bar subjected to gravity load (two segments)

the bar in the fig has constant cross sectional area A .the top half of the bar is made of material with mass density and youngs module, the bottom half of the bar made of density and young module total length of bar is 2 when bar is hung with ceiling it stretches its own weight no other loads are applied neglect gravity load on the top half of the bar

To analyze the behavior of the bar subjected to gravity load, we need to consider the following:

1. Cross-Sectional Area (A): The bar has a constant cross-sectional area (A) throughout its length. This means that the area of the top and bottom halves of the bar is the same.

2. Material Properties:
a. Top Half of the Bar: The top half of the bar is made of a material with a mass density (ρ) and Young's modulus (E).
b. Bottom Half of the Bar: The bottom half of the bar is made of a material with a different mass density (ρ') and Young's modulus (E').

3. Bar Length (L): The total length of the bar is 2.

4. Hanging from Ceiling: The bar is hung from the ceiling, which means it stretches under its own weight, with no other loads applied.

To determine the behavior of the bar under the gravity load, we can follow these steps:

Step 1: Calculate the Total Weight of the Bar:
- The gravity load on the entire bar will be equal to its weight.
- Weight (W) = Mass (m) x Acceleration due to Gravity (g)
- The mass can be calculated using the density and cross-sectional area:
- Mass (m) = Density (ρ or ρ') x Volume (V)
- The volume of the bar is given by the product of its cross-sectional area and length:
- Volume (V) = Cross-sectional Area (A) x Length (L)
- Therefore, the weight of the entire bar is:
- W = Density (ρ or ρ') x Cross-sectional Area (A) x Length (L) x Acceleration due to Gravity (g)

Step 2: Determine the Load Distribution:
- Since the gravity load is acting on the entire length of the bar, the load is evenly distributed along the whole bar.

Step 3: Calculate Elongation or Change in Length of the Bar:
- The elongation or change in length of the bar can be calculated using Hooke's Law:
- Elongation (ΔL) = Load (F) x Length (L) / (Cross-Sectional Area (A) x Young's Modulus (E or E'))
- In this case, we consider the load as the weight of the bar, which we calculated in Step 1.

By following these steps, we can analyze the behavior of the bar subjected to the gravity load.