Center of mass variable ambiguity

[QD311]

Homework Statement

I'm trying to understand the equations given in my textbook (Principles of Physics, Ninth International Edition) for finding the center of mass. The equations are given below. They're used to find the x, y, and z components of the center of mass for objects. I'm trying to interpret them in the framework of a problem given to us, where we had to calculate the moment of inertia of a rod of mass M, with a solid sphere of mass 2M attached at the end, with the axis of rotation through the center of the rod and sphere (a rough approximation of a baseball bat).

Homework Equations

$${ x }_{ com }=\frac { 1 }{ M } \int { x\quad dm } ,\quad y_{ com }=\frac { 1 }{ M } \int { y\quad dm } ,\quad { z }_{ com }=\frac { 1 }{ M } \int { z\quad dm }$$

The Attempt at a Solution

The problem is, that I have no idea what to put in for the 'x', 'y' and 'z' variables in the integral equations.

 

[tiny-tim]

Welcome to PF!

Hi QD311!Welcome to PF!

… I have no idea what to put in for the 'x', 'y' and 'z' variables in the integral equations.

That's just the x y and z coordinates of each point in the body.

But why do you want the centre of mass (and why are you integrating)?

Why not just use the standard moment of inertia formulas for a rod and for a sphere, together with the parallel axis theorem?

 

[QD311]

The Library gives this equation:

$$I\quad =\quad { I }_{ C }\quad +\quad m{ d }^{ 2 }$$

Where d is the distance from the combined center of mass. That's where my problem lies, I don't know how to find that combined center of mass. Especially with two objects that aren't the same (a rod and solid sphere in this case). Could I simply treat each object as a particle, of mass M and 2M on an axis (which is the axis of rotation) to find the combined center of mass?

 

[tiny-tim]

… we had to calculate the moment of inertia of a rod of mass M, with a solid sphere of mass 2M attached at the end, with the axis of rotation through the center of the rod and sphere (a rough approximation of a baseball bat).

The Library gives this equation:

$$I\quad =\quad { I }_{ C }\quad +\quad m{ d }^{ 2 }$$

Where d is the distance from the combined center of mass. That's where my problem lies, I don't know how to find that combined center of mass.

Yes, but that's assuming you already know what I_c is.

You know I_c for a sphere, and you know I_c for a rod, so find I for each separately (about the given axis), and add.

(the moment of inertia of a composite body is the sum of the moments of inertia of its parts, about the same axis)