Problem 81 - A Complete Resource Book in Mathematics for JEE Main 2017

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 \([\mathbf{2 0 0 4}]\) (A) 40 (B) 30 (C) 25 (D) 15

Step-by-Step Solution

Answer

The work done by the forces is 40 units, corresponding to option (A).

Step 1: Understand the Formula for Work

The work done by a force on an object when it is displaced is given by the dot product of the force vector and the displacement vector. Mathematically, this is expressed as \( W = \mathbf{F} \cdot \mathbf{d} \) where \( \mathbf{F} \) is the total force vector and \( \mathbf{d} \) is the displacement vector.

Step 2: Calculate the Total Force Vector

The total force vector \( \mathbf{F}_{total} \) is the sum of the individual force vectors. Thus, we have:\( \mathbf{F}_{total} = (4 \hat{i}+\hat{j}-3 \hat{k}) + (3 \hat{i}+\hat{j}-\hat{k}) = (4+3) \hat{i} + (1+1) \hat{j} + (-3-1) \hat{k} = 7 \hat{i} + 2 \hat{j} - 4 \hat{k} \).

Step 3: Calculate the Displacement Vector

The displacement vector \( \mathbf{d} \) is found by subtracting the initial position vector from the final position vector. Thus, we have: \( \mathbf{d} = (5 \hat{i}+4 \hat{j}+\hat{k}) - (\hat{i} + 2 \hat{j} + 3 \hat{k}) = (5-1) \hat{i} + (4-2) \hat{j} + (1-3) \hat{k} = 4 \hat{i} + 2 \hat{j} - 2 \hat{k} \).

Step 4: Calculate the Dot Product

The work done \( W \) is the dot product of \( \mathbf{F}_{total} \) and \( \mathbf{d} \). Calculate this as follows:\[ W = (7 \hat{i} + 2 \hat{j} - 4 \hat{k}) \cdot (4 \hat{i} + 2 \hat{j} - 2 \hat{k}) \]\[ = (7)(4) + (2)(2) + (-4)(-2) \]\[ = 28 + 4 + 8 = 40 \].

Step 5: Conclusion with Verification

The resulting work done is 40. This matches option (A) from the given answers. Thus, the work done by the forces during the displacement is indeed \(40\) units.

Key Concepts

Vector Addition

When dealing with forces acting upon a particle, vector addition is crucial. It allows us to combine multiple vectors to find a single resultant vector representing the total effect. Consider two forces given by their vectors:

• First force: \(4 \hat{i} + \hat{j} - 3 \hat{k}\)
• Second force: \(3 \hat{i} + \hat{j} - \hat{k}\)

To find the total force acting on the particle, you simply add the corresponding components of these vectors. The result is:

• In the \(\hat{i}\) direction: \(4 + 3 = 7\)
• In the \(\hat{j}\) direction: \(1 + 1 = 2\)
• In the \(\hat{k}\) direction: \(-3 - 1 = -4\)

Thus, the total force vector \( \mathbf{F}_{total} \) becomes \(7 \hat{i} + 2 \hat{j} - 4 \hat{k}\). This new vector reflects the combined effect of both forces acting simultaneously. Vector addition helps in simplifying multiple force scenarios and is foundational in solving physics problems involving multiple vectors.

Dot Product

The dot product is a mathematical operation performed between two vectors. It calculates a single scalar value rather than a vector. This is achieved by multiplying the components of two vectors and summing them. It is often used to find work done by a force when an object is displaced. For vectors \( \mathbf{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \) and \( \mathbf{B} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \), the dot product is computed as follows:\[ \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3 \]In our scenario, you seek the work done \( W \) by the forces acting on the particle. The total force vector is \( 7 \hat{i} + 2 \hat{j} - 4 \hat{k} \) and the displacement vector is \( 4 \hat{i} + 2 \hat{j} - 2 \hat{k} \). The calculation of the dot product is:\[ W = (7 \cdot 4) + (2 \cdot 2) + (-4 \cdot -2) \]\[ W = 28 + 4 + 8 = 40 \]Here, each term arises from multiplying corresponding components of the force and displacement vectors. This operation yields the scalar value of work done as 40 units.

Displacement Vector

A displacement vector represents how an object moves from one point to another in a vector form. It captures the change in position. When calculating displacement, the initial position vector is subtracted from the final position vector. In this exercise:

• Initial position: \( \hat{i} + 2 \hat{j} + 3 \hat{k} \)
• Final position: \( 5 \hat{i} + 4 \hat{j} + \hat{k} \)

To find the displacement vector \( \mathbf{d} \), perform the following calculations:

• \( \hat{i} \) component: \(5 - 1 = 4\)
• \( \hat{j} \) component: \(4 - 2 = 2\)
• \( \hat{k} \) component: \(1 - 3 = -2\)

So, the displacement vector is \( 4 \hat{i} + 2 \hat{j} - 2 \hat{k} \). This vector signifies the overall movement from the start point to end point in its three-dimensional components. Understanding displacement vectors is vital for analyzing how objects shift in space, particularly when calculating work as in this problem.