Finding B When Work Done Is Zero A JEE Main 2025 Physics Problem

Introduction

In physics, the concept of work is crucial for understanding how forces cause displacement. When a force acts on an object, and the object moves, work is said to be done. This article delves into a specific problem concerning the work done by a force on a particle, focusing on determining the value of an unknown component within the force vector. This kind of problem is typical for the JEE Main exam, requiring a strong understanding of vector algebra and the definition of work. Our main keyword is the concept of work done by a force, which is a fundamental topic in mechanics. We aim to provide a comprehensive explanation and a step-by-step solution to help you grasp the underlying principles. In this article, we'll explore how the dot product of force and displacement vectors determines the work done, and we'll apply this knowledge to solve a problem where the work done is zero. This scenario helps us understand the conditions under which a force, even when applied, does not result in any work. The problem we're addressing is representative of the type of questions asked in the JEE Main exam, making it a valuable exercise for aspirants. By understanding the relationship between force, displacement, and work, you'll be better prepared to tackle similar problems and enhance your problem-solving skills. This knowledge extends beyond theoretical calculations and is applicable in various real-world scenarios, making it a core concept in physics.

Problem Statement

The problem presented is as follows: A force F = 2i + bj + k is applied to a particle, causing it to undergo a displacement of s = i - 2j - k. Given that the work done on the particle is zero, we need to find the value of 'b'.

This problem encapsulates several key concepts in physics, including vector representation of forces and displacements, the definition of work as a dot product, and the conditions under which work done is zero. The core challenge lies in using the given information to set up an equation and solve for the unknown variable 'b'. This involves a strong understanding of vector algebra and the ability to apply it in a physical context. To solve this, we'll first need to recall the definition of work done by a force. Work (W) is defined as the dot product of the force vector (F) and the displacement vector (s). Mathematically, this is represented as W = F · s. The dot product essentially measures the component of the force that acts in the direction of the displacement. When the work done is zero, it implies that either the force is perpendicular to the displacement, or one of the vectors (force or displacement) is a null vector. In our case, the force and displacement vectors are non-zero, so the condition of zero work implies their perpendicularity. This condition is crucial for setting up the equation needed to solve for 'b'. The problem also serves as an excellent example of how vector concepts are applied in mechanics. It requires not just the knowledge of the formula for work, but also the understanding of what the dot product represents and how it relates to the physical scenario. By solving this problem, we reinforce our understanding of these concepts and develop our problem-solving abilities in physics.

Key Concepts: Work Done by a Force

To tackle this problem effectively, it's essential to have a firm grasp of the concept of work done by a force. In physics, work is defined as the energy transferred to or from an object by the application of a force along with a displacement. Mathematically, work (W) is expressed as the dot product of the force vector (F) and the displacement vector (s):

W = F · s

Where:

• F is the force vector.
• s is the displacement vector.
• The dot product (·) is a scalar quantity calculated as the sum of the products of the corresponding components of the two vectors. If F = Fxi + Fyj + Fzk and s = sxi + syj + szk, then:

F · s = Fx * sx + Fy * sy + Fz * sz

The dot product is a crucial mathematical operation in physics, particularly in mechanics, as it allows us to extract the component of one vector that lies along the direction of another. In the context of work, the dot product gives us the component of the force that acts along the displacement, which is the effective force contributing to the work done. When the angle between the force and displacement vectors is 90 degrees, their dot product is zero, indicating that no work is done. This is a key concept for understanding scenarios where a force is applied, but no work results. In the problem at hand, we are given that the work done is zero. This implies that the dot product of the force and displacement vectors must be zero. This condition forms the basis for our solution, allowing us to set up an equation and solve for the unknown variable 'b'. Understanding the dot product and its significance in the definition of work is crucial for solving this problem and similar problems in mechanics. It is also essential to remember that work is a scalar quantity, meaning it has magnitude but no direction. This distinguishes it from force and displacement, which are vector quantities and have both magnitude and direction. By mastering these fundamental concepts, you will be well-equipped to tackle a wide range of problems involving work, energy, and forces.

Step-by-Step Solution

1. Write down the given vectors:

   • Force vector: F = 2i + bj + k
   • Displacement vector: s = i - 2j - k

2. Apply the work formula:

   Work (W) = F · s = (2i + bj + k) · (i - 2j - k)

3. Calculate the dot product:

   W = (2 * 1) + (b * -2) + (1 * -1) W = 2 - 2b - 1

4. Set the work equal to zero:

   Since the work done is zero, we have: 0 = 2 - 2b - 1

5. Solve for b:

   0 = 1 - 2b 2b = 1 b = 1/2

Therefore, the value of b is 1/2. This step-by-step solution illustrates the application of the work formula and the use of the dot product to solve for an unknown variable. The key here is the understanding that zero work implies the dot product of the force and displacement vectors is zero. This allows us to set up a simple algebraic equation and find the value of 'b'. By breaking down the problem into smaller, manageable steps, we can avoid errors and ensure a clear understanding of the solution process. This approach is particularly useful for complex problems in physics, where careful application of formulas and concepts is essential. Furthermore, this solution highlights the importance of vector algebra in physics. Being comfortable with vector operations such as the dot product is crucial for solving many problems in mechanics and other areas of physics. Practicing such problems regularly will help solidify your understanding and improve your problem-solving speed and accuracy. In summary, the solution involves applying the definition of work, calculating the dot product, setting up an equation based on the given condition of zero work, and solving for the unknown variable. Each step is crucial and contributes to the overall understanding of the problem and its solution.

Conclusion

In conclusion, by applying the concept of work done as the dot product of force and displacement vectors, we successfully determined the value of 'b' to be 1/2. This problem demonstrates the importance of understanding the relationship between force, displacement, and work, as well as the application of vector algebra in physics. The core takeaway is that when work done is zero, it implies that the dot product of the force and displacement vectors is zero, which can be used to solve for unknowns. This principle is crucial for tackling similar problems in mechanics. Understanding and mastering these fundamental concepts is essential for success in exams like JEE Main. The problem-solving approach we've used – breaking down the problem into smaller steps, applying relevant formulas, and carefully performing calculations – is a valuable strategy for tackling complex physics problems. It emphasizes the importance of not just knowing the formulas, but also understanding the underlying concepts and how to apply them in different scenarios. Furthermore, this exercise highlights the significance of vector algebra in physics. The ability to work with vectors, calculate dot products, and interpret their physical meaning is a crucial skill for any physics student. By practicing such problems, you'll not only improve your problem-solving speed and accuracy but also deepen your understanding of the fundamental principles of mechanics. This will equip you to confidently approach a wide range of problems and build a strong foundation in physics. Therefore, continue practicing similar problems, focusing on understanding the concepts behind each step, and you'll be well on your way to mastering physics.