About pendulum

Question

It is basically a question of pendulum. the question is A bob of mass ‘m’ is hanging vertically with the help of a weightless, inextensible and flexible
string from a rigid support. It is given a small angular displacement ‘φ’. Derive the expression of
the time period and frequency of oscillation and hence show that the displacement and velocity
graph of the oscillator is elliptical.

Answer 1

We begin by looking at the free-body diagram of the bob. Gravity acts on the bob thus we have F_g=mg. Part of this gravitational force is balanced with the tension of the string, T, such that T=mg(cosφ) while the remaining, known as the restoring force, is perpendicular thus F_restore=mg(sinφ). The displacement of the bob as it swings, s, is directly proportional to the string length by s=Lφ where L is the string length. Since pendulums follow simple harmonic motion, we can substitute s into the simple harmonic motion equation: F_restore=-ks. Since we also know F_restore=mg(sinφ), then mg(sinφ)=-ks. Plugging in s=Lφ and solving for k, we get k=-mg(sinφ)/Lφ. We notice that for small angles, φ, the value of (sinφ)/φ≈1 so we can simplify the equation to k=-mg/L. The simple harmonic motion equation for the period of an oscillating system is T=2π(m/k)^0.5 so plugging in our k for the pendulum gives: T=2π(L/g)^0.5. The frequency is merely the reciprocal of the period: f=1/T.