This problem applies the parallel axis theorem.

Parallel axis theorem:

$\overline{){\mathbf{I}}{\mathbf{=}}{{\mathbf{I}}}_{\mathbf{c}\mathbf{m}}{\mathbf{+}}{\mathbf{m}}{{\mathbf{d}}}^{{\mathbf{2}}}}$ Where I is the moment of inertia of the object about a parallel axis, I_{cm} is the moment of inertia about the center of mass, and m is the **total mass**, and d is the distance of the parallel axis.

Moment of inertia of a point mass:

$\overline{){\mathbf{I}}{\mathbf{=}}{{\mathbf{mR}}}^{{\mathbf{2}}}}$ Where m is the mass of the object and r is the distance from the mass to the point of rotation.

X-coordinate of the center of mass:

$\overline{){{\mathbf{x}}}_{\mathbf{c}\mathbf{m}}{\mathbf{=}}\frac{{\mathbf{m}}_{\mathbf{1}}{\mathbf{x}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}{\mathbf{x}}_{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{m}}_{\mathbf{n}}{\mathbf{x}}_{\mathbf{n}}}{{\mathbf{m}}_{\mathbf{1}}\mathbf{+}{\mathbf{m}}_{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{m}}_{\mathbf{n}}}}$ where m_{1}, m_{2} are masses of particles/objects in the system and x_{1}, x_{2}, are the coordinates of the system.

To solve many problems about rotational motion, it is important to know the moment of inertia of each object involved. Calculating the moments of inertia of various objects, even highly symmetrical ones, may be a lengthy and tedious process. While it is important to be able to calculate moments of inertia from the definition , in most cases it is useful simply to recall the moment of inertia of a particular type of object. The moments of inertia of frequently occurring shapes (such as a uniform rod, a uniform or a hollow cylinder, a uniform or a hollow sphere) are well known and readily available from any mechanics text, including your textbook. However, one must take into account that an object has not one but an infinite number of moments of inertia. One of the distinctions between the moment of inertia and mass (the latter being the measure of translational inertia) is that the moment of inertia of a body depends on the axis of rotation. The moments of inertia that you can find in the textbooks are usually calculated with respect to an axis passing through the center of mass of the object. However, in many problems the axis of rotation does not pass through the center of mass. Does that mean that one has to go through the lengthy process of finding the moment of inertia from scratch? It turns out that in many cases, calculating the moment of inertia can be done rather easily if one uses the parallel-axis theorem. Mathematically, it can be expressed as *I = I _{cm} + md*

Part A. Using the definition of moment of inertia, calculate** I**_{cm}, the moment of inertia about the center of mass, for this object. Express your answer in terms of * m* and

** I_{cm} **= ?

Part B. Using the definition of moment of inertia, calculate *I*_{B}, the moment of inertia about an axis through point B, for this object. Point B coincides with the center of) one of the spheres (see the figure). Express your answer in terms of ** m** and

* I_{B}* =

Part C. Now express I

*I*_{B} = I_{cm} +

Part D. Using the definition of moment of inertia, calculate *I*_{C}, the moment of inertia about an axis through point C, for this object. Point C is located a distance ** r** from the center of mass (see the figure). Express your answer in terms of

*I*_{C}=

Part E Now express *I*_{C} for this object using the parallel-axis theorem. Express your answer in terms of ** m** and

*I*_{C} = I_{cm} +

Consider an irregular object of mass ** m**. Its moment of inertia measured with respect to axis A (parallel to the plane of the page), which passes through the center of mass (see the second diagram), is given by I

Part F. Which moment of inertia is the smallest?

a. I_{A}

b. I_{B}

c. I_{C}

d. I_{D}

e. I_{E}

Part G. Which moment of inertia is the largest?

a. I_{A}

b. I_{B}

c. I_{C}

d. I_{D}

e. I_{E}

Part H. Which moments of inertia are equal?

a. I_{A} and I_{D}

b. I_{B} and I_{C}

c. I_{C} and I_{E}

d. No two moments of inertia are equal.

Part I. Which moment of inertial equals 4.64mr^{2}?

a. I_{B}

b. I_{C}

c. I_{D}

d. I_{E}

Part J. Axis X, not shown in the diagram, is parallel to the axes shown. It is known that I_{X} = 6mr^{2}. Which of the following is a possible location for axis X?

a. between axes A and C

b. between axes C and D

c. between axes D and E

e. to the right of axis E

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