How can I derive the equation X(cm) = int(X dm)/M?

In summary, the conversation discusses the process of deriving the equation X(cm)=int(X dm)/M and the meaning of the symbols used in the equation. It is explained that the integral symbol represents the limit of a sum as the differences tend to zero, and this is how the equation is derived. The concept of using an integral to calculate energy and moment of mass is also discussed.
  • #1
Dweirdo
174
0
OMG second time I'm opening a thread in the wrong forum FFS!
Damn bookmarks! MODS move it please.

Homework Statement


Not a home work question, just something i cam across and need a clarification.
Could One show me how to derive to the equation that X(cm)=int(X dm)/M
int=the deformed S of the integral(2 lazy to write in Latex XD).

Homework Equations


X(cm)=sigma(Xi Dmi)/M

The Attempt at a Solution


I know It's simple, But I can't imagine how sigma(Xi Dmi) becomes int(X dm),
I don't understand what it means , trying to convert it to words just doesn't work for me,so could some 1 explain that for me?
AFAIK sigma(Xi Dmi) means the mass distribution,but how does the integral takes place here?
I really need to understand the math part in physics.

Thanks a lot in advanced !
 
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  • #2
Hi Dweirdo! :smile:

(have a sigma: ∑ and a delta: ∆ and an integral: ∫ and try using the X2 tag just above the Reply box :wink:)
Dweirdo said:
… I can't imagine how sigma(Xi Dmi) becomes int(X dm),
I don't understand what it means , trying to convert it to words just doesn't work for me,so could some 1 explain that for me?
AFAIK sigma(Xi Dmi) means the mass distribution,but how does the integral takes place here?

How does ∑ Xi ∆mi become ∫ X dm ?

Because that's what an ∫ is …

it's defined as the limit of a ∑ as the ∆s tend to zero. :smile:
 
  • #3
tiny-tim said:
Hi Dweirdo! :smile:

(have a sigma: ∑ and a delta: ∆ and an integral: ∫ and try using the X2 tag just above the Reply box :wink:)


How does ∑ Xi ∆mi become ∫ X dm ?

Because that's what an ∫ is …

it's defined as the limit of a ∑ as the ∆s tend to zero. :smile:

But why?? Like I know that in Energy, if you make a graph of force and distance and it is curved than integral calculates the plot.
but wtf is it here?
thanks :}
 
  • #4
Because energy (= work done ) = force x distance, so it's the limit of ∑ (force x ∆distance)

Similarly, moment of mass = distance x mass, so it's the limit of ∑ (distance x ∆mass) :smile:
 

What is the center of mass?

The center of mass is a point at which the entire mass of an object can be considered to be concentrated for the purpose of calculating its motion.

How is the center of mass calculated?

The center of mass can be calculated by taking the weighted average of the positions of all the particles that make up an object. This is usually done by using the mass and position of each particle in a mathematical formula.

Why is the concept of center of mass important?

The concept of center of mass is important because it is used to describe the overall motion of an object, regardless of its shape or orientation. It also allows us to simplify complex systems and make predictions about their behavior.

What is the difference between center of mass and center of gravity?

The center of mass is the average location of an object's mass, while the center of gravity is the point where the force of gravity is considered to act on an object. In most cases, these points are the same, but they may differ in certain situations, such as when an object is in a non-uniform gravitational field.

How is center of mass used in real-world applications?

The concept of center of mass is used in various real-world applications, such as in designing stable structures, predicting the motion of objects in space, and analyzing the flight of aircraft. It is also used in sports, such as determining the balance point of a surfboard or skateboard, and in robotics to ensure stability and control of movement.

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