A sphere a disk and a hoop made of homogeneous materials have the same radius (10 cm) and mass (3kg) They are released from rest at the top of a `30^(@)` incline and roll down without slipping through a vertical distance of 2m. `(g=9.8m//s^(2))`
(a). What are their speeds at the bottom?
(b). findt the friction force `f` each case
(c). if they start together at `t=0`, at what time does each reach the bottom?

Explanation

`(I)/(mR^(2))=(2)/(5)=0.4` for sphere
`=(1)/(2)=0.5` for disc and `=1` for help
`s=(2)/(sin30^(@))=4m`
for sphere
`a=(gsintheta)/(1+(I)/(mR^(2)))=(9.8xx(1)/(2))/(1+0.4)=3.5m//s`
`thereforev=sqrt(2sa)=sqrt(2xx3.5xx4)=5.29m//s`
`f=(mgsintheta)/(1+mR^(2)//I)=(3xx9.8xx(1)/(2))/(1+((1)/(0.4)))=4.2N`
`t=sqrt((2s)/(a))=sqrt((2xx4)/(3.5))=1.51s`
Similarly the value of disk and hoop can be obtained.