A body constrained to move along the z-axis of a coordinate system is subject to a constant force F given by `F = hati + 2hatj + 3hatk`. What is the work done by this force in moving the body a distance of 4 m along the z-axis ?

Options

A. 12 J
B. 24 J
C. 48 J
D. 6 J

Answer

To solve the problem of calculating the work done by a constant force \( \vec{F} = \hat{i} + 2\hat{j} + 3\hat{k} \) when a body moves along the z-axis for a distance of 4 m, we can follow these steps:

### Step 1: Identify the Force Vector and Displacement Vector
The force vector is given as:
\[ \vec{F} = \hat{i} + 2\hat{j} + 3\hat{k} \]
The displacement vector when the body moves 4 m along the z-axis can be expressed as:
\[ \vec{s} = 4\hat{k} \]

### Step 2: Use the Work Done Formula
The work done \( W \) by a force when it moves an object is given by the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{s} \]

### Step 3: Calculate the Dot Product
Now, we calculate the dot product:
\[ \vec{F} \cdot \vec{s} = (\hat{i} + 2\hat{j} + 3\hat{k}) \cdot (4\hat{k}) \]
Using the properties of the dot product:
\[ \vec{F} \cdot \vec{s} = \hat{i} \cdot (4\hat{k}) + 2\hat{j} \cdot (4\hat{k}) + 3\hat{k} \cdot (4\hat{k}) \]
Since \( \hat{i} \cdot \hat{k} = 0 \) and \( \hat{j} \cdot \hat{k} = 0 \), we have:
\[ \vec{F} \cdot \vec{s} = 0 + 0 + 3 \cdot 4 = 12 \]

### Step 4: Conclusion
Thus, the work done by the force in moving the body a distance of 4 m along the z-axis is:
\[ W = 12 \text{ Joules} \]