A solid wooden sphere rolls down two different inclined planes of the same height but of different inclinations. (a) Will it reach the bottom with same speed in each case ?
(b) Will it take longer to roll down one inclined plane than other ? Explain.

Explanation

(a) Using law of conservation of energy,
`(1)/(2)mv^(2)+(1)/(2)Iomgea^(2)=mgh`
or `(1)/(2)mv^(2)+(1)/(2)((2)/(5)mR^(2))(v^(2))/(R^(2))=mgh`
or `(7)/(10)v^(2)=gh " or" v=sqrt((10gh)/(7))`
Since h is same for both the inclined planes therfore v is the same.
(b) `l=(1)/(2)((g sin theta)/(1+(K^(2))/(R^(2))))t^(2)=(g sin theta)/(2(1+(2)/(5)))t^(2)=(5g sin theta)/(14) t^(2)`
or `t=sqrt((14 l)/(5 g sin theta))`
Now, `sin theta =(h)/(l) " or " l =(h)/(sin theta)`
`therefore t=(1)/(sin theta) sqrt((14 h)/(5g))`
Lesser the value of `theta`, more will be t.
( c) Clearly, the solid sphere will take longer to roll down the plane with smaller inclination.