A force `vec F = 2 hat i + 3 hat j N` is applied to an object that is pivoted about a fixed axle aligned along the `z` coordinate axis. If the force is applied at the point `vec r = 4 hat i + 5 hat j m`, (a) the magnitude of the net torque about the `z` and (b) the direction of the torque vector `tau`.

Options

A. 4
B. 1
C. 2
D. 10

Answer

To solve the problem, we need to calculate the torque \(\vec{\tau}\) produced by the force \(\vec{F}\) applied at the position \(\vec{r}\) about the z-axis. The torque is defined as the cross product of the position vector \(\vec{r}\) and the force vector \(\vec{F}\).

### Given:
- Force vector: \(\vec{F} = 2 \hat{i} + 3 \hat{j}\) N
- Position vector: \(\vec{r} = 4 \hat{i} + 5 \hat{j}\) m

### Step 1: Calculate the Torque
The torque \(\vec{\tau}\) is given by the formula:
\[ \vec{\tau} = \vec{r} \times \vec{F} \]

Substituting the values:
\[ \vec{\tau} = (4 \hat{i} + 5 \hat{j}) \times (2 \hat{i} + 3 \hat{j}) \]

### Step 2: Use the Determinant Method
To compute the cross product, we can use the determinant of a matrix:
\[ \vec{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 & 5 & 0 \\ 2 & 3 & 0 \end{vmatrix} \]

### Step 3: Calculate the Determinant
Calculating the determinant, we have:
\[ \vec{\tau} = \hat{i} \begin{vmatrix} 5 & 0 \\ 3 & 0 \end{vmatrix} - \hat{j} \begin{vmatrix} 4 & 0 \\ 2 & 0 \end{vmatrix} + \hat{k} \begin{vmatrix} 4 & 5 \\ 2 & 3 \end{vmatrix} \]

Calculating each of the 2x2 determinants:
1. \(\begin{vmatrix} 5 & 0 \\ 3 & 0 \end{vmatrix} = 5 \cdot 0 - 0 \cdot 3 = 0\)
2. \(\begin{vmatrix} 4 & 0 \\ 2 & 0 \end{vmatrix} = 4 \cdot 0 - 0 \cdot 2 = 0\)
3. \(\begin{vmatrix} 4 & 5 \\ 2 & 3 \end{vmatrix} = 4 \cdot 3 - 5 \cdot 2 = 12 - 10 = 2\)

Now substituting back into the torque equation:
\[ \vec{\tau} = 0 \hat{i} - 0 \hat{j} + 2 \hat{k} = 2 \hat{k} \]

### Step 4: Magnitude of the Torque
The magnitude of the torque is given by:
\[ |\vec{\tau}| = 2 \, \text{N m} \]

### Step 5: Direction of the Torque Vector
The direction of the torque vector \(\vec{\tau}\) is along the positive z-axis, represented by \(\hat{k}\).

### Final Answers:
(a) The magnitude of the net torque about the z-axis is \(2 \, \text{N m}\).

(b) The direction of the torque vector \(\vec{\tau}\) is along the positive z-axis.