Question

A straight rod of length \(L\) has one of its ends at the origin and the other end at \(\mathrm{x}=\mathrm{L}\) If the mass per unit length of rod is given by Ax where \(A\) is constant where is its centre of mass. \(\\{\mathrm{A}\\} \mathrm{L} / 3\) \(\\{\mathrm{B}\\} \mathrm{L} / 2\) \(\\{\mathrm{C}\\} 2 \mathrm{~L} / 3\) \(\\{\mathrm{D}\\} 3 \mathrm{~L} / 4\)

Short answer

The center of mass of the rod is located at \(\frac{2}{3}L\).

Step-by-step solution

Understanding the center of mass formula for continuous objects

In the case of continuous mass distribution, the center of mass coordinates can be calculated as follows. For the x-coordinate of the center of mass: \[ x_{cm} = \frac{\int xdm}{\int dm} \] Here, \(dm = Axdx\), where Ax is the mass per unit length and dx is an infinitesimal length element on the rod.

Calculate the mass of the rod

To find the total mass (M) of the rod, we integrate the dm function over the length of the rod: \[ M = \int_0^L dm = \int_0^L Axdx \]

Integrate

Integrate the mass function over the length of the rod: \[ M = A\int_0^L xdx = A\left[\frac{1}{2}x^2\right]_0^L = \frac{1}{2}AL^2\]

Calculate the numerator of the center of mass formula

Now we need to calculate the integral in the numerator of the center of mass formula: \[ \int_0^L xdm = \int_0^L x(Axdx) = A\int_0^L x^2dx\]

Integrate

Integrate the function over the length of the rod: \[ A\int_0^L x^2dx = A\left[\frac{1}{3}x^3\right]_0^L = \frac{1}{3}AL^3\]

Calculate the center of mass

Now we can plug our results into the center of mass formula: \[ x_{cm} = \frac{\frac{1}{3}AL^3}{\frac{1}{2}AL^2} \]

Simplify the result

Simplify the expression to find the x-coordinate of the center of mass: \[ x_{cm} = \frac{\frac{1}{3}AL^3}{\frac{1}{2}AL^2} = \frac{2}{3}L \] So, the center of mass of the rod is located at \(\frac{2}{3}L\). The correct answer is option C.