A particle of mass `2 kg` located at the position `(hati + hatj)m` has velocity `2(hati - hatj + hatk) m//s` . Its angualr momentum about Z-axis in `kg m^(2)//s` is

Options

A. `+4`
B. `- 8`
C. `+ 8`
D. `- 4`

Explanation

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

### Step 1: Identify the given quantities
- Mass of the particle, \( m = 2 \, \text{kg} \)
- Position vector, \( \mathbf{r} = \hat{i} + \hat{j} \, \text{m} \)
- Velocity vector, \( \mathbf{v} = 2(\hat{i} - \hat{j} + \hat{k}) \, \text{m/s} \)

### Step 2: Calculate the momentum of the particle
The momentum \( \mathbf{p} \) of the particle can be calculated using the formula:
\[ \mathbf{p} = m \cdot \mathbf{v} \]
Substituting the values:
\[ \mathbf{p} = 2 \, \text{kg} \cdot (2 \hat{i} - 2 \hat{j} + 2 \hat{k}) \, \text{m/s} = 4 \hat{i} - 4 \hat{j} + 4 \hat{k} \, \text{kg m/s} \]

### Step 3: Calculate the angular momentum
The angular momentum \( \mathbf{L} \) about the Z-axis is given by the cross product of the position vector \( \mathbf{r} \) and the momentum vector \( \mathbf{p} \):
\[ \mathbf{L} = \mathbf{r} \times \mathbf{p} \]
Substituting the values:
\[ \mathbf{L} = (\hat{i} + \hat{j}) \times (4 \hat{i} - 4 \hat{j} + 4 \hat{k}) \]

### Step 4: Compute the cross product
Using the distributive property of the cross product:
\[ \mathbf{L} = \hat{i} \times (4 \hat{i} - 4 \hat{j} + 4 \hat{k}) + \hat{j} \times (4 \hat{i} - 4 \hat{j} + 4 \hat{k}) \]
Calculating each term:
1. \( \hat{i} \times 4 \hat{i} = 0 \)
2. \( \hat{i} \times (-4 \hat{j}) = -4 \hat{k} \)
3. \( \hat{i} \times 4 \hat{k} = 4 \hat{j} \)

So, the first part gives:
\[ \hat{i} \times (4 \hat{i} - 4 \hat{j} + 4 \hat{k}) = 0 - 4 \hat{k} + 4 \hat{j} = 4 \hat{j} - 4 \hat{k} \]

Now for the second term:
1. \( \hat{j} \times 4 \hat{i} = -4 \hat{k} \)
2. \( \hat{j} \times (-4 \hat{j}) = 0 \)
3. \( \hat{j} \times 4 \hat{k} = 4 \hat{i} \)

So, the second part gives:
\[ \hat{j} \times (4 \hat{i} - 4 \hat{j} + 4 \hat{k}) = -4 \hat{k} + 0 + 4 \hat{i} = 4 \hat{i} - 4 \hat{k} \]

### Step 5: Combine the results
Now, combine both results:
\[ \mathbf{L} = (4 \hat{j} - 4 \hat{k}) + (4 \hat{i} - 4 \hat{k}) = 4 \hat{i} + 4 \hat{j} - 8 \hat{k} \]

### Step 6: Extract the Z-component of angular momentum
The Z-component of angular momentum \( L_z \) is:
\[ L_z = -8 \, \text{kg m}^2/\text{s} \]

Thus, the angular momentum about the Z-axis is:
\[ \mathbf{L} = 4 \hat{i} + 4 \hat{j} - 8 \hat{k} \quad \text{and} \quad L_z = -8 \, \text{kg m}^2/\text{s} \]

### Final Answer
The angular momentum about the Z-axis is \( -8 \, \text{kg m}^2/\text{s} \).

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