Prove that the center of mass of a rod having uniformly distributed mass lies in the middle of the rod.

Question

Accepted Answer

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> [!TLDR]
> Integrating $x\,dm$ over a rod with uniform linear mass density gives $x_{cm} = L/2$ — the middle of the rod.
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> Consider a thin uniform rod of total mass $M$ and length $L$, lying along the x-axis from $x = 0$ to $x = L$. Because the mass is distributed *uniformly*, the linear mass density (mass per unit length) is constant:
> $$\lambda = \frac{M}{L}$$
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> Consider a small element of the rod of length $dx$ at position $x$; its mass is $dm = \lambda\,dx$.
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> The x-coordinate of the centre of mass is defined as:
> $$x_{cm} = \frac{1}{M}\int_0^L x \, dm = \frac{1}{M}\int_0^L x \, \lambda \, dx = \frac{\lambda}{M}\left[\frac{x^2}{2}\right]_0^L = \frac{\lambda}{M}\cdot\frac{L^2}{2}$$
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> Substituting $\lambda = M/L$:
> $$x_{cm} = \frac{M/L}{M}\cdot\frac{L^2}{2} = \frac{L^2}{2L} = \frac{L}{2}$$
> …

Q.Prove that the center of mass of a rod having uniformly distributed mass lies in the middle of the rod.

Concept understanding — Center of Mass

What is the Center of Mass?

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The precise definition

Breaking it down with an example

Key properties to remember

Q.Prove that the center of mass of a rod having uniformly distributed mass lies in the middle of the rod.

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