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

Options

A. zero
B. `+8`
C. 12
D. `-8`

Answer

To find the angular momentum of a particle about the Z-axis, we can use the formula:

\[ \vec{L} = \vec{r} \times \vec{p} \]

where \(\vec{L}\) is the angular momentum, \(\vec{r}\) is the position vector, and \(\vec{p}\) is the linear momentum of the particle. The linear momentum \(\vec{p}\) can be calculated as:

\[ \vec{p} = m \vec{v} \]

where \(m\) is the mass of the particle and \(\vec{v}\) is its velocity.

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

### Step 2: Calculate the linear momentum \(\vec{p}\)
Using the formula for linear momentum:

\[ \vec{p} = m \vec{v} = 2 \, \text{kg} \cdot 2(\hat{i} - \hat{j} + \hat{k}) \, \text{m/s} \]

Calculating this gives:

\[ \vec{p} = 4(\hat{i} - \hat{j} + \hat{k}) \, \text{kg m/s} = 4\hat{i} - 4\hat{j} + 4\hat{k} \, \text{kg m/s} \]

### Step 3: Calculate the angular momentum \(\vec{L}\)
Now we can calculate the angular momentum using the cross product:

\[ \vec{L} = \vec{r} \times \vec{p} = (\hat{i} + \hat{j}) \times (4\hat{i} - 4\hat{j} + 4\hat{k}) \]

### Step 4: Expand the cross product
Using the distributive property of the cross product:

\[ \vec{L} = \hat{i} \times (4\hat{i} - 4\hat{j} + 4\hat{k}) + \hat{j} \times (4\hat{i} - 4\hat{j} + 4\hat{k}) \]

Calculating each term:

1. \(\hat{i} \times 4\hat{i} = 0\)
2. \(\hat{i} \times (-4\hat{j}) = -4(\hat{i} \times \hat{j}) = -4\hat{k}\)
3. \(\hat{i} \times 4\hat{k} = 4(\hat{i} \times \hat{k}) = 4(-\hat{j}) = -4\hat{j}\)

Now for the second part:

1. \(\hat{j} \times 4\hat{i} = 4(\hat{j} \times \hat{i}) = -4\hat{k}\)
2. \(\hat{j} \times (-4\hat{j}) = 0\)
3. \(\hat{j} \times 4\hat{k} = 4(\hat{j} \times \hat{k}) = 4\hat{i}\)

Combining all these results:

\[ \vec{L} = (0 - 4\hat{k} - 4\hat{j}) + (4\hat{i} - 4\hat{k} + 0) = 4\hat{i} - 4\hat{j} - 8\hat{k} \]

### Step 5: Find the angular momentum about the Z-axis
The angular momentum about the Z-axis corresponds to the coefficient of \(\hat{k}\):

\[ L_z = -8 \, \text{kg m}^2/\text{s} \]

### Final Answer
The angular momentum of the particle about the Z-axis is:

\[ \vec{L}_z = -8 \, \text{kg m}^2/\text{s} \]