The problem describes a beam of length 10 meters that supports a load with a variable weight along its length. The load at the left end is 1200 N/m and decreases to 200 N/m at the right end. The weight per unit length is given as a function of $x$, where $x$ is the distance from the left end of the beam:

$w(x) = 10(120 - x^2) \, \text{N/m}$

We are tasked with solving two parts:

1. Find the total weight of the load: To find the total weight of the load, we need to integrate the weight function $w(x)$ over the length of the beam (from $x = 0$ to $x = 10$).

2. Find the x-coordinate of the centroid of the load: The centroid $\bar{x}$ is the point where the total load can be considered to act. It is calculated by the formula:

$\bar{x} = \frac{\int_0^{10} x \cdot w(x) \, dx}{\int_0^{10} w(x) \, dx}$

Let me calculate these values for you.

Step 1: Total Weight of the Load

To find the total weight, we integrate the weight function over the length of the beam:

$\text{Total Weight} = \int_0^{10} w(x) \, dx = \int_0^{10} 10(120 - x^2) \, dx$

Step 2: x-Coordinate of the Centroid

To find the centroid, we calculate:

$\bar{x} = \frac{\int_0^{10} x \cdot w(x) \, dx}{\int_0^{10} w(x) \, dx}$

Let me compute these integrals for you. sympy as sp

Define the variable

x = sp.symbols('x')

Define the weight function

w = 10 * (120 - x**2)

Step 1: Total weight of the load

total_weight = sp.integrate(w, (x, 0, 10))

Step 2: x-coordinate of the centroid

centroid_numerator = sp.integrate(x * w, (x, 0, 10)) centroid_denominator = total_weight

centroid = centroid_numerator / centroid_denominator

total_weight, centroidThe results of the calculations are:

1. Total weight of the load: The total weight of the load is $\frac{26000}{3} \, \text{N}$, which is approximately 8666.67 N.

2. x-coordinate of the centroid: The x-coordinate of the centroid is $\frac{105}{26} \, \text{m}$, which is approximately 4.04 meters from the left end of the beam.

Would you like further details or clarification on these calculations?

Here are some related questions you might find useful:

1. How do changes in the load distribution affect the centroid?
2. What is the physical meaning of the centroid of a load distribution?
3. How would the calculations change if the beam had a different length?
4. How can you calculate the bending moment for a distributed load?
5. What are the key factors to consider when designing a beam to withstand a varying load?

Tip: When calculating centroids for distributed loads, breaking the load into small segments and integrating over those intervals can help solve complex load distributions.