A solid sphere rolls down two different inclined planes of the same heights but different angles of inclination. (a) Will it reach the bottom with the same speed in each case? (b) Will it take longer to roll down one plane than the other? (c ) If so, which one and why?

Explanation

(a) In figure two inclined planes AB and AC is shown with angle of inclination `theta_(1)andtheta_(2)` respectively and have equal height h. Here `theta_(1)gttheta_(2)`.

The velocity of solid sphere at the bottom of the inclined plane of angle `theta`.
`v=sqrt((2gh)/((1+(K^(2))/(R^(2)))))`
Radius of gyration for sphere `K^(2)=(2)/(5)R^(2)`
`therefore v=sqrt((2gh)/(1+(2)/(5)))`
`therefore v=sqrt((10)/(7)gh)`
In this formula there is no term of angle, means the velocity of sphere at the bottom does not depend on the angle and as h is same in the two cases velocity must be same. Time taken to roll down the two planes will also be the same.
(b) Yes, if solid sphere rolls down from both the slopes (1) rolling time from slope (1) is less and rolling time from slope (2) is more
(c ) It will take more time for slope (2) because acceleration of body parallel to the surface of slope of angle `theta`
`a=(gsintheta)/(sqrt(1+(K^(2))/(R^(2))))`
but for solid sphere `K^(2)=(2)/(5)R^(2)`
`therefore a=(gsintheta)/(1+(2)/(5))=(5gsintheta)/(7)`,
Here `(5)/(7)` and g are constant
`therefore apropsintheta....(1)`
`therefore (a_(1))/(a_(2))=(sintheta_(1))/(sintheta_(2))` [From eqn. (1)]
but in first phase sin is increasing function and `theta_(1)gttheta_(2)impliessintheta_(1)gtsintheta_(2)`
`implies (sintheta_(1))/(sintheta_(2))gt1`
`therefore (a_(1))/(a_(2))gt1....(2)`
Now in equation of motion `v=v_(0)+at,v` is same and `v_(0)=0`
`therefore` at is constant
`therefore a_(1)t_(1)=a_(2)t_(2)`
`therefore (a_(1))/(a_(2))=(t_(2))/(t_(1))`
`therefore (t_(2))/(t_(1))gt1` [`because` from eqn. (2)]
`therefore t_(2)gtt_(1)`
(c) Short method :
Suppose `d_(1)` = distance (AB) of slope-1
`d_(2)` = distance (AC) of slope-2
`t_(1)` = time taken by slope AB
`t_(2)` = time taken by slope AC
Here, solid sphere rolls at the same velocity from both the slope. Hence, time taken depend on the distance of slope.
If distance is more, time taken is also more.
Here `d_(2)gtd_(1) [because theta_(1)gttheta_(2)]`
`therefore t_(2)gtt_(1)`