Derive the expression T=2π√l/g
Sure! The expression T=2π√l/g is derived from the formula for the period of a simple pendulum. The period T is the time it takes for one complete oscillation of the pendulum, l is the length of the pendulum, and g is the acceleration due to gravity.
The formula for the period of a simple pendulum is given by:
T = 2π√(l/g)
This formula can be derived using the equation for the period of a simple harmonic oscillator:
T = 2π√(m/k)
Where m is the mass of the object attached to the spring and k is the spring constant.
For a simple pendulum, the restoring force acting on the pendulum bob is provided by gravity, so the force equation can be written as:
F = -mg sin(θ)
Where θ is the angle the pendulum makes with the vertical. The negative sign indicates that the force is acting in the opposite direction to the displacement of the pendulum bob.
Using Newton's second law, F = ma, and substituting the expression for the force into this equation, we get:
-mg sin(θ) = ml(d^2θ/dt^2)
Where m is the mass of the pendulum bob and d^2θ/dt^2 is the angular acceleration of the pendulum.
Since the angle θ is small (θ << 1), sin(θ) ≈ θ, and the equation simplifies to:
-mgθ = ml(d^2θ/dt^2)
Rearranging this equation, we get:
d^2θ/dt^2 + (g/l)θ = 0
This is a second-order differential equation that describes simple harmonic motion with angular frequency ω = √(g/l).
The period T of a simple harmonic oscillator is related to the angular frequency by the equation:
T = 2π/ω = 2π√(l/g)
Therefore, the expression for the period of a simple pendulum is T = 2π√(l/g).