A very thin circular disk with radius R and mass M lies in the xy-plane with its

Question

A very thin circular disk with radius R and mass M lies in the xy-plane with its center at the origin.

a) Assuming that the mass of the disk is distributed uniformly over its circular area, determine the elements of the inertia tensor. Note that the disk has azimuthal symmetry.

b) The disk is now caused to rotate with a constant angular velocity ? about an axis that lies in the xz- plane and makes an angle ? with the z-axis. Derive a relation between ? and the angle ? that the angular momentum vector of the disk makes with the z-axis.

c) Show that the rotational kinetic energy of the disk is proportional to 1+cos2? and find the constant of proportionality.

Explanation / Answer

The moment of inertia about this axis is a measure

of how di?cult it is to rotate the lamina. It plays the same role for rotating bodies that the mass

of an object plays when dealing with motion in a line. An object with large mass needs a large

force to achieve a given acceleration. Similarly, an object with large moment of inertia needs

a large turning force to achieve a given angular acceleration. Thus knowledge of the moments

of inertia of laminas, and also of solid bodies, is essential for understanding their rotational

properties.

The element has mass ?m, and is located a distance r from the axis through O. The moment of

inertia of this small piece about the given axis is de?ned to be ?mr2

, that is, the mass multiplied

by the square of its distance from the axis of rotation. To ?nd the total moment of inertia we

sum the individual contributions to give

r

2

?m

where the sum must be taken in such a way that all parts of the lamina are included. As ?m ? 0

we obtain the following integral as the de?nition of moment of inertia, I:

Related Questions

Navigate

• Browse (All)
• Browse A
• Subjects

Integrity-first tutoring: explanations and feedback only — we do not complete graded work. Learn more.