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?
Answers
EXPLANATION.
Solid sphere rolls down two different
inclined planes of the same height
but different angle of inclination.
1) = will it reach the bottom with the
same speed in each case.
Total energy of the sphere at top =
potential energy = mgh
if at the bottom the sphere has both
translation and rotation energy.
translation energy = 1/mv²
rotational energy = 1/2iw²
by using conservation of energy
decrease in P. E = increase in K. E
=> mgh = 1/2mv² + 1/2iw² ......(1)
For the solid sphere = I = 2/5MR²
angular speed = (w) = v/R
=> (1/2mv²) + 1/2 X ( 2/5MR²) X v²/R² = mgh
=> 7/10 Mv² = mgh
=> v = √10gh/7
speed is independent of mass therefore,
speed is same in each case
2) = will it take longer to roll down
one plane then the other.
=> yes, the sphere will take longer time
to roll down because one plane is smaller
angle of inclination.
3) = if so which one and why?
=> because, acceleration for the less
inclined plane is less.
therefore, it take to longer time to reach the
bottom angle the plane with ø inclination.