The angular momentum of a particle about origin is varying as L =4t+8(SI units) whern its moves along a straight line y=x-4(x,y in metres). The magnitude of force acting on the particle will be

Options

A. 1 N
B. 2 N
C. `sqrt2 N`
D. `sqrt3 N`

Answer

To solve the problem, we need to find the magnitude of the force acting on the particle given its angular momentum \( L \) and the path it follows. Let's break it down step by step.

### Step 1: Understanding Angular Momentum
The angular momentum \( L \) of the particle about the origin is given by the equation:
\[ L = 4t + 8 \]
where \( t \) is time in seconds.

### Step 2: Finding the Rate of Change of Angular Momentum
The torque \( \tau \) is defined as the rate of change of angular momentum:
\[ \tau = \frac{dL}{dt} \]
Calculating the derivative:
\[ \frac{dL}{dt} = \frac{d}{dt}(4t + 8) = 4 \]
Thus, the torque \( \tau \) is:
\[ \tau = 4 \, \text{N m} \]

### Step 3: Analyzing the Particle's Motion
The particle moves along the straight line given by the equation:
\[ y = x - 4 \]
This can be rewritten in standard form:
\[ y - x + 4 = 0 \]
The slope of this line is 1, which indicates that the angle \( \theta \) with respect to the x-axis is:
\[ \tan \theta = 1 \implies \theta = 45^\circ \]

### Step 4: Relating Torque and Force
The relationship between torque, force, and the position vector \( \mathbf{r} \) is given by:
\[ \tau = \mathbf{F} \cdot \mathbf{r} \sin \theta \]
In this case, since \( \theta = 45^\circ \):
\[ \tau = F \cdot r \cdot \sin(45^\circ) \]
We know that \( \sin(45^\circ) = \frac{1}{\sqrt{2}} \).

### Step 5: Solving for Force
Rearranging the equation to solve for the magnitude of the force \( F \):
\[ F = \frac{\tau}{r \cdot \sin(45^\circ)} \]
Substituting \( \tau = 4 \, \text{N m} \) and \( \sin(45^\circ) = \frac{1}{\sqrt{2}} \):
\[ F = \frac{4}{r \cdot \frac{1}{\sqrt{2}}} = \frac{4\sqrt{2}}{r} \]
To find the magnitude of the force, we need the value of \( r \).

### Step 6: Finding the Position Vector \( r \)
The position vector \( \mathbf{r} \) can be expressed in terms of \( x \) and \( y \). For the line \( y = x - 4 \), at any point \( (x, y) \):
\[ r = \sqrt{x^2 + (x - 4)^2} = \sqrt{x^2 + (x^2 - 8x + 16)} = \sqrt{2x^2 - 8x + 16} \]
However, we can simplify our calculations by noting that we need the force magnitude, which is independent of \( r \) in this case.

### Final Calculation
Since we are looking for the magnitude of the force, we can conclude that:
\[ F = 4 \sqrt{2} \text{ N} \]

### Conclusion
Thus, the magnitude of the force acting on the particle is:
\[ \boxed{4\sqrt{2} \, \text{N}} \]