Question

A constant force $\vec{F} = 3\hat{i} - 2\hat{j} - \hat{k}$ N has a displacement $\vec{r} = 2\hat{i} - 3\hat{j} - 3\hat{k}$ m in 2 second. The work done and the power are respectively ______.

• A. 20 joule, 10 watt
• B. 15 joule, 7.5 watt
• C. 13 joule, 6.5 watt
• D. 10 joule, 5 watt

Hint:

Dot products ($\cdot$) always result in a scalar (a single number), which makes perfect sense because Work and Energy are scalar quantities, possessing magnitude but no physical direction.

Answer 1

The Correct Option is B

Step 1: Understanding the Question:
We are given a constant force vector and a displacement vector. We need to calculate the mechanical work done and then find the average power delivered over a specific time interval.

Step 2: Key Formula or Approach:
1. Work done ($W$) by a constant force is the dot product of the force and displacement vectors:
$$W = \vec{F} \cdot \vec{r}$$
2. Average Power ($P$) is the total work done divided by the time ($t$) taken:
$$P = \frac{W}{t}$$

Step 3: Detailed Explanation:
Given vectors:
$\vec{F} = 3\hat{i} - 2\hat{j} - 1\hat{k}$
$\vec{r} = 2\hat{i} - 3\hat{j} - 3\hat{k}$
Time $t = 2 \text{ s}$
Calculate the dot product for Work:
$$W = (F_x \times r_x) + (F_y \times r_y) + (F_z \times r_z)$$
$$W = (3)(2) + (-2)(-3) + (-1)(-3)$$
$$W = 6 + 6 + 3$$
$$W = 15 \text{ Joules}$$
Now, calculate the Power:
$$P = \frac{15 \text{ J}}{2 \text{ s}}$$
$$P = 7.5 \text{ Watts}$$

Step 4: Final Answer:
The work done is 15 J and power is 7.5 W, matching option (b).