A simple pendulum can be described by the equation d2 ?(t) / dt2 = -(g/l)?(t) wh

Question

A simple pendulum can be described by the equation

d2 ?(t) / dt2 = -(g/l)?(t)

where ? is the angular displacement from the vertical, l is the length of the pendulum and g = 9.81 m s-2

3) Explain what is meant by the phase of an oscillator, and write down anexpression for this quantity in the case of the simple pendulum described byEquation 1. Find the difference in phase at t = 10 s, between pendulum 1 andanother swinging pendulum, call it pendulum 2, with length 0.35 m andangular displacement ? = 0.00 rad at t = 0.00 s. Express your answer as avalue lying between 0 and 2p.

Explanation / Answer

For a simple pendulum the time period is given by
T = 2pi(l/g)^1/2
where l is length of pendulum and g is acceleration due to gravity and has value g = 9.8 m/s^2
let the angular displacement of pendulum be A
we know that
S = l * A
S is length of arc
differentiating the above equation we get
dS/dt = l * dA/dt
or v = l * w
where w is angular velocity of pendulum
the speed of pendulum is
v = A * w * sin(wt)
when wt = 90o then
v_max = A * w

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