Force of 1, 2 units act along the lines x=0 and y= 0, the equation of the line of acting of the resultants is

Options

A. `y-2x=0`
B. `2y-x=0`
C. `y+x=0`
D. `y-x=0`

Answer

To solve the problem step by step, we will analyze the forces acting along the x and y axes and find the equation of the line of action of the resultant force.

### Step 1: Identify the Forces
We have two forces acting:
- A force of 1 unit acting along the line \( x = 0 \) (the y-axis).
- A force of 2 units acting along the line \( y = 0 \) (the x-axis).

### Step 2: Represent the Forces as Vectors
- The force along the y-axis can be represented as \( \vec{F_1} = 0\hat{i} + 1\hat{j} \) (1 unit in the positive y-direction).
- The force along the x-axis can be represented as \( \vec{F_2} = 2\hat{i} + 0\hat{j} \) (2 units in the positive x-direction).

### Step 3: Calculate the Resultant Force
The resultant force \( \vec{R} \) can be found by vector addition:
\[ \vec{R} = \vec{F_1} + \vec{F_2} = (0 + 2)\hat{i} + (1 + 0)\hat{j} = 2\hat{i} + 1\hat{j} \]

### Step 4: Determine the Direction of the Resultant Force
To find the direction of the resultant force, we can calculate the angle \( \theta \) it makes with the x-axis using the tangent function:
\[ \tan(\theta) = \frac{R_y}{R_x} = \frac{1}{2} \]
Thus, we can find \( \theta \):
\[ \theta = \tan^{-1}\left(\frac{1}{2}\right) \]

### Step 5: Write the Equation of the Line of Action
The slope \( m \) of the line can be calculated as:
\[ m = \frac{R_y}{R_x} = \frac{1}{2} \]
Using the point-slope form of the line equation, where we can take the point (0,0) (origin) as the point through which the line passes:
\[ y - 0 = \frac{1}{2}(x - 0) \]
This simplifies to:
\[ y = \frac{1}{2}x \]

### Step 6: Rearranging the Equation
To express the equation in standard form, we can rearrange it:
\[ 2y - x = 0 \]

### Final Result
The equation of the line of action of the resultant force is:
\[ 2y - x = 0 \]

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