What is fluid? Show that fluid exerts pressure and prove that the force acting on a fluid in equilibrium at rest have to be perpendicular to its surface.

Options

A. Volume of fluid displaced
B. Density of body
C. Density of fluid
D. Acceleration due to gravity

Answer

### Step-by-Step Solution:

**Step 1: Definition of Fluid**
A fluid is defined as a substance that can flow continuously when subjected to stress. This includes liquids and gases, which can change shape and move easily.

**Step 2: Showing that Fluids Exert Pressure**
To demonstrate that fluids exert pressure, consider a container filled with fluid (e.g., water). If we create a small hole at the bottom of the container, the fluid will flow out of the hole. This outflow indicates that there is pressure within the fluid that pushes it out through the hole.

Mathematically, we can express pressure (P) as force (F) per unit area (A):
\[ P = \frac{F}{A} \]
In this case, the pressure exerted by the fluid inside the container is greater than the atmospheric pressure outside, which causes the fluid to flow out.

**Step 3: Proving that Force on Fluid in Equilibrium is Perpendicular to its Surface**
Consider a fluid in a container at rest (in equilibrium). Let’s analyze the forces acting on a small area of the fluid surface.

1. **Force Acting at an Angle**: If a force \( F \) is applied at an angle \( \theta \) to the surface, it can be resolved into two components:
- A vertical component: \( F \cos \theta \)
- A horizontal component: \( F \sin \theta \)

2. **Equilibrium Condition**: For the fluid to be in equilibrium, the net force acting on it must be zero. This means that the horizontal component of the force must be balanced by the internal stresses within the fluid.

3. **Stress Analysis**: The horizontal component of the force creates a shear stress given by:
\[ \tau = \frac{F \sin \theta}{A} \]
Since the fluid is at rest, there can be no net shear stress acting on it. Therefore, for equilibrium:
\[ F \sin \theta = 0 \]

4. **Conclusion**: The only way for this equation to hold true (given that \( F \) is not zero) is if \( \sin \theta = 0 \), which implies that \( \theta = 0^\circ \). This means that the force must act perpendicular to the surface of the fluid.

Thus, we conclude that for a fluid in equilibrium at rest, the force acting on it must be perpendicular to its surface.