1. the density of a thin rod of length l varies with the distance x from one end as​​​ row=rownot x^2/l^2 rownot is a constant.find the position of centre of mass?

To find the center of mass of a thin rod with a varying density, we need to consider how the density changes along its length. In this case, the density ρ at a distance x from one end of the rod is given by the equation ρ = ρ₀ (x²/l²), where ρ₀ is a constant and l is the total length of the rod. Let's break this down step by step.

Understanding the Problem

The center of mass (CM) of an object is the point where its mass is evenly distributed. For a rod with uniform density, the CM is at its midpoint. However, since the density varies in this scenario, we need to calculate the CM using the varying density function.

Setting Up the Integral

To find the center of mass, we use the formula:

x_cm = (1/M) ∫ x dm

Here, M is the total mass of the rod, and dm is the mass element at a distance x. Since the density varies, we express dm in terms of the density function:

dm = ρ(x) dx = ρ₀ (x²/l²) dx

Calculating the Total Mass

First, we need to find the total mass M of the rod by integrating the mass element from 0 to l:

M = ∫ dm = ∫₀^l ρ₀ (x²/l²) dx

Now, let's compute this integral:

• Factor out the constant ρ₀/l²:
• M = (ρ₀/l²) ∫₀^l x² dx
• The integral of x² from 0 to l is (l³/3):
• M = (ρ₀/l²) * (l³/3) = (ρ₀ l)/3

Finding the Center of Mass

Now that we have the total mass, we can find the center of mass using the earlier formula:

x_cm = (1/M) ∫ x dm

Substituting for dm:

x_cm = (1/M) ∫₀^l x (ρ₀ (x²/l²)) dx

Now, substituting M = (ρ₀ l)/3:

x_cm = (3/l) ∫₀^l (x³/l²) dx

Factor out the constants:

x_cm = (3/l³) ∫₀^l x³ dx

The integral of x³ from 0 to l is (l⁴/4):

x_cm = (3/l³) * (l⁴/4) = (3l)/4

Final Result

Thus, the position of the center of mass of the rod is located at:

x_cm = (3l)/4

This means that the center of mass is positioned three-quarters of the way along the length of the rod from the end where the density is measured. This result illustrates how the distribution of mass affects the center of mass, especially in cases of non-uniform density.