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So , I broke down the meaning of this equation and have got stuck at one point.

I have not done rotational mechanics and have started with centre of mass newly .Getting a bit confused here with this new equation.

Let us say say we have a bar if length L , Now take a small segment dx in it.

We know mass per unit length = $M/L$.This means if M=10 and L=2. It tells that there are 5 rods or segments of distance 2 metres with mass 2Kg.

If we say say dx as L for a moment.It would mean that M*L that is $10*2$ gives me 2 rods that have mass 10Kg. This meaning gives me 2 new rods.It gives mass of each rod.

Now,with dx =0.1 now. $10/2=5$

5 number of segments of 2m.

Now multiplying them by 0.1 gives 0.5 Kg for dm. Now I sent relate it with what I wrote earlier since M/L and not M.

Does it mean 0.1 distance for a rod of length 2m has mass =0.5kg.

I am not completely sure but please help in understand this.

Please tell me how can I improve the quality of my question if not good.

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  • $\begingroup$ Can you edit the question and add math formatting for clarity? $\endgroup$ – JAlex Nov 19 at 16:20
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Understanding the physical meaning behind math expressions can be confusing in physics because there are many ways to arrange terms. One way to deduce the meaning is by looking at units.

This means if M=10 and L=2. It tells that there are 5 rods or segments of distance 2 metres with mass 2Kg.

This would not be the case because joining 5 segments each of distance 2 metres would produce 10 metres, which is longer than the initial bar of length L=2. In terms of units, we cannot say there are 5 rods of 2 metres because "5" has units of $\frac{M (kilogram)}{L(metre)} = kg/m$, while "number of rods" should not have units because it is a number.

If we say say dx as L for a moment.It would mean that ML that is 102 gives me 2 rods that have mass =10Kg. This meaning gives me 2 new rods.It gives mass of each rod.

If dx were L, ML would not represent 2 rods each of mass 10kg. This is also because of units where "ML" has units of $M(kilogram) \times L (metre) = kg \cdot m$, while "2 rods of mass 10kg" should have units of kilogram because "number of rods" is a unitless number and mass has units of kilogram.

The way I understand dm = M/Ldx is that there is a bar with uniform mass per unit length M/L, and dx is some length within the bar. When we multiply the length of dx with mass per unit length, we get mass dm. When we add all the segments of length dx we get back the orginal length L, and when we add all the mass of each segment dm we get M.

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The the basic equation here is $${\rm d}m = \rho\, {\rm d} V$$ The interpretation is the mass contained inside a small infinitesimal volume is calculated from the mass density $\rho$ at any location.

For a slender bar of length $\ell$ and cross section area $A$ you can specify ${\rm d}V = A \, {\rm d}x$ with $x=0 \ldots \ell$, which makes the infinitesimal mass

$$ {\rm d}m = \rho A\, {\rm d} x $$

This literaly means that a slice ${\rm d}x$ contains mass of ${\rm d}m$.

If the cross section is constant then the above is integrated into

$$ m = \rho A \ell $$ or $$ \rho A = \frac{m}{\ell}$$

So for those cases, you can rewrite the infinitesimal mass as

$$ {\rm d} m = \frac{m}{\ell} {\rm d}x $$

So this still means the mass inside a slice. The $m\ell$ part is sometimes called a linear mass density which has units of $\text{mass / length}$. The generic way of looking at this relationship is $$ {\rm d}m = \lambda \, {\rm d}x$$

To relate this with math, consider a line with $y = a x + b$ and take a small slice in $x$ to find the rise as $${\rm d}y = a\, {\rm d}x$$ Now the meaning is: amount of $y$ contained in a slice of $x$

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