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?
What is the formula for speed of pendulum at any point using
1 Answer
A simple pendulum consists of a bob of mass
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=-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
#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=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)#