Problem 6 - Master Resource Book in JEE Main Physics

Question

The \(x\) and \(y\) components of a force are \(2 \mathrm{~N}\) and \(-3 \mathrm{~N}\). The force is (a) \(2 \hat{\mathrm{i}}-3 \hat{\mathrm{j}}\) (b) \(2 \hat{\mathrm{i}}+3 \hat{\mathrm{j}}\) (c) \(-2 \hat{i}-3 \hat{j}\) (d) \(3 \hat{\mathrm{i}}+2 \hat{\mathrm{j}}\)

Step-by-Step Solution

Answer

The correct option is (a) \(2 \hat{\mathrm{i}} - 3 \hat{\mathrm{j}}\).

Step 1: Interpret the Problem

In this task, we need to express a force vector using its components. The given components are in the x-direction and y-direction, where the x-component is \(2 \text{ N}\) and the y-component is \(-3 \text{ N}\).

Step 2: Construct the Vector

To represent a force vector given its components, combine the unit vectors with their corresponding components. Therefore, the force vector will be expressed as \(2 \hat{\mathrm{i}} - 3 \hat{\mathrm{j}}\). The vector combines the contributions in both the x- and y-directions.

Step 3: Match with Options

Look at the provided options to see which one matches our constructed vector. Our constructed vector is \(2 \hat{\mathrm{i}} - 3 \hat{\mathrm{j}}\). The option that corresponds to this vector is option (a) \(2 \hat{\mathrm{i}} - 3 \hat{\mathrm{j}}\).

Step 4: Verify and Conclude

Double-check the components match the description provided in the problem. The x-component \(2 \text{ N}\) and y-component \(-3 \text{ N}\) match exactly with the vector \(2 \hat{\mathrm{i}} - 3 \hat{\mathrm{j}}\). This confirms that option (a) is the correct representation.

Key Concepts

Force Vector

A force vector represents the magnitude and direction of a force in two-dimensional or three-dimensional space. It is an essential concept in physics and engineering, allowing us to analyze forces acting on an object in a structured way.

In two dimensions, a force vector is often expressed using the unit vectors \(\hat{i}\) and \(\hat{j}\), which represent the directions along the x-axis and y-axis respectively. This makes it easier to understand how much of the force is acting in each direction:

• \(\hat{i}\) represents the x-direction.
• \(\hat{j}\) represents the y-direction.

By breaking down the force into its components, we can easily sum up multiple forces acting on an object to get the resultant force. This is crucial for solving problems in statics and dynamics. The given problem involves a force vector with x and y components of \(2 \, \text{N}\) and \(-3 \, \text{N}\) respectively.

Unit Vector Notation

Unit vectors are essential in vector notation. A unit vector has a magnitude of 1 and indicates direction. By expressing a vector in terms of unit vectors, we can clearly express both direction and magnitude in a standardized form.

In two-dimensional space, unit vectors consist of \(\hat{i}\) and \(\hat{j}\):

• \(\hat{i}\) has a value of 1 in the x-direction.
• \(\hat{j}\) has a value of 1 in the y-direction.

For the given problem, the force vector is expressed as \(2 \hat{i} - 3 \hat{j}\). This representation implies that the force has a magnitude of 2 units in the positive x-direction and 3 units in the negative y-direction.
This notation simplifies calculations by breaking up complex problems into manageable components. Each component is multiplied by its respective unit vector, allowing precise representation and calculation of forces in multiple directions.

X and Y Components

The x and y components of a vector tell us how much of the vector's magnitude lies along each axis. These components are crucial for completely understanding the vector's impact or effect in each direction.

In the given problem, the force vector is initially broken into x and y components of \(2 \, \text{N}\) and \(-3 \, \text{N}\) respectively.

• The x-component is \(2 \, \text{N}\): this component acts along the positive x-axis.
• The y-component is \(-3 \, \text{N}\): this acts along the negative y-axis.

Understanding these components allows for an accurate depiction of how the force interacts with objects, making them indispensable when constructing the vector. These components can be resolved using trigonometric relationships if needed, providing flexibility in problem-solving scenarios.