Moment of a force of magnitude 20 N acting along positive x-direction at point `(3m,0,0)` about the point `(0,2,0)` (in N-m) is

Options

A. 20
B. 60
C. 40
D. 30

Explanation

To solve the problem of finding the moment of a force about a point, we can follow these steps:

### Step 1: Identify the Given Information
- **Force (F)**: 20 N acting along the positive x-direction. This can be represented as a vector:
\[ \vec{F} = 20 \hat{i} \]
- **Point of application of force**: \( (3 \, \text{m}, 0, 0) \)
- **Point about which we are calculating the moment**: \( (0, 2, 0) \)

### Step 2: Calculate the Position Vector (r)
The position vector \( \vec{r} \) from the point about which we are calculating the moment to the point of application of the force is given by:
\[ \vec{r} = (3 - 0) \hat{i} + (0 - 2) \hat{j} + (0 - 0) \hat{k} \]
This simplifies to:
\[ \vec{r} = 3 \hat{i} - 2 \hat{j} \]

### Step 3: Use the Moment Formula
The moment \( \vec{M} \) of a force about a point is given by the cross product of the position vector \( \vec{r} \) and the force vector \( \vec{F} \):
\[ \vec{M} = \vec{r} \times \vec{F} \]

### Step 4: Set Up the Cross Product
We can express this in determinant form:
\[ \vec{M} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 3 & -2 & 0 \\ 20 & 0 & 0 \end{vmatrix} \]

### Step 5: Calculate the Determinant
Calculating the determinant, we have:
\[ \vec{M} = \hat{i} \begin{vmatrix} -2 & 0 \\ 0 & 0 \end{vmatrix} - \hat{j} \begin{vmatrix} 3 & 0 \\ 20 & 0 \end{vmatrix} + \hat{k} \begin{vmatrix} 3 & -2 \\ 20 & 0 \end{vmatrix} \]
Calculating each of these 2x2 determinants:
- For \( \hat{i} \): \( (-2)(0) - (0)(0) = 0 \)
- For \( \hat{j} \): \( (3)(0) - (20)(0) = 0 \)
- For \( \hat{k} \): \( (3)(0) - (-2)(20) = 40 \)

Putting it all together:
\[ \vec{M} = 0 \hat{i} - 0 \hat{j} + 40 \hat{k} = 40 \hat{k} \]

### Step 6: Find the Magnitude of the Moment
The magnitude of the moment \( |\vec{M}| \) is:
\[ |\vec{M}| = 40 \, \text{N-m} \]

### Final Answer
The moment of the force about the point \( (0, 2, 0) \) is:
\[ \boxed{40 \, \text{N-m}} \]