Exhibits, Shows and Displays

How can I explain and show mathematically that a pendulum exhibits simple harmonic motion?

Public Comments

1. There are only two external forces acting on the bob of simple pendulum whether in mean position or any other position at rest or moving. The bob is constrained to move along the curved path of a circle with point of support as centre and the distance from support to the centre of gravity of bob as point particle. at any instant the bob can have velocity in the direction of tangent drawn from it to this circle the sense being the direction in which it is tending to move. The only force of the two which can contribute to this motion of teh bob at any position 'theta' defined by the angle which the thread makes with the vertical, is "mg" and that too only its component mg sin (theta) F= mg sin (theta) = m a where a is the acceleration this force will produce itheeh direction of force. Let x be the shortest distance from mean position of the bob when its position is described by theta. Then x = l tan (thehta) = l sin (theta) if 'theta' is small. Or sin (theta) = x/l So,F = ma = mg sin (theta) = (mg/l) x. If we consider the direction, then F is always in the opposite direction of displacement x. So we have F = - (mg/l) x So we get for teh motion of the pendulum for small theta F(x) = -kx. Here k = mg/l.So the bob must execute SHM with period, T T = 2pi sqrt(m/k) = 2pi sqrt[m/(mg/l)] = 2pi sqrt(l/g)