Understanding Moment and Center of Gravity in One Dimension

In summary, the book says that the moment about the origin is mx. The center of gravity of a point mass is the moment (M) divided by the total mass (m).
  • #1
hndalama
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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.
 
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  • #2
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.
 
  • #3
haruspex said:
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?
 
  • #4
hndalama said:
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".
 
  • #5
haruspex said:
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?
 
  • #6
hndalama said:
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.
hndalama said:
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.
 
  • #7
hndalama said:
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.
hndalama said:
...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.
 
  • #8
David Lewis said:
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.
 
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Likes David Lewis
  • #9
Thanks for your help
 

1. What is the definition of moment?

The moment of a physical quantity is a measure of its tendency to cause a body to rotate around a specific point or axis. It is calculated by multiplying the magnitude of the quantity by the distance from the reference point to the line of action of the quantity.

2. How is moment different from force?

While both moment and force are measures of physical quantities, they differ in their effects on a body. A force causes a body to move in a straight line, while a moment causes a body to rotate around a point or axis.

3. Can you give an example of a moment?

An example of a moment is the turning effect of a wrench on a bolt. The force applied by the wrench creates a moment around the bolt, causing it to rotate.

4. What are the units of moment?

The units of moment depend on the physical quantity being measured. For example, the moment of force is measured in newton-meters (Nm), while the moment of inertia is measured in kilogram-meter squared (kg*m^2).

5. How is moment used in engineering and physics?

Moment is a crucial concept in both engineering and physics. In engineering, it is used to design structures that can withstand forces and moments without collapsing. In physics, it is used to analyze and understand the motion of objects, particularly in rotational motion and equilibrium.

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