What is the formula for speed of pendulum at any point?

What is the formula for speed of pendulum at any point using #theta# where #theta# is the angle made by the string of the pendulum with the vertical at the given point?

1 Answer
Jun 6, 2018

A simple pendulum consists of a bob of mass #m# suspended from a friction-less and fixed pivot with the help of a mass-less, rigid, inextensible rod of length #L#. Its position with respect to time #t# can be described by the angle #theta# (measured against a reference line, usually vertical line).
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As shown in the figure above the driving force is

#F=-mgsintheta#
where the #-ve# sign implies that the restoring force acts opposite to the direction of motion of the bob.

Using Newton's Second Law of motion we get linear acceleration #a# as

#a=-gsintheta# .....(1)

As the bob is moving along the arc of a circle, its angular acceleration is given by

#alpha=(d^2theta)/dt^2 = a/L# .....(2)

from (1) and (2) we get differential equation of motion as

#(d^2theta)/dt^2 = -g/L sintheta#

Given the initial conditions #θ(0) = θ_0 and (dθ)/dt(0) = 0#, the solution becomes

#theta (t)=theta _0 cos (sqrt (g/L)t)#

Angular velocity is given by

#dottheta (t)=-theta _0 sqrt (g/L)sin (sqrt (g/L)t)#

Linear velocity is given by #v=romega#. Hence,

#v=Lxx(-theta _0 sqrt (g/L)sin (sqrt (g/L)t))#
#=>v=-theta _0 sqrt (Lg)sin (sqrt (g/L)t)#

Speed is given as

#|v|=theta _0 sqrt (Lg)sin (sqrt (g/L)t)#