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 (hatk - hati)`
B. `-48 hatk`
C. `36hatk`
D. `48(hati + hatj)`

Explanation

To solve the problem, we need to find the angular momentum of a particle with mass \( m = 2 \) kg, whose position vector as a function of time is given by:

\[ \vec{r}(t) = 2t \hat{i} - 3t^2 \hat{j} \]

We will calculate the angular momentum \( \vec{L} \) with respect to the origin at time \( t = 2 \) seconds.

### Step 1: Calculate the position vector at \( t = 2 \) seconds

Substituting \( t = 2 \) into the position vector:

\[ \vec{r}(2) = 2(2) \hat{i} - 3(2^2) \hat{j} = 4 \hat{i} - 12 \hat{j} \]

### Step 2: Calculate the velocity vector

The velocity \( \vec{v} \) is the derivative of the position vector with respect to time:

\[ \vec{v}(t) = \frac{d\vec{r}}{dt} = \frac{d}{dt}(2t \hat{i} - 3t^2 \hat{j}) = 2 \hat{i} - 6t \hat{j} \]

Now substituting \( t = 2 \):

\[ \vec{v}(2) = 2 \hat{i} - 6(2) \hat{j} = 2 \hat{i} - 12 \hat{j} \]

### Step 3: Calculate the angular momentum

The angular momentum \( \vec{L} \) is given by the formula:

\[ \vec{L} = m (\vec{r} \times \vec{v}) \]

Substituting \( m = 2 \) kg, \( \vec{r} = 4 \hat{i} - 12 \hat{j} \), and \( \vec{v} = 2 \hat{i} - 12 \hat{j} \):

First, we need to calculate the cross product \( \vec{r} \times \vec{v} \):

\[ \vec{r} \times \vec{v} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & -12 & 0 \\ 2 & -12 & 0 \end{vmatrix} \]

Calculating the determinant:

\[ \vec{r} \times \vec{v} = \hat{k} \left( 4(-12) - (-12)(2) \right) = \hat{k} \left( -48 + 24 \right) = -24 \hat{k} \]

### Step 4: Calculate \( \vec{L} \)

Now substituting back into the angular momentum formula:

\[ \vec{L} = 2 \cdot (-24 \hat{k}) = -48 \hat{k} \]

### Final Answer

Thus, the angular momentum of the particle with respect to the origin at \( t = 2 \) seconds is:

\[ \vec{L} = -48 \hat{k} \text{ kg m}^2/\text{s} \]

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