The time dependence of the position of a particle of mass m = 2 is given by `vec(r) (t) = 2t hat(i) - 3 t^(2) hat(j)` Its angular momentum, with respect to the origin, at time t = 2 is :

Options

A. `-34 (hat(k) -hat(i))`
B. `-48hat(k)`
C. `36hat(k)`
D. `48(hat(i) + hat(j))`

Explanation

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

### Step 1: Write down the position vector
The position vector of the particle is given by:
\[ \vec{r}(t) = 2t \hat{i} - 3t^2 \hat{j} \]

### Step 2: Substitute the time \( t = 2 \)
To find the position vector at \( t = 2 \):
\[ \vec{r}(2) = 2(2) \hat{i} - 3(2^2) \hat{j} = 4 \hat{i} - 12 \hat{j} \]

### Step 3: Differentiate the position vector to find the velocity vector
The velocity vector \( \vec{v}(t) \) is the derivative of the position vector with respect to time:
\[ \vec{v}(t) = \frac{d\vec{r}}{dt} = \frac{d}{dt}(2t \hat{i} - 3t^2 \hat{j}) = 2 \hat{i} - 6t \hat{j} \]

### Step 4: Substitute the time \( t = 2 \) into the velocity vector
Now, substitute \( t = 2 \) into the velocity vector:
\[ \vec{v}(2) = 2 \hat{i} - 6(2) \hat{j} = 2 \hat{i} - 12 \hat{j} \]

### Step 5: Calculate the angular momentum
The angular momentum \( \vec{L} \) with respect to the origin is given by:
\[ \vec{L} = m \vec{r} \times \vec{v} \]
where \( m = 2 \) kg. First, we need to calculate the cross product \( \vec{r} \times \vec{v} \):
\[ \vec{r} = 4 \hat{i} - 12 \hat{j}, \quad \vec{v} = 2 \hat{i} - 12 \hat{j} \]

Using the determinant to find the cross product:
\[ \vec{r} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & -12 & 0 \\ 2 & -12 & 0 \end{vmatrix} \]

Calculating the determinant:
\[ = \hat{k} \left( 4(-12) - (-12)(2) \right) = \hat{k} \left( -48 + 24 \right) = -24 \hat{k} \]

### Step 6: Multiply by the mass to get angular momentum
Now, multiply by the mass:
\[ \vec{L} = 2 \times (-24 \hat{k}) = -48 \hat{k} \]

### Final Answer
Thus, the angular momentum of the particle with respect to the origin at \( t = 2 \) seconds is:
\[ \vec{L} = -48 \hat{k} \]

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