Given, radius of sphere is a, mass of sphere is
m, horizontal speed of sphere is
v0.
Moment of inertia of solid sphere,
I=2∕5ma2 Let,
h be the height of inclined plane and
∕ be the length of inclined plane.
By using law of conservation of energy,
Total energy at
1= Total energy at 2
∴‌‌PE1+KE1=PE2+KE2 ⇒‌‌0+1∕2mv02+1∕2∕ω2=mgh. . . (i)
As,
‌‌v0=aω⇒ω=‌ ∴‌‌‌mv02+‌(‌ma2)‌=mgh ⇒‌‌‌mv02(1+2∕5)=mgh ⇒‌‌‌=gh . . . (ii)
As,
‌‌sin‌θ=h∕l ⇒‌‌h=I‌sin‌θ Put this value in Eq. (ii), we get
⇒ ‌v02=gl‌sin‌θ⇒l=‌