A body constrained to move in y direction is subjected to a force given by `vec(F)=(-2hat(i)+15hat(j)+6hat(k))` N . What is the work done by this force in moving the body through a distance of `10 m` along y-axis ?

Options

A. 150 J
B. 20 J
C. 190 J
D. 160 J

Answer

To solve the problem, we will calculate the work done by the force on the body as it moves along the y-axis. The work done \( W \) by a force \( \vec{F} \) when moving an object through a displacement \( \vec{d} \) is given by the dot product:

\[ W = \vec{F} \cdot \vec{d} \]

### Step 1: Identify the force vector and displacement vector
The force vector is given as:
\[ \vec{F} = -2 \hat{i} + 15 \hat{j} + 6 \hat{k} \, \text{N} \]

Since the body is constrained to move only in the y-direction, the displacement vector \( \vec{d} \) for a distance of 10 m along the y-axis is:
\[ \vec{d} = 10 \hat{j} \, \text{m} \]

### Step 2: Calculate the dot product
Now, we will calculate the dot product \( \vec{F} \cdot \vec{d} \):
\[ W = \vec{F} \cdot \vec{d} = (-2 \hat{i} + 15 \hat{j} + 6 \hat{k}) \cdot (10 \hat{j}) \]

Using the property of dot product, we can simplify this:
\[ W = (-2 \hat{i} \cdot 10 \hat{j}) + (15 \hat{j} \cdot 10 \hat{j}) + (6 \hat{k} \cdot 10 \hat{j}) \]

Since \( \hat{i} \cdot \hat{j} = 0 \) and \( \hat{k} \cdot \hat{j} = 0 \), we have:
\[ W = 0 + (15 \cdot 10) + 0 \]
\[ W = 150 \, \text{J} \]

### Step 3: Conclusion
Thus, the work done by the force in moving the body through a distance of 10 m along the y-axis is:
\[ W = 150 \, \text{J} \]