Question

a force \(\underline{F}=(2,1,-1)\) act on a practical and displaces it from the point \(\mathrm{A}(2,-1,0)\) to the point \(\mathrm{B}(2,1,0)\) then work done by force is equal to (a) 2 (b) 4 (c) 6 (d) 5

Short answer

The short answer is: The work done by the force \(\underline{F}\) when it displaces the particle from point A to point B is 2.

Step-by-step solution

Calculate the displacement vector

To find the displacement vector \(\underline{d}\), subtract the coordinates of point A from the coordinates of point B: \(\underline{d} = \mathrm{B} - \mathrm{A} = (2 - 2, 1 - (-1), 0 - 0) = (0, 2, 0)\)

Calculate the dot product between the force vector and displacement vector

Now, we will find the dot product between the force vector \(\underline{F}\) and the displacement vector \(\underline{d}\): \(\underline{F} \cdot \underline{d} = (2, 1, -1) \cdot (0, 2, 0) = 2 \times 0 + 1 \times 2 + (-1) \times 0 = 0 + 2 + 0 = 2\)

Match the result with the given options

Since we have calculated the work done as 2, the correct answer is: (a) 2

Key concepts

Displacement Vector

Understanding displacement vectors is crucial in physics for representing the change in position of an object. A displacement vector points from the initial position to the final position, indicating the straight-line distance along with the direction. For example, in the given exercise, the formula to determine the displacement vector \( \underline{d} \) is a simple example of vector subtraction. We subtract the coordinates of the initial point A from the coordinates of the final point B. Mathematically, this looks like \( \underline{d} = B - A \) which in our case involves subtracting \( (2, -1, 0) \) from \( (2, 1, 0) \) to get the resultant displacement vector \( \underline{d} = (0, 2, 0) \). This vector tells us that the object has moved 2 units in the positive y-direction without any movement in the x or z-direction.

Dot Product

The dot product, also known as the scalar product, is a critical operation when working with vectors in physics. It allows us to calculate the amount of one vector that lies in the direction of another. The dot product of two vectors \( \underline{A} \) and \( \underline{B} \) can be calculated by multiplying their corresponding components and summing those products. The formula for the dot product is represented as \( \underline{A} \cdot \underline{B} = A_xB_x + A_yB_y + A_zB_z \), where subscript x, y, and z denote the components of the vectors.

In the context of our problem, the dot product of the force vector \( \underline{F} \) and the displacement vector \( \underline{d} \) determines the work done by the force when displacing an object. Calculating the dot product \( \underline{F} \cdot \underline{d} = (2, 1, -1) \cdot (0, 2, 0) \), we only consider the y-components as the other components will be multiplied by zero, resulting in \( 2 \cdot 0 + 1 \cdot 2 + (-1) \cdot 0 = 2 \). This result shows the work done, only accounting for the force in the direction of the displacement.

Vector Subtraction

Vector subtraction is the process of finding a vector that represents the difference between two points in space. It is similar to vector addition but involves flipping the direction of the second vector and then performing addition. In physics problems, we use vector subtraction to identify displacements, force differences, and more. For instance, the solution to the given exercise uses vector subtraction to determine the displacement vector. By calculating \( \underline{d} = B - A \) which involves subtracting the components of point A from B, as in \( (2 - 2, 1 - (-1), 0 - 0) \), we have the new vector \( (0, 2, 0) \). This method is efficient for finding the direction and magnitude of the change in position, which is indispensable for solving physics problems relating to forces and movements.