A body is constrained to move along z-axis of the co-ordinate system is being applied by a constant force `vec(F)=(2hat(j)+3hat(j)+4hat(k))`. Find the work done by this force in moving the body over a distance of 5m along z-axis.

Options

A. 30 J
B. 20 J
C. 40 J
D. 61 J

Answer

To solve the problem of finding the work done by the force on the body moving along the z-axis, we can follow these steps:

### Step 1: Identify the Force Vector
The force vector is given as:
\[ \vec{F} = 2\hat{i} + 3\hat{j} + 4\hat{k} \]

### Step 2: Identify the Displacement Vector
Since the body is moving along the z-axis for a distance of 5 meters, the displacement vector can be represented as:
\[ \vec{D} = 0\hat{i} + 0\hat{j} + 5\hat{k} = 5\hat{k} \]

### Step 3: Use the Work Done Formula
The work done \( W \) by a force is calculated using the dot product of the force vector and the displacement vector:
\[ W = \vec{F} \cdot \vec{D} \]

### Step 4: Calculate the Dot Product
Now, we will calculate the dot product:
\[ W = (2\hat{i} + 3\hat{j} + 4\hat{k}) \cdot (0\hat{i} + 0\hat{j} + 5\hat{k}) \]
Breaking this down:
- The dot product of the \( \hat{i} \) components: \( 2 \cdot 0 = 0 \)
- The dot product of the \( \hat{j} \) components: \( 3 \cdot 0 = 0 \)
- The dot product of the \( \hat{k} \) components: \( 4 \cdot 5 = 20 \)

Adding these results together:
\[ W = 0 + 0 + 20 = 20 \text{ Joules} \]

### Step 5: Conclusion
The work done by the force in moving the body over a distance of 5 meters along the z-axis is:
\[ \boxed{20 \text{ Joules}} \]

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