**Comparison of Stresses in a Rectangular Bar**

In this question, we are given a rectangular bar with dimensions of 50 by 50 mm. The bar is subjected to end moments of 2083 N·m in three special cases. We need to compare the stresses in these different cases and explain the results in detail.

**Case (a): Straight Beam**
In the first case, the bar is straight, meaning it is not curved or bent. When a straight beam is subjected to bending moments, the maximum stress occurs at the extreme fibers of the beam. This stress is given by the equation:

σ = (M * c) / I

Where:
σ is the stress
M is the bending moment
c is the distance from the neutral axis to the extreme fiber
I is the moment of inertia of the beam's cross-sectional area

Since the bar is square, the moment of inertia can be calculated as:

I = (b * h^3) / 12

Where:
b is the width of the beam
h is the height of the beam

Using the given dimensions, we have:
b = h = 50 mm

Plugging these values into the equation for moment of inertia, we get:
I = (50 * 50^3) / 12

Therefore, the stress in the straight beam can be calculated using the given moment of 2083 N·m and the calculated moment of inertia.

**Case (b): Curved Beam with F = 250 mm**
In the second case, the bar is curved to a radius of 250 mm along the centroidal axis. This means that the beam is curved in a circular shape. When a curved beam is subjected to bending moments, the stress distribution is not uniform across the cross-section. The stress at any point on the curved beam can be calculated using the equation:

σ = (M * y) / I

Where:
σ is the stress at a given point
M is the bending moment at that point
y is the distance from the neutral axis to the point
I is the moment of inertia of the beam's cross-sectional area

Since the beam is curved, the distance y is different for different points on the beam. Therefore, the stress distribution is non-uniform.

**Case (c): Curved Beam with r = 75 mm**
In the third case, the bar is curved to a radius of 75 mm. Similar to case (b), the stress distribution in a curved beam is non-uniform. The stress at any point on the curved beam can be calculated using the same equation as in case (b).

However, since the radius is smaller in this case, the stress distribution is expected to be higher compared to case (b). This is because a smaller radius results in a higher curvature, leading to higher stresses in the beam.

In summary, when comparing the stresses in the three special cases, we can expect the following:

- In case (a), the straight beam will have a uniform stress distribution with the maximum stress occurring at the extreme fibers of the beam.
- In case (b), the curved beam with F = 250 mm will have a non-uniform stress distribution, with higher stresses on the outer fibers of the beam compared to the inner fibers.
- In case (c), the curved beam with r = 75 mm will have a non-uniform stress distribution, with higher stresses compared to case (b)