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Masteringphysics: Mechanics 2 - Assessed

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James Moore
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Question 7

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 workenergy 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 desirable to obtain a formula for kinetic energy of rotational motion that is
similar to the one for translational motion; such a formula would include the term instead of .

Such a formula can, indeed, be written: for rotational motion of a system of small particles or for a rigid object with continuous
mass distribution, the kinetic energy can be written as
.
Here, is called the moment of inertia of the object (or of the system of particles). It is the quantity representing the inertia
with respect to rotational motion.
It can be shown that for a discrete system, say of particles, the moment of inertia (also known as rotational inertia) is given
by
.
In this formula, is the mass of the ith particle and is the distance of that particle from the axis of rotation.
For a rigid object, consisting of infinitely many particles, the analogue of such summation is integration over the entire object:
.
In this problem, you will answer several questions that will help you better understand the moment of inertia, its properties,
and its applicability. It is recommended that you read the corresponding sections in your textbook before attempting these
questions.

Part A
On which of the following does the moment of inertia of an object depend?

Part B
What is the moment of inertia of particle a?

Part C
Find the moment of inertia of particle a with respect to the x axis (that is, if the x axis is the axis of rotation), the
moment of inertia of particle a with respect to the y axis, and the moment of inertia of particle a with respect to the
z axis (the axis that passes through the origin perpendicular to both the x and y axes).

Part D
Find the total moment of inertia of the system of two particles shown in the diagram with respect to the y-axis

Part E
Using the total moment of inertia of the system found in Part D, find the total kinetic energy of the system.
Remember that both particles rotate about the y axis

Part F
Using the formula for kinetic energy of a moving particle , find the kinetic energy of particle a and the
kinetic energy of particle b. Remember that both particles rotate about the y axis.

Part G
Now, using the results of Part F, find the total kinetic energy of the system. Remember that both particles rotate about
the y axis.

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