A particle of mass 300 g is moving with a speed of 20 `ms^-1` along the straight line y = x - `4sqrt2`. The angular momentum of the particle about the origin is (where y & x are in metres)

Answer

To find the angular momentum of a particle about the origin, we can follow these steps:

### Step 1: Identify the mass and speed of the particle
The mass \( m \) of the particle is given as 300 g, which we need to convert to kilograms:
\[ m = 300 \, \text{g} = 0.3 \, \text{kg} \]
The speed \( v \) of the particle is given as:
\[ v = 20 \, \text{m/s} \]

### Step 2: Determine the equation of the line and find points of intersection with axes
The particle is moving along the line given by the equation:
\[ y = x - 4\sqrt{2} \]
To find the points where this line intersects the axes, we can set \( x = 0 \) to find the y-intercept:
\[ y = 0 - 4\sqrt{2} = -4\sqrt{2} \]
So, the point of intersection with the y-axis is \( (0, -4\sqrt{2}) \).

Next, we set \( y = 0 \) to find the x-intercept:
\[ 0 = x - 4\sqrt{2} \implies x = 4\sqrt{2} \]
So, the point of intersection with the x-axis is \( (4\sqrt{2}, 0) \).

### Step 3: Find the slope of the line
The slope \( m \) of the line can be calculated using the two points found:
- Point A: \( (0, -4\sqrt{2}) \)
- Point B: \( (4\sqrt{2}, 0) \)

The slope \( m \) is given by:
\[ m = \frac{y_2 - y_1}{x_2 - x_1} = \frac{0 - (-4\sqrt{2})}{4\sqrt{2} - 0} = \frac{4\sqrt{2}}{4\sqrt{2}} = 1 \]

### Step 4: Find the coordinates of the particle's position
The equation of the line can be rewritten in slope-intercept form:
\[ y = x - 4\sqrt{2} \]
To find the coordinates of the particle at a specific time, we can use the fact that it moves with a speed of 20 m/s along this line.

### Step 5: Calculate the position vector \( \vec{R} \)
Assuming the particle is at point \( (x, y) \) at time \( t \), we can express \( R \) as:
\[ R = \sqrt{x^2 + y^2} \]
Using the coordinates \( (2\sqrt{2}, -2\sqrt{2}) \) (found in the video transcript), we calculate:
\[ R = \sqrt{(2\sqrt{2})^2 + (-2\sqrt{2})^2} = \sqrt{8 + 8} = \sqrt{16} = 4 \, \text{m} \]

### Step 6: Calculate the angular momentum \( L \)
The angular momentum \( L \) about the origin is given by the formula:
\[ L = m \cdot v \cdot R \]
Substituting the known values:
\[ L = 0.3 \, \text{kg} \cdot 20 \, \text{m/s} \cdot 4 \, \text{m} \]
Calculating this gives:
\[ L = 0.3 \cdot 20 \cdot 4 = 24 \, \text{kg m}^2/\text{s} \]

### Final Answer
The angular momentum of the particle about the origin is:
\[ \boxed{24 \, \text{kg m}^2/\text{s}} \]