Understanding Moment and Center of Gravity in One Dimension

[hndalama]

Homework Statement

The book I am reading says that given a point mass(m) at the point x, the quantity mx is the "moment about the origin)"
It then defines the moment of a collection of points as M = m(1)x(1) + m(2)x(2) + ... m(n)x(n)
where m(1) = mass of first point and x(1)=distance of first point from origin

It then defines that the center of gravity (X) of the point masses is the moment (M) divided by the total mass(m). X=M/m
This is all in one dimension,i.e. like the point masses are on a seesaw(x axis)

Homework Equations

The Attempt at a Solution

Previously I learned that moment = torque. In one dimension, torque is defined as the force times the distance from the pivot point. hence torque of a point mass from the origin is Fx.
if moment= Fx then how can the "moment about the origin" be mx? mass is not a force.

 

[haruspex]

The term "moment" is used in more than one way. The book you are reading is describing moment of mass. Torque is the moment of a force. Moment of inertia is also called the second moment of mass.

 

[hndalama]

The term "moment" is used in more than one way. The book you are reading is describing moment of mass. Torque is the moment of a force. Moment of inertia is also called the second moment of mass.

Okay, but when we say "moment" of mass, "moment" of force or "moment" of inertia, what does the word moment mean in these phrases?

 

[haruspex]

Okay, but when we say "moment" of mass, "moment" of force or "moment" of inertia, what does the word moment mean in these phrases?

They all come from the common English usage of 'moment' to mean importance, significance. Similarly momentum.
The extension to different orders of moment is common to mechanics and statistics. Given any quantity f(x) which might vary according to a parameter x, we may speak of the zeroth moment, ∫f(x).dx, the first moment, ∫f(x)x.dx, the second moment ∫f(x).x².dx, etc. E.g. "second moment of area".

 

[hndalama]

They all come from the common English usage of 'moment' to mean importance, significance. Similarly momentum.
The extension to different orders of moment is common to mechanics and statistics. Given any quantity f(x) which might vary according to a parameter x, we may speak of the zeroth moment, ∫f(x).dx, the first moment, ∫f(x)x.dx, the second moment ∫f(x).x².dx, etc. E.g. "second moment of area".

Thank you, I understand that. I have one more question if you don't mind. When the book talks about the moment of mass in two dimensions, it defines M(x) as the moment of mass about the "x axis" and M(y) about the "y axis." so by my understanding, on cartesian coordinates, two points of equal mass located at the points (3,3) and (3,6). will have the same moment of mass about the y-axis despite having different distances from points on the y axis.

This is different to moment of force and moment of inertia which are defined with distances from a point. so my question is why is moment of mass defined with distances from an axis instead of a point?

 

[haruspex]

two points of equal mass located at the points (3,3) and (3,6). will have the same moment of mass about the y-axis despite having different distances from points on the y axis.

Is that what you meant? (3,3) and (3,6) are the same distance from the y axis.

This is different to moment of force and moment of inertia which are defined with distances from a point. so my question is why is moment of mass defined with distances from an axis instead of a point?

No, moment of inertia is also defined in relation to an axis, namely, an actual or potential axis of rotation.
Moment of force is often about a specified axis, but can be about a point. The other difference is that it is a vector, whereas the other two are scalars. This is related to the parity of the exponent on the displacement vector: mr⁰ and mr² are necessarily scalar, whereas mr¹ is naturally a vector.

 

[David Lewis]

This is different to moment of force and moment of inertia which are defined with distances from a point. so my question is why is moment of mass defined with distances from an axis instead of a point?

Moments are always taken with respect to a line. With polar moments (as opposed to rectangular moments) the moment arm can be measured from the point where the reference axis intersects the plane of rotation.

...given a point mass (m) at the point x

You could have the moment of the mass about some point. In elementary situations x can be the magnitude of a displacement vector running from the reference axis to the mass.

 

[haruspex]

Moments are always taken with respect to a line.

Only for scalar moments. As I wrote in post #6, vector moments, such as torque, are about a point. Similarly, the first moment of mass about a point gives the mass times the displacement vector from the point to the mass centre.
In many situations, though, we are only interested in the component of the vector parallel to a given axis.

 

[hndalama]

Thanks for your help