{α} = - m g L θ$

$α = - \frac{m g L θ}{I}$

$- ω_{02} θ = - \frac{m g L θ}{I}$

$ω_{0}^{2} = (m g L) / I$

$ω_{0} = \sqrt{\frac{m g L}{I}}$

Using I = ML², [where I denotes the moment of inertia of the bob]

We get, $ω_{0} = \sqrt{\frac{g}{L}}$

Therefore, the time period of a simple pendulum is given by:

$$T = \frac{2\pi}{\omega_0} = 2\pi \times \sqrt{\frac{L}{g}}$$

Energy of Simple Pendulum

Potential Energy

The potential energy is expressed by the fundamental equation

Potential Energy = mgh

M is the mass of the object.

G is the acceleration due to gravity.

H is the height of the object.

The height of the pendulum is not a result of free fall, but is instead determined by the angle θ and length L of the rod or string. Therefore, h = L(1 – cos θ).

Therefore, when $θ = 90 °$ , the pendulum is at the highest point, $c o s 90 ° = 0$ , and h = L.

Potential Energy = m x g x L

When θ = 0°, the pendulum is at the lowest point. Then cos θ°= 1. Therefore h = L (1-1) = 0

Potential Energy = mgL (1-1) = 0

The potential energy at all points in between is given as $m g L (1 - \cos θ)$ .

Kinetic Energy

K.E. = (1/2) mv²

m is the mass of the pendulum.

V is the velocity of the pendulum.

At the lowest point, the kinetic energy is at its maximum, while at the highest point, it is zero. Nonetheless, the total energy remains constant over time.

Mechanical Energy of the Bob:

The mechanical energy of the bob is the sum of its potential energy and kinetic energy.

The mechanical energy of a simple pendulum is conserved, according to the Law of Conservation of Energy.

$$E = KE + PE = 1/2 mv^2 + mgL(1 - cosθ) = constant$$

⇒ Note: This statement has been rewritten.

If the temperature of a system changes, then the length of the simple pendulum will also change, resulting in a change in the time period of the pendulum.

A simple pendulum placed in a non-inertial frame of reference, such as an accelerated lift, a horizontally accelerated vehicle, or a vehicle moving along an inclined plane, can be used to study the effects of acceleration.

The mean position of the pendulum may change, in which case g is replaced by g effective for determining the time period (T).

For Example:

The lift moving upwards with acceleration ‘a’, then, $T = 2 π \times \sqrt [L / (g + a)] = 2 π \sqrt (L / g_{e f f})$

If the lift is moving downward with acceleration ‘a’, then $T = 2 π \times \sqrt [L / (g - a)]$

The time period of a simple pendulum with length equal to the radius of the earth (L = R = 6.4 x 10⁶ m) is $T = 2 π \sqrt R / 2 g$ .

$T = 2 π \times \sqrt (L / g)$ , where $L >> R$ near the earth surface.

Physical Pendulum

A simple pendulum is an idealized model, which is not achievable in reality. On the other hand, a physical pendulum is a real pendulum in which a body of finite shape oscillates. From its frequency of oscillation, we can calculate the moment of inertia of the body about the axis of rotation.

Simple Pendulum image 1

A body of irregular shape and mass (m) is free to oscillate in a vertical plane about a horizontal axis passing through a point, with the weight of its centre of gravity (G) being acted upon by the force of gravity (mg) downwards.

⇒ Check: Center of Mass of a System of Particles

If the body is displaced through a small angle (θ) and released from this position, a torque is exerted by the weight of the body to restore it to its equilibrium.

τ = mgd sinθ

τ = αI

$$I\alpha = -mg\sin\theta$$

$$d^2θ/dt^2 = -mgsinθ$$

Where I = Moment of Inertia of a Body about the Axis of Rotation

$$d\frac{d^2\theta}{dt^2} = \frac{mgd}{I} \theta [Since, \sin\theta \approx \theta]$$

$$\omega_0 = \sqrt{\frac{mgd}{I}}$$

Period of a Physical Pendulum

$T = 2 π / ω ₀ = 2 π \sqrt [m / g I]$

For I, using the Parallel Axis Theorem,

$I = I_{c} m + m d^{2}$

Therefore, the time period of a physical pendulum is given by:

$T = 2 π \sqrt [(I_{c m} + m d^{2}) / m g d]$

Frequently Asked Questions on Simple Pendulum

What is a simple pendulum?

A simple pendulum is a mass suspended from a fixed point that swings back and forth under the influence of gravity.

A simple pendulum is a point mass suspended by a weightless and inextensible string fixed rigidly to a support.

What is an expression for the time period of a simple pendulum?

$T = 2 π \sqrt (l / g)$

The length of the pendulum is l.

g is the acceleration due to gravity.

What Factors Affect the Energy of a Particle Executing Simple Harmonic Motion?

Mass of the particle.

The amplitude of the particle squared.

The Frequency Squared of the Vibrating Particle

What is a Second’s Pendulum?

A second’s pendulum is a pendulum that has a period of exactly two seconds; that is, it takes exactly two seconds to complete one full swing. It is used in clocks and other timekeeping devices to keep accurate time.

A pendulum with a time period of 2 seconds is referred to as a “second’s pendulum”.

When does the motion of a simple pendulum become simple harmonic motion?

The motion of a simple pendulum will be simple harmonic motion if its angular displacement θ is very small.

When a pendulum is taken on top of the mountain will the time period increase or decrease?

The time period increases as g decreases, so it gains time.

How Does the Earth’s Gravity Affect a Simple Pendulum?

At the center of the Earth, g = 0

Therefore, T = 2π√(l/g) = ∞

The pendulum will take an infinite amount of time to complete one vibration; in other words, the pendulum will not oscillate at the centre of the earth.

A vibrating simple pendulum of time period T is placed in a lift which is accelerating upwards. What will be the effect on the time period?

The weight of the pendulum increases when the lift is accelerated upwards, and the new weight (mg’) is given as

‘mg’ = mg + ma

‘g’ = g + a

#The Period of the Pendulum

T = 2π√(l/g) becomes T = 2π √(l/g)

$$T = 2\pi\sqrt{\frac{l}{g’}}$$

Since g' > g, so T' < T

Therefore, the time period decreases.

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