The linear mass density d of a rod of length L kept along x axis varies as d= a + bx. where a and b are positive constants. The centre of mass of the rod is at????

To find the center of mass of a rod where the linear mass density varies along its length, we can use the formula for the center of mass (CM) of a continuous body. In this case, the linear mass density is given by the equation \( d(x) = a + bx \), where \( a \) and \( b \) are positive constants, and \( x \) is the position along the rod from one end.

Understanding the Center of Mass Calculation

The center of mass \( x_{CM} \) for a rod can be calculated using the following formula:

\( x_{CM} = \frac{1}{M} \int_0^L x \cdot d(x) \, dx \)

Here, \( M \) is the total mass of the rod, and \( d(x) \) is the linear mass density at position \( x \). We will need to calculate both the total mass \( M \) and the integral of \( x \cdot d(x) \) over the length of the rod.

Step 1: Calculate the Total Mass \( M \)

The total mass \( M \) of the rod can be found by integrating the linear mass density over the length of the rod:

\( M = \int_0^L d(x) \, dx = \int_0^L (a + bx) \, dx \)

Now, let's compute this integral:

• First, integrate \( a \): \( \int_0^L a \, dx = aL \)
• Next, integrate \( bx \): \( \int_0^L bx \, dx = \frac{bL^2}{2} \)

Combining these results gives:

\( M = aL + \frac{bL^2}{2} \)

Step 2: Calculate the Integral for the Center of Mass

Next, we need to compute the integral \( \int_0^L x \cdot d(x) \, dx \):

\( \int_0^L x \cdot d(x) \, dx = \int_0^L x(a + bx) \, dx = \int_0^L (ax + bx^2) \, dx \)

Now, we can break this down:

• For \( ax \): \( \int_0^L ax \, dx = \frac{aL^2}{2} \)
• For \( bx^2 \): \( \int_0^L bx^2 \, dx = \frac{bL^3}{3} \)

Combining these results gives:

\( \int_0^L x \cdot d(x) \, dx = \frac{aL^2}{2} + \frac{bL^3}{3} \)

Step 3: Putting It All Together

Now we can substitute these results back into the center of mass formula:

\( x_{CM} = \frac{1}{M} \left( \frac{aL^2}{2} + \frac{bL^3}{3} \right) \)

Substituting \( M \) from our earlier calculation:

\( x_{CM} = \frac{1}{aL + \frac{bL^2}{2}} \left( \frac{aL^2}{2} + \frac{bL^3}{3} \right) \)

Final Expression for the Center of Mass

This expression can be simplified further, but it already gives a clear idea of how the center of mass depends on the parameters \( a \), \( b \), and \( L \). The center of mass will shift towards the end of the rod where the mass density is greater, which is influenced by the linear variation of the density.

In summary, the center of mass of the rod is determined by integrating the varying mass density and considering the total mass, leading to a clear understanding of how mass distribution affects the center of mass location. If you have any further questions about this topic or related concepts, feel free to ask!