Question

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?

Answer 1

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)`