Question

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?

Answer 1

Let v is the speed of the solid sphere at the bottom of the inclined plane.

Now apply law of conservation of energy theorem,
translational kinetic energy + rotational kinetic energy = potential energy

or, 1/2 mv² + 1/2 Iw² = mgh

Where I is moment of inertia , w is angular speed, h is height of inclined plane from ground.

for solid sphere, I = 2/5 mr²
and w = v/r

So, 1/2 mv² + 1/2 (2/5)mr² × (v²/r²) = mgh

or, v² = 10gh/7 => v = √{10gh/7}.

Here it is clear that final speed is directly proportional to height .
as h is same in two cases, v must be same.
e.g., it will reach the bottom with the same speed. time taken to roll down the two plane will also e the same as their height is the same.

Answer 2

Explanation:

Given, moment of inertia of a disc of mass M and radius R about any of its diameters = \frac{MR^2}{4}

4

MR

2

we have to find moment of inertia about an axis normal to the disc and passing through a point on its edge.

first , find M.I about an axis normal to the plane and passing through its centre.

From perpendicular axis theorem,

M.I about centre of disc , I = 2 × moment of inertia of disc about diameter.

= 2 × MR²/4 = MR²/2

Now, use parallel axis theorem,

Moment of inertia about an axis normal to the plane and passing through its centre = MR² + MR²/2

= 3/2 MR²