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

[tex]{ 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 }[/tex]

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:

[tex] I\quad =\quad { I }_{ C }\quad +\quad m{ d }^{ 2 } [/tex]

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:

[tex] I\quad =\quad { I }_{ C }\quad +\quad m{ d }^{ 2 } [/tex]

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)

 

[asteriatic]

As a scientist, it is important to understand the principles and equations involved in solving a problem. In this case, the equations given in your textbook are used to find the center of mass for objects, which is an important concept in physics. The equations may seem complex, but they are essential in solving problems involving the center of mass.

In order to understand the equations better, it is important to first understand the concept of center of mass. The center of mass is the point at which an object can be balanced or the point where the mass of an object is evenly distributed. This point is often located at the geometric center of an object, but can also vary depending on the distribution of mass.

Now, in the problem given to you, you are asked to find the moment of inertia of a rod with a solid sphere attached to the end. The moment of inertia is a measure of an object's resistance to rotational motion around a particular axis. In this case, the axis of rotation is through the center of the rod and sphere.

To use the equations given in your textbook, you need to understand that the variables 'x', 'y', and 'z' represent the coordinates of each infinitesimal mass element (dm) that makes up the object. In other words, you need to take into account the distribution of mass in the object and integrate over the entire object to find the center of mass.

In summary, the equations given in your textbook are essential in finding the center of mass of an object and can be applied to solve problems involving rotational motion. It is important to carefully consider the distribution of mass in an object and integrate over the entire object to find the center of mass. With a clear understanding of these principles, you should be able to successfully solve the problem given to you.