Show that the motion of simple pendulum is simple harmonic and hence derive an equation for its time period. What is seconds pendulum?

To demonstrate that the motion of a simple pendulum is simple harmonic, we first need to understand the characteristics of simple harmonic motion (SHM). In SHM, the restoring force acting on the object is directly proportional to the displacement from its equilibrium position and acts in the opposite direction. Let's break down the motion of a simple pendulum and derive the equation for its time period.

Understanding the Simple Pendulum

A simple pendulum consists of a mass (or bob) attached to a string or rod of negligible mass, swinging back and forth under the influence of gravity. The key factors influencing its motion are the length of the pendulum and the acceleration due to gravity.

Deriving the Equation of Motion

When the pendulum is displaced from its equilibrium position (the lowest point), it experiences a restoring force due to gravity. Let's denote:

• θ: the angle of displacement from the vertical
• L: the length of the pendulum
• g: the acceleration due to gravity

For small angles (typically less than 15 degrees), we can use the small angle approximation, where sin(θ) ≈ θ (in radians). The restoring force can be expressed as:

F = -mg sin(θ) ≈ -mgθ

Using Newton's second law, F = ma, where a is the angular acceleration, we can relate the force to the motion of the pendulum:

ma = -mgθ

Since a = -L(d²θ/dt²), we substitute this into the equation:

-mL(d²θ/dt²) = -mgθ

By simplifying, we find:

d²θ/dt² + (g/L)θ = 0

This is the standard form of the equation for simple harmonic motion, where the term (g/L) is the square of the angular frequency (ω²). Therefore, we can identify:

ω² = g/L

Time Period of the Pendulum

The time period (T) of a simple harmonic oscillator is given by:

T = 2π/ω

Substituting our expression for ω, we have:

T = 2π√(L/g)

This equation shows that the time period of a simple pendulum depends only on its length and the acceleration due to gravity, not on the mass of the bob or the amplitude of the swing (as long as the angle is small).

What is a Seconds Pendulum?

A seconds pendulum is a specific type of pendulum that takes exactly two seconds to complete one full oscillation (one swing back and forth). This means that it has a time period of 2 seconds. To find the length of a seconds pendulum, we can use the time period formula:

2 = 2π√(L/g)

Squaring both sides and rearranging gives:

L = g/π²

Using the standard value of g (approximately 9.81 m/s²), we can calculate the length of a seconds pendulum:

L ≈ 9.81/(π²) ≈ 0.994 m

This means that a seconds pendulum has a length of about 0.994 meters. In summary, the motion of a simple pendulum is indeed simple harmonic, and its time period can be derived based on its length and the acceleration due to gravity. The seconds pendulum, with its specific time period of 2 seconds, serves as a fascinating example of this principle in action.