Answer to Question #282161 in Mechanics | Relativity for rai

Question

(a) Find the value of the moment of inertia of the sphere about its axis of rotation.

(b) What is the magnitude of angular velocity of the sphere when it touches the

inclined plane?

(c) How far along the second inclined plane (45º) on the right will the center of the

sphere travel before it comes to a stop?

Answer 1

Solution:

1) The expression for the moment of inertia of a sphere can be developed by summing the moments of infinitesimally thin disks about the z-axis. The moment of inertia of a thin disk is

2) A vector quantity describing the motion of an object in a circular motion; its magnitude is equal to the angular speed (ω) of the particle, and the direction is perpendicular to the plane of its circular motion.

3) From the conservation of energy between the initial and final states:

$$\frac{1}{2}mv^2+\frac{1}{2}I\omega ^2=mgh$$

and $I=\frac{1}{2}mr^2$

$$\implies \frac{1}{2}mv^2+\frac{1}{2}(\frac{1}{2}mr^2)\omega^2=mgh$$

Also for rolling, $v=r\omega$

Thus

$$h=\frac{3v^2}{4g}$$