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Moment of Inertia of Two Particles Located in the X-Y Plane

In this problem, you will answer several questions that will help you better understand the moment of inertia, its properties, and its applicability.


1. On which of the following does the moment of inertia of an object depend?
A. linear speed
B. linear acceleration
C. angular speed
D. angular acceleration
E. total mass
F. shape and density of the object
G. location of the axis of rotation
Type the letters corresponding to the correct answers. Do not use commas. For instance, if you think that only assumptions C and D are correct, type CD.

2. Find the moment of inertia Ix of particle a with respect to the x axis (that is, if the x axis is the axis of rotation), the moment of inertia Iy of particle a with respect to the y axis, and the moment of inertia Iz of particle a with respect to the z axis (the axis that passes through the origin perpendicular to both the x and y axes).
Express your answers in terms of m and r separated by commas.

3. It is useful to see how the formula for rotational kinetic energy agrees with the formula for the kinetic energy of an object that is not rotating. To see the connection, let us find the kinetic energy of each particle.
Using the formula for kinetic energy of a moving particle , find the kinetic energy of particle a and the kinetic energy of particle b.
Express your answers in terms of m , w , and r separated by a comma.


Solution Preview


Learning Goal: To understand the definition and the meaning of moment of inertia; to be able to calculate the moments of inertia for a group of particles and for a continuous mass distribution with a high degree of symmetry.
By now, you may be familiar with a set of equations describing rotational kinematics. One thing that you may have noticed was the similarity between translational and rotational formulas. Such similarity also exists in dynamics and in the work-energy domain.
For a particle of mass moving at a constant speed , the kinetic energy is given by the formula . If we consider instead a rigid object of mass rotating at a constant angular speed , the kinetic energy of such an object cannot be found by using the formula directly: different parts of the object have different linear speeds. However, they all have the same angular speed. It would be ...

Solution Summary

Two particles are located in the x-y plane. Moment of inertia of the two particles with respect to the x, y and z coordinate axes are found. Further, kinetic energy of the particles are derived and expressed in terms of the particle's mass, distance and angular speed.