A particle of mass `m=5` units is moving with a uniform speed `v = 3 sqrt(2)` units in the XY-plane along the `y=x+4`. The magnitude of the angular momentum about origin is

Options

A. zero
B. 60 unit
C. 7.5 unit
D. `40 sqrt(2)` unit

Explanation

To find the magnitude of the angular momentum of a particle moving along a line in the XY-plane, we can follow these steps:

### Step 1: Understand the motion of the particle
The particle has a mass \( m = 5 \) units and is moving with a uniform speed \( v = 3\sqrt{2} \) units along the line given by the equation \( y = x + 4 \).

### Step 2: Determine the slope and angle of the line
The line \( y = x + 4 \) has a slope \( m = 1 \). This means that the angle \( \theta \) that the line makes with the x-axis can be calculated using:
\[ \tan(\theta) = 1 \implies \theta = 45^\circ \]

### Step 3: Find the perpendicular distance from the origin to the line
To find the perpendicular distance \( r_{\perp} \) from the origin (0,0) to the line \( y = x + 4 \), we can use the formula for the distance from a point to a line:
\[ \text{Distance} = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}} \]
For the line \( y = x + 4 \), we can rewrite it in the form \( Ax + By + C = 0 \):
\[ x - y + 4 = 0 \quad \Rightarrow \quad A = 1, B = -1, C = 4 \]
Now, substituting the origin coordinates \( (x, y) = (0, 0) \):
\[ \text{Distance} = \frac{|1(0) - 1(0) + 4|}{\sqrt{1^2 + (-1)^2}} = \frac{4}{\sqrt{2}} = 2\sqrt{2} \]

### Step 4: Calculate the angular momentum
The angular momentum \( L \) about the origin is given by the formula:
\[ L = m \cdot v \cdot r_{\perp} \]
Substituting the known values:
\[ L = 5 \cdot (3\sqrt{2}) \cdot (2\sqrt{2}) \]
Calculating this step by step:
1. Calculate \( 3\sqrt{2} \cdot 2\sqrt{2} \):
\[ 3\sqrt{2} \cdot 2\sqrt{2} = 3 \cdot 2 \cdot 2 = 12 \]
2. Now multiply by the mass:
\[ L = 5 \cdot 12 = 60 \]

### Final Answer
The magnitude of the angular momentum about the origin is \( L = 60 \) units.

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