Question

A force of \((2 \mathrm{i} \wedge+3 j \wedge-\mathrm{k} \wedge) \mathrm{N}\) acts on a body for 5 second, produces a displacement of \((3 i \wedge+5 j \wedge+k \wedge)\). What was the power used? (A) \(4 \mathrm{~W}\) (B) \(20 \mathrm{~W}\) (C) \(21 \mathrm{~W}\) (D) \(4.2 \mathrm{~W}\)

Short answer

The power used by the force is 4 W.

Step-by-step solution

Calculate dot product of force and displacement vectors

To calculate the dot product of the force and displacement vectors, multiply the respective components and add them together: Dot product = (Force in x) × (Displacement in x) + (Force in y) × (Displacement in y) + (Force in z) × (Displacement in z) Dot product = (2)(3) + (3)(5) + (-1)(1) = 6 + 15 - 1 = 20 J

Calculate power used by the force

Now that we have the work done by the force (20 J), we can divide that by the time (5 seconds) to calculate the power: Power = \(\frac{\text{Work done}}{\text{Time}}\) Power = \(\frac{20 \text{ J}}{5 \text{ s}}\) Power = 4 W So, the correct answer is (A) 4 W.

Key concepts

Dot product of vectors

The dot product, also known as the scalar product, is an operation that multiplies two vectors and results in a scalar (a single number). It is a crucial concept in physics, especially when dealing with forces and movements. To find the dot product, you need to multiply the corresponding components of each vector together and then sum the results.
For a force vector \((\vec{F} = 2\hat{i} + 3\hat{j} - \hat{k})\) and a displacement vector \((\vec{d} = 3\hat{i} + 5\hat{j} + \hat{k})\), the dot product is calculated as follows:

• Multiply the x-components: \(2 \times 3 = 6\)
• Multiply the y-components: \(3 \times 5 = 15\)
• Multiply the z-components: \((-1) \times 1 = -1\)

Now, add these three products together: \(6 + 15 - 1 = 20\). The dot product gives the work done by a force acting over a distance.

Work done

Work done in physics refers to the energy transferred when a force is applied to an object to move it over a distance. It is particularly interesting because it's directly related to how forces interact with objects. When calculating work done (W), we use the dot product of the force and displacement vectors. This is because work is defined as the component of force in the direction of displacement multiplied by the magnitude of displacement.
In our case:

• A force of \(2\hat{i} + 3\hat{j} - \hat{k} \) N is applied
• The work done during movement across a displacement of \(3\hat{i} + 5\hat{j} + \hat{k} \) m is 20 J

This number, 20 joules, indicates the total energy transferred to the body through the applied force.

Force and displacement

Force and displacement are fundamental concepts in physics that describe the interaction between an object and the effect on its motion. Force (\( \vec{F} \)) is a vector that represents a push or a pull and has both magnitude and direction. Displacement (\( \vec{d} \)) is also a vector that shows how far and in which direction an object has moved from its initial position.
In our exercise:

• The force is \(2\hat{i} + 3\hat{j} - \hat{k}\) N, which acts as a push or pull along the i, j, and k directions
• Displacement caused by this force is \(3\hat{i} + 5\hat{j} + \hat{k}\) m

Together, these vectors describe the action (force) and the effect (displacement) resulting in work done, as mentioned earlier. Understanding how force and displacement relate through vector arithmetic helps you solve problems involving work and energy.

Vector operations

Vector operations involve the basic mathematical manipulations of vectors, such as addition, subtraction, and multiplication, including the dot product. These operations are essential because they allow you to solve problems involving physical quantities that have direction and magnitude.
Consider the following for our problem:

• Vector addition: Although not directly used here, it helps relate force and response vectors
• Dot product: \(\vec{F} \cdot \vec{d} = 20\), finds the component of one vector along another
• Scalar multiplication: Dot product themselves are scalar quantities derived from vectors

By mastering vector operations, you'll be better equipped to handle various physics-based questions, especially those involving force, displacement, and energy. Understanding that vectors must be manipulated with direction in mind is crucial in physics problem-solving.