In the figure shown, `u=sqrt(6gR)(gtsqrt(5gR))`
Find `h,v,a_(r),a_(t),T` and `F_("net")` when
(a) `theta=60^(@)`
(b) `theta=90^(@)`
(c) `theta=180^(@)`

Answer

A, B

Explanation

When `theta=60^(@)`

In the figure, we can see that,
`h=PM=OP-OM=R-Rcostheta`
`=R-Rcos60^(@)=R-(R)/(2)`
`v=sqrt(u^(2)-2gh)=sqrt(6gR-2g((R)/(2))=sqrt(5gR)`
`a_(r)=(v^(2))/(R)=((sqrt(5gR))^(2))/(R)=5g`
`a_(t)=(F_(t))/(m)=(sqrt(3)mg)/(2m)=(sqrt(3)g)/(2)`
`a=sqrt(a_(r)^(2)+a_(t)^(2))=sqrt((5g)^(2)+((sqrt(g))/(2))^(2))=sqrt(103)/(2)g` ltbr. `T=(mg)/(2)=ma_(r)=(5g)`
`T=5.5mg`
`F_("net")=ma=(sqrt(103))/(2)mg`
(b) When `theta=90^(@)`

`h=R`
`v=sqrt(u^(2)-2gh)=sqrt(6gR-2gR)`
`=2sqrt(gR)`
`a_(r)=(v^(2))/(R)=((2sqrt(gR)^(2))/(R))=4g`
`a_(t)=(F_(t))/(m)=(mg)/(m)=g`
`a=sqrt(a_(r)^(2)+a_(t)^(2))=sqrt((4g)^(2)+(g)^(2))=sqrt(17)g`
`T=ma_(r)=m(4g)`
`=4mg`
`F_("net")=ma`
`=sqrt(17)mg`
(c) Where `theta=180^(@)`
`h=2R`
`v=sqrt(u^(2)-2gh)`
`=sqrt(6gR-2gxx2R)`
`=sqrt(2gR)`
`a_(r)=(v^(2))/(R)=((sqrt2gR)^(2))/(R)` `=2g`
`a_(t)=(F_(t))/(m)=0`
`a=a_(r)`
`2g`
`T+mg=ma_(r)=m(2g)`
`T=mg`