An isosceles triangular plates of base 3 m and altitude 3 m is immersed in oil vertically with its base coinciding with its base coinciding with the free surface of the oil of relative density 0.8. Determine the total thrust.

Options

A. 24 N
B. 48 N
C. 36 N
D. None of these

Explanation

To determine the total thrust on the isosceles triangular plate immersed in oil, we can follow these steps:

### Step 1: Identify the dimensions of the triangle
The base \( b \) of the triangle is given as 3 m, and the altitude \( h \) is also given as 3 m.

### Step 2: Calculate the area of the triangle
The area \( A \) of a triangle can be calculated using the formula:
\[ A = \frac{1}{2} \times \text{base} \times \text{height} \]
Substituting the values:
\[ A = \frac{1}{2} \times 3 \, \text{m} \times 3 \, \text{m} = \frac{9}{2} \, \text{m}^2 = 4.5 \, \text{m}^2 \]

### Step 3: Determine the depth of the center of mass
For an isosceles triangle, the center of mass is located at a distance of \( \frac{h}{3} \) from the base. Since the altitude \( h \) is 3 m:
\[ \text{Depth of center of mass} = \frac{3}{3} = 1 \, \text{m} \]

### Step 4: Calculate the pressure at the center of mass
The pressure \( P \) at a depth \( h \) in a fluid is given by:
\[ P = \rho \cdot g \cdot h \]
Where:
- \( \rho \) is the density of the fluid (oil in this case),
- \( g \) is the acceleration due to gravity (approximately \( 10 \, \text{m/s}^2 \)),
- \( h \) is the depth of the center of mass.

Given that the relative density of oil is 0.8, the density \( \rho \) of oil can be calculated as:
\[ \rho = 0.8 \times 1000 \, \text{kg/m}^3 = 800 \, \text{kg/m}^3 \]
Now substituting the values into the pressure formula:
\[ P = 800 \, \text{kg/m}^3 \times 10 \, \text{m/s}^2 \times 1 \, \text{m} = 8000 \, \text{Pa} \]

### Step 5: Calculate the total thrust (force)
The total thrust \( F \) on the triangular plate can be calculated using the formula:
\[ F = P \cdot A \]
Substituting the values we have:
\[ F = 8000 \, \text{Pa} \times 4.5 \, \text{m}^2 = 36000 \, \text{N} = 36 \, \text{kN} \]

### Final Answer
The total thrust on the isosceles triangular plate is **36 kN**.

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