The problem in the image involves finding the center of mass of a non-uniform rod, where the linear mass density varies along its length. Let's go step-by-step through the solution.

Step 1: Understanding the Linear Mass Density Function

The mass density, $\lambda(x)$, varies linearly from 1.02 g/cm at one end to 5.25 g/cm at the other end of a 0.450 m rod. This means the linear mass density can be written as:

$\lambda(x) = \lambda_0 + kx$

where:

• $\lambda_0 = 1.02 \, \text{g/cm}$ is the mass density at $x = 0$ (the less massive end),
• $\lambda(L) = 5.25 \, \text{g/cm}$ is the mass density at $x = L = 0.450 \, \text{m}$,
• $x$ is the position along the rod from the origin.

We can find $k$, the rate of change of density, using the formula for a line:

$k = \frac{\lambda(L) - \lambda_0}{L}$

Substituting the values:

$k = \frac{5.25 \, \text{g/cm} - 1.02 \, \text{g/cm}}{0.450 \, \text{m}}$

Let's compute this value.

Step 2: Computing $k$ and Substituting it Back

Once we compute $k$, the function for $\lambda(x)$ will be complete, and we can proceed to find the center of mass. The x-coordinate of the center of mass is found using the formula:

$x_{cm} = \frac{\int_0^L x \lambda(x) dx}{\int_0^L \lambda(x) dx}$

This requires integrating the function $\lambda(x)$ over the length of the rod.

I'll compute this step now.The value of $k$, the rate of change of the linear mass density, is $9.4 \, \text{g/cm}^2$.

Thus, the linear mass density as a function of $x$ is:

$\lambda(x) = 1.02 + 9.4x$

Step 3: Center of Mass Formula

The x-coordinate of the center of mass, $x_{cm}$, is given by:

$x_{cm} = \frac{\int_0^L x \lambda(x) dx}{\int_0^L \lambda(x) dx}$

We need to evaluate these two integrals:

1. $\int_0^L \lambda(x) dx$
2. $\int_0^L x \lambda(x) dx$

Let's compute these integrals to find $x_{cm}$.The x-coordinate of the center of mass, $x_{cm}$, is approximately:

$x_{cm} = 0.276 \, \text{m} = 27.6 \, \text{cm}$

Step 4: $y_{cm}$ Coordinate

Since the rod lies along the $+x$-axis and no variation is given in the $y$-direction, the $y$-coordinate of the center of mass is simply:

$y_{cm} = 0 \, \text{cm}$

Final Answers:

• $x_{cm} = 27.6 \, \text{cm}$
• $y_{cm} = 0 \, \text{cm}$

Would you like more details or have any questions about the solution?

Here are some related questions for further exploration:

1. How would the center of mass change if the mass density varied quadratically instead of linearly?
2. How can you calculate the total mass of the rod from the linear mass density?
3. What would happen to the center of mass if the rod were bent into a curve?
4. How is the center of mass affected if the rod is not uniform in two dimensions?
5. What is the physical significance of the center of mass in a system?

Tip: When dealing with non-uniform objects, always express the mass distribution as a function and use integrals to determine physical properties like the center of mass.