A sphere of radius `0.1m` and mass `8pikg` is attached to the lower end of a steel wire of length `5.0m` and diameter `10^(-3)m`. The wire is suspended from `5.22m` high ceiling of a room . When the sphere is made to swing as a simple pendulum, it just grazes the floor at its lower point. Calculate the velocity of the sphere at the lowest position . Y for steel `=1.994xx10^(11)N//m^(2)`.

Explanation

As the length of the wire is `5m` and diameter of sphere `2xx0.1=0.2m` and at lowest point it grazes the floor which is at a distance `5.22m` from the roof, the increase in length of the wire at lowest point

`DeltaL=5.22-(5+0.2)=0.02m`
So tension in the wire (due to elasticity)
`T=(YA)/(L)DeltaL=(1.994xx10^(11)xxpi(5xx10^(-4))^(2)xx0.02)/(5)=199.4piN`
and as equation of circular motion of a mass m tied to a string in a vertical plane is
`(mv^(2)//r)=T-mgcostheta`
So, at lowest point
`(mv^(2)//r)=T-mg` [as `theta=0`]
But here `r=5+0.02+0.1=5.12m`
So, `(8piv^(2)//5.12)=(109.4pi-8pixx9.8)`
i.e., `v^(2)=(121xx5.12//8)=77.44`
So, `v=8.8ms//s`