Four forces of magnitude P, 2P, 3P and 4P act along the four sides of a square ABCD in cyclic order. Use the vector method to find the magnitude of resultant force.

Explanation

To find the magnitude of the resultant force acting on the square ABCD due to the forces P, 2P, 3P, and 4P, we can follow these steps:

### Step 1: Define the Coordinate System
We will place the square ABCD in a coordinate system. Let:
- Point A be at (0, 0)
- Point B be at (a, 0)
- Point C be at (a, a)
- Point D be at (0, a)

Here, 'a' is the length of the sides of the square.

### Step 2: Represent the Forces as Vectors
The forces acting along the sides of the square can be represented as vectors:
- Force F1 (P) acts along AB: \( \vec{F_1} = P \hat{i} \)
- Force F2 (2P) acts along BC: \( \vec{F_2} = 2P \hat{j} \)
- Force F3 (3P) acts along CD: \( \vec{F_3} = -3P \hat{i} \)
- Force F4 (4P) acts along DA: \( \vec{F_4} = -4P \hat{j} \)

### Step 3: Calculate the Resultant Force
Now, we will add these vectors to find the resultant force \( \vec{R} \):
\[ \vec{R} = \vec{F_1} + \vec{F_2} + \vec{F_3} + \vec{F_4} \]
Substituting the values:
\[ \vec{R} = P \hat{i} + 2P \hat{j} - 3P \hat{i} - 4P \hat{j} \]
Combining the i and j components:
\[ \vec{R} = (P - 3P) \hat{i} + (2P - 4P) \hat{j} \]
\[ \vec{R} = -2P \hat{i} - 2P \hat{j} \]

### Step 4: Find the Magnitude of the Resultant Force
The magnitude of the resultant vector \( \vec{R} \) can be calculated using the Pythagorean theorem:
\[ |\vec{R}| = \sqrt{(-2P)^2 + (-2P)^2} \]
\[ |\vec{R}| = \sqrt{4P^2 + 4P^2} = \sqrt{8P^2} = 2\sqrt{2}P \]

### Final Answer
The magnitude of the resultant force is \( 2\sqrt{2}P \).

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