A ball of mass `m` is released from rest from a height `H` along a smooth, light and fixed tube having a semicircular portion so that the ball just reaches the top of the semicircle.
a. Find the radius of the semi-circle.
b. Find the maximum force imparted by the ball on the ground.

Explanation

a. Since the ball moves inside the tube, it can stay at the top of the tube with zero speed.
`implies v_(B)=0`

Applying `COE`
`1/2m(v_(B)^(2)-v_(A)^(2))=mgh=0`
Putting `v_(A)=v_(B)=0` and `h=H-2r`
we obtain `r=H/2`
b. At the lowest positon `C`, let the (maximm) thrust exerted on the ground be `N`.
`N-mg=ma_(r)=(mv_(c)^(2))/rimpliesN=m[v_(c)^(2)/r+g]`
where `v_(c)` can be obtained by coserving energy between `A` and `C` as
`1/2mV_(c)^(2)=mgH=0implies v_(c)^(2)=2gH`
`impliesN=m[(sqrt(2g)^(2))/r+g]`
Putting `r=H/2` we obtain `N=m[(2gH)/(H/2)+g]=5mg`