A uniform steel bar of cross-sectional area A and length L is suspended, so that it hangs vertically. The stress at the middle point of the bar is (`rho` is the density of stell)

Options

A. `(L)/(2A) rho g`
B. `(L rho g)/(2)`
C. `(LA)/(rho g)`
D. `L rho g`

Explanation

To find the stress at the middle point of a uniform steel bar of cross-sectional area \( A \) and length \( L \) that is suspended vertically, we can follow these steps:

### Step 1: Understand the Concept of Stress
Stress is defined as the force applied per unit area. Mathematically, it is expressed as:
\[ \text{Stress} = \frac{\text{Force}}{\text{Area}} \]

### Step 2: Identify the Force at the Midpoint
When the bar is suspended, the force acting on the midpoint is due to the weight of the lower half of the bar. The weight of an object is given by:
\[ \text{Weight} = \text{mass} \times g \]
where \( g \) is the acceleration due to gravity.

### Step 3: Calculate the Mass of the Lower Half of the Bar
The mass of the entire bar can be expressed in terms of its density \( \rho \) and volume. The volume \( V \) of the bar is given by:
\[ V = A \times L \]
Thus, the mass \( m \) of the entire bar is:
\[ m = \rho \times V = \rho \times A \times L \]
The mass of the lower half of the bar is:
\[ m_{\text{lower}} = \frac{m}{2} = \frac{\rho \times A \times L}{2} \]

### Step 4: Calculate the Force at the Midpoint
The force acting at the midpoint due to the weight of the lower half of the bar is:
\[ F = m_{\text{lower}} \times g = \left(\frac{\rho \times A \times L}{2}\right) \times g = \frac{\rho \times A \times L \times g}{2} \]

### Step 5: Substitute into the Stress Formula
Now, we can substitute the force \( F \) into the stress formula:
\[ \text{Stress} = \frac{F}{A} = \frac{\frac{\rho \times A \times L \times g}{2}}{A} \]

### Step 6: Simplify the Expression
The area \( A \) cancels out:
\[ \text{Stress} = \frac{\rho \times L \times g}{2} \]

### Final Answer
Thus, the stress at the midpoint of the bar is:
\[ \text{Stress} = \frac{\rho \times L \times g}{2} \]