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Hi,

I appreciate that we can calculate the x-coordinate of the centre of mass of a system of bodies like this:

[tex]x = (m_{1}x_{1} + m_{2}x_{2} + m_{3}x_{3} + ... + m_{n}x_{n}) / M[/tex]

where M is the total mass.

This could be rewritten as

[tex] x = \frac {1} {M} \sum_{i=1}^n m_{i}x_{i} [/tex]

My next step would be to say that

[tex] x = \frac {1} {M} \int_{a}^{b} m dx[/tex]

(I am imagining a scanner sweeping horizontally through an object X and recording its mass at very small intervals).

However, looking in the textbook, it is expressed as

[tex] x = \frac {1} {M} \int x dm[/tex]

I'm a bit of a beginner as far as calculus goes, but it seems to me that[tex] \int x dm[/tex] is not the same as [tex]\int m dx[/tex].

Looking at the problem geometrically, if you plot m against x, then [tex]\int_{a}^{b} m dx[/tex] will give you an area between the graph of m and the x-axis, whereas [tex]\int_{a}^{b} x dm[/tex] will give you an area between the graph of m and the y-axis.

I appreciate that we can calculate the x-coordinate of the centre of mass of a system of bodies like this:

[tex]x = (m_{1}x_{1} + m_{2}x_{2} + m_{3}x_{3} + ... + m_{n}x_{n}) / M[/tex]

where M is the total mass.

This could be rewritten as

[tex] x = \frac {1} {M} \sum_{i=1}^n m_{i}x_{i} [/tex]

My next step would be to say that

[tex] x = \frac {1} {M} \int_{a}^{b} m dx[/tex]

(I am imagining a scanner sweeping horizontally through an object X and recording its mass at very small intervals).

However, looking in the textbook, it is expressed as

[tex] x = \frac {1} {M} \int x dm[/tex]

I'm a bit of a beginner as far as calculus goes, but it seems to me that[tex] \int x dm[/tex] is not the same as [tex]\int m dx[/tex].

Looking at the problem geometrically, if you plot m against x, then [tex]\int_{a}^{b} m dx[/tex] will give you an area between the graph of m and the x-axis, whereas [tex]\int_{a}^{b} x dm[/tex] will give you an area between the graph of m and the y-axis.

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