How do you derive the formula for the time period of a pendulum?

The formula for the time period of a pendulum is derived using simple harmonic motion principles and the acceleration due to gravity.

The motion of a pendulum is a classic example of simple harmonic motion (SHM). In SHM, the restoring force is directly proportional to the displacement and acts in the direction opposite to that of displacement. For a pendulum, the restoring force is the component of the weight of the bob acting along the arc.

Let's consider a simple pendulum, which consists of a small bob of mass 'm' suspended from a light inextensible string of length 'l'. When the bob is displaced from its equilibrium position and released, it starts oscillating back and forth. The displacement 'x' of the bob from its equilibrium position can be represented as x = lθ, where θ is the angle made by the string with the vertical.

The restoring force acting on the bob is -mg sinθ, where 'g' is the acceleration due to gravity. According to Newton's second law, this force equals the mass times the acceleration of the bob, i.e., -mg sinθ = m * (d²x/dt²).

For small angles, sinθ can be approximated as θ. So, the equation becomes -mgθ = m * (d²x/dt²). Substituting x = lθ, we get -mg/l * x = d²x/dt². This is a differential equation of SHM, whose solution gives the time period of the pendulum.

The time period 'T' of the pendulum is given by T = 2π √(l/g). This formula tells us that the time period of a simple pendulum is directly proportional to the square root of its length and inversely proportional to the square root of the acceleration due to gravity. It's important to note that this formula is an approximation that assumes small oscillations and an ideal, frictionless system. In real-world situations, factors like air resistance and the size and shape of the bob can affect the time period.

IB Physics Tutor Summary: To derive the formula for a pendulum's time period, we use simple harmonic motion and gravity. A pendulum swings due to a restoring force proportional to its displacement. Assuming small swings and no air resistance, we get the formula T = 2π √(l/g), where T is the time period, l is the string length, and g is gravity. This shows the pendulum's time is linked to its length and gravity.