Obtain the expression for the period of a simple pendulum performing S.H.M.

To derive the expression for the period of a simple pendulum performing simple harmonic motion (S.H.M.), we need to consider the forces acting on the pendulum and how they relate to its motion. A simple pendulum consists of a mass (or bob) attached to a string of fixed length, swinging back and forth under the influence of gravity.

Understanding the Forces at Play

When the pendulum is displaced from its equilibrium position, gravity acts to restore it back to that position. The restoring force is proportional to the sine of the angle (θ) between the string and the vertical. For small angles, we can use the small-angle approximation, where sin(θ) is approximately equal to θ (in radians).

Setting Up the Equation of Motion

The restoring force (F) acting on the pendulum can be expressed as:

• F = -mg sin(θ)

Using the small-angle approximation, this simplifies to:

• F ≈ -mgθ

According to Newton's second law, F = ma, where 'a' is the acceleration of the pendulum. The acceleration can also be expressed in terms of angular displacement:

• a = -g/L * θ

Here, 'L' is the length of the pendulum. Thus, we can write:

• m(-g/L)θ = m * a

By canceling 'm' from both sides, we arrive at the equation:

• θ'' + (g/L)θ = 0

This is a second-order linear differential equation characteristic of simple harmonic motion.

Finding the Period of the Pendulum

The general solution to this type of equation is of the form:

• θ(t) = θ₀ cos(ωt + φ)

where 'ω' is the angular frequency. The angular frequency is given by:

• ω = √(g/L)

The period (T) of the pendulum, which is the time taken for one complete cycle of motion, is related to the angular frequency by the formula:

• T = 2π/ω

Substituting the expression for ω, we get:

• T = 2π√(L/g)

Final Expression for the Period

Thus, the period of a simple pendulum performing simple harmonic motion is:

• T = 2π√(L/g)

This formula shows that the period depends only on the length of the pendulum 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). This is a fascinating aspect of simple harmonic motion, illustrating the uniformity of gravitational effects on pendular motion.

Practical Implications

In practical terms, if you were to increase the length of the pendulum, the period would increase, meaning it would take longer to complete a swing. Conversely, a shorter pendulum would swing back and forth more quickly. This relationship is crucial in designing pendulum clocks and understanding various physical systems that exhibit harmonic motion.