A particle of mass 0.2 kg is moving with linear velocity `(i-j+2k)`. If the radius vector `r=4i+j-k`, the angular momentum of the particle is

Options

A. `2.14` units
B. `sqrt(4.28)` units
C. `sqrt(107)` units
D. `5` units

Answer

To solve the problem of finding the angular momentum of a particle with a given mass, linear velocity, and radius vector, we can follow these steps:

### Step 1: Identify the given quantities
- Mass of the particle, \( m = 0.2 \, \text{kg} \)
- Linear velocity vector, \( \mathbf{v} = \mathbf{i} - \mathbf{j} + 2\mathbf{k} \)
- Radius vector, \( \mathbf{r} = 4\mathbf{i} + \mathbf{j} - \mathbf{k} \)

### Step 2: Use the formula for angular momentum
The angular momentum \( \mathbf{L} \) of a particle is given by the formula:
\[ \mathbf{L} = m \, \mathbf{r} \times \mathbf{v} \]
where \( \times \) denotes the cross product.

### Step 3: Calculate the cross product \( \mathbf{r} \times \mathbf{v} \)
To compute the cross product, we can use the determinant method:
\[ \mathbf{r} \times \mathbf{v} = \begin{vmatrix} \mathbf{i} & \mathbf{j} & \mathbf{k} \\ 4 & 1 & -1 \\ 1 & -1 & 2 \end{vmatrix} \]

### Step 4: Expand the determinant
Calculating the determinant:
\[ \mathbf{r} \times \mathbf{v} = \mathbf{i} \begin{vmatrix} 1 & -1 \\ -1 & 2 \end{vmatrix} - \mathbf{j} \begin{vmatrix} 4 & -1 \\ 1 & 2 \end{vmatrix} + \mathbf{k} \begin{vmatrix} 4 & 1 \\ 1 & -1 \end{vmatrix} \]

Calculating the individual 2x2 determinants:
1. For \( \mathbf{i} \):
\[ 1 \cdot 2 - (-1) \cdot (-1) = 2 - 1 = 1 \]
2. For \( \mathbf{j} \):
\[ 4 \cdot 2 - (-1) \cdot 1 = 8 + 1 = 9 \]
3. For \( \mathbf{k} \):
\[ 4 \cdot (-1) - 1 \cdot 1 = -4 - 1 = -5 \]

Putting it all together:
\[ \mathbf{r} \times \mathbf{v} = \mathbf{i}(1) - \mathbf{j}(9) + \mathbf{k}(-5) = \mathbf{i} - 9\mathbf{j} - 5\mathbf{k} \]

### Step 5: Calculate the angular momentum
Now, substituting back into the angular momentum formula:
\[ \mathbf{L} = m \, \mathbf{r} \times \mathbf{v} = 0.2 \, (\mathbf{i} - 9\mathbf{j} - 5\mathbf{k}) \]
\[ \mathbf{L} = 0.2\mathbf{i} - 1.8\mathbf{j} - 1.0\mathbf{k} \]

### Step 6: Find the magnitude of angular momentum
The magnitude of angular momentum is given by:
\[ |\mathbf{L}| = \sqrt{(0.2)^2 + (-1.8)^2 + (-1.0)^2} \]
Calculating:
\[ |\mathbf{L}| = \sqrt{0.04 + 3.24 + 1.00} = \sqrt{4.28} \]

### Final Result
Thus, the angular momentum of the particle is:
\[ \mathbf{L} = 0.2\mathbf{i} - 1.8\mathbf{j} - 1.0\mathbf{k} \quad \text{with magnitude} \quad |\mathbf{L}| = \sqrt{4.28} \]