Problem 81 - A Complete Resource Book in Mathematics for JEE Main

Question

A particle is acted upon by constant forces \(4 \hat{i}+\hat{j}-3 \hat{k}\) and \(3 \hat{i}+\hat{j}-\hat{k}\) which displace it from a point \(\hat{i}+2 \hat{j}+3 \hat{k}\) to the point \(5 \hat{i}+4 \hat{j}+\hat{k}\). The work done in standard units by the forces is given by (A) 40 (B) 30 (C) 25 (D) 15

Step-by-Step Solution

Answer

The work done by the forces is 40 (option A).

Step 1: Understand the forces

The particle is acted upon by two constant forces, \( \vec{F_1} = 4 \hat{i} + \hat{j} - 3 \hat{k} \) and \( \vec{F_2} = 3 \hat{i} + \hat{j} - \hat{k} \). We need to find the work done by these forces when they displace the particle.

Step 2: Calculate the net force

The net force \( \vec{F} \) acting on the particle is the vector sum of \( \vec{F_1} \) and \( \vec{F_2} \). This gives us: \( \vec{F} = (4 + 3) \hat{i} + (1 + 1) \hat{j} + (-3 - 1) \hat{k} = 7 \hat{i} + 2 \hat{j} - 4 \hat{k} \).

Step 3: Determine the displacement vector

The initial position vector is \( \vec{R_1} = \hat{i} + 2 \hat{j} + 3 \hat{k} \) and the final position vector is \( \vec{R_2} = 5 \hat{i} + 4 \hat{j} + \hat{k} \). The displacement \( \vec{d} \) is given by: \( \vec{d} = \vec{R_2} - \vec{R_1} = (5 - 1) \hat{i} + (4 - 2) \hat{j} + (1 - 3) \hat{k} = 4 \hat{i} + 2 \hat{j} - 2 \hat{k} \).

Step 4: Calculate the work done

Work done \( W \) by the net force is calculated using the dot product of force and displacement: \( W = \vec{F} \cdot \vec{d} \). Calculate this as follows: \( W = (7 \hat{i} + 2 \hat{j} - 4 \hat{k}) \cdot (4 \hat{i} + 2 \hat{j} - 2 \hat{k}) = 7 \times 4 + 2 \times 2 + (-4)\times(-2) = 28 + 4 + 8 = 40 \).

Key Concepts

Vector Addition

When dealing with multiple forces, vector addition is a crucial step required to determine the net force acting on an object. Forces are vectors, which means they have both magnitude and direction. To add these force vectors, you simply add their corresponding components along each axis.

• For example, if you have forces described by the vectors \(4 \hat{i} + \hat{j} - 3 \hat{k}\) and \(3 \hat{i} + \hat{j} - \hat{k}\), you add the \(\hat{i}\), \(\hat{j}\), and \(\hat{k}\) components separately.
• In this scenario, \(4 + 3 = 7\) along the \(\hat{i}\) axis; \(1 + 1 = 2\) along the \(\hat{j}\) axis; and \(-3 - 1 = -4\) along the \(\hat{k}\) axis.

These individual sums give you the resultant force vector, \(7 \hat{i} + 2 \hat{j} - 4 \hat{k}\), which we then use for further calculations.

Dot Product

The dot product, or scalar product, of two vectors is a mathematical operation that results in a scalar quantity. It is used to calculate the work done when a force moves a particle. The dot product is computed by multiplying together corresponding components of the vectors and then summing those products.

• In the work done calculation, the force vector and the displacement vector's dot product gives the work done by the net force.
• For instance, the dot product of \(7 \hat{i} + 2 \hat{j} - 4 \hat{k}\) and \(4 \hat{i} + 2 \hat{j} - 2 \hat{k}\) involves multiplying the corresponding components: \(7 \times 4\), \(2 \times 2\), and \((-4) \times (-2)\).
• Adding these calculated values gives the total work done: \(28 + 4 + 8 = 40\) standard units.

Displacement Vector

Displacement is a vector that denotes the change in position of an object. It is not the same as the distance traveled but rather the straight-line distance between the starting and ending points, considering direction.

• The displacement vector \(\vec{d}\) is found by subtracting the initial position vector from the final position vector.
• Here, the particle moves from \(\vec{R_1} = \hat{i} + 2 \hat{j} + 3 \hat{k}\) to \(\vec{R_2} = 5 \hat{i} + 4 \hat{j} + \hat{k}\). The displacement vector is \(\vec{d} = \vec{R_2} - \vec{R_1}\).
• This simplifies to \(4 \hat{i} + 2 \hat{j} - 2 \hat{k} \), providing the necessary vector for the work computation.

Constant Forces

Constant forces have both a fixed magnitude and direction. This means they do not change over time or with the path taken by an object. Because of this, calculating work done in situations with constant forces becomes straightforward.

• The work done is dependent solely on the displacement and direction of the force.
• In our example, even though the particle hikes from one point to another, the forces \(\vec{F_1} = 4 \hat{i} + \hat{j} - 3 \hat{k}\) and \(\vec{F_2} = 3 \hat{i} + \hat{j} - \hat{k}\) remain unchanged.
• Such constancy simplifies the calculation, since we can use the derived net force vector directly with the displacement vector in the dot product.