A rod of length L has a non-uniform density λ = λ0
s
L
, where λ0 is constant and ‘s’ is the
distance from the end marked zero. Fins the center of mass of the road.

To find the center of mass of a rod with a non-uniform density, we need to consider how the density varies along its length. In this case, the density is given by the equation λ = λ0 s L, where λ0 is a constant and 's' is the distance from one end of the rod. Let's break this down step by step.

Understanding the Problem

The rod has a length L, and its density increases linearly from the end marked zero. This means that as you move along the rod from one end to the other, the mass per unit length increases. To find the center of mass, we will use the formula for the center of mass of a continuous body:

Formula for Center of Mass

The center of mass (CM) for a continuous distribution can be calculated using the formula:

• CM = (1/M) ∫ x dm

where M is the total mass of the rod, x is the position along the rod, and dm is the differential mass element.

Calculating the Total Mass

First, we need to find the total mass of the rod. The mass element dm can be expressed in terms of the density:

• dm = λ(s) ds = λ0 s L ds

Now, we can find the total mass M by integrating dm from 0 to L:

• M = ∫(from 0 to L) λ0 s L ds

Calculating this integral:

• M = λ0 L ∫(from 0 to L) s ds
• M = λ0 L [s²/2] (from 0 to L)
• M = λ0 L (L²/2) = (λ0 L³)/2

Finding the Center of Mass

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

• CM = (1/M) ∫(from 0 to L) x dm

Substituting dm:

• CM = (1/M) ∫(from 0 to L) x (λ0 s L ds)

Now, we need to express 's' in terms of 'x' for the integration:

• CM = (1/M) λ0 L ∫(from 0 to L) x s ds

We can rewrite this integral as:

• CM = (1/M) λ0 L ∫(from 0 to L) x s ds

Now we need to evaluate the integral:

• ∫(from 0 to L) x s ds = (1/2) s² x (from 0 to L) = (1/2) L² x

Substituting this back into the CM formula gives:

• CM = (1/M) λ0 L [(1/2) L² x]

Final Calculation

Now, substituting M back into the equation:

• CM = (1/((λ0 L³)/2)) λ0 L [(1/2) L² x]

After simplifying, we find:

• CM = (2/3) L

Thus, the center of mass of the rod is located at a distance of (2/3) L from the end marked zero. This result illustrates how the distribution of mass affects the center of mass, especially in non-uniform density scenarios.