Problem 70 - Master Resource Book in JEE Main Physics

Question

The gravitational field in a region is given by \(\mathbf{I}=(4 \hat{\mathbf{i}}+\hat{\mathbf{j}}) \mathbf{N k g}^{-1}\). Work done by this field is zero when a particle is moved along the line (a) \(x+y=6\) (b) \(x+4 y=6\) (c) \(y+4 x=6\) (d) \(x-y=6\)

Step-by-Step Solution

Answer

Work done is zero along line (c) \(y+4x=6\).

Step 1: Understand the Problem

We are given a gravitational field \(\mathbf{I}=(4 \hat{\mathbf{i}}+\hat{\mathbf{j}}) \mathbf{N \, kg}^{-1}\) and need to find a line along which the work done by this field is zero on a particle.

Step 2: Know when Work Done is Zero

Work done by a field along a path is zero if the field vector is perpendicular to the direction vector of the path. Mathematically, this means the dot product between the field vector and direction vector is zero.

Step 3: Determine the Direction Vector of Each Line

For a line given by \(ax + by = c\), the direction vector is \((-b, a)\). Let's find the direction vectors:- (a) For \(x+y=6\), the direction vector is \((-1, 1)\).- (b) For \(x+4y=6\), the direction vector is \((-4, 1)\).- (c) For \(y+4x=6\), rewrite it as \(4x+y=6\), getting direction vector \((-1, 4)\).- (d) For \(x-y=6\), the direction vector is \((1, 1)\).

Step 4: Compute Dot Product for Each Line

Calculate the dot product of the gravitational field vector \((4, 1)\) with each direction vector:- (a) Dot with \((-1, 1)\): \(4(-1) + 1(1) = -4 + 1 = -3\)- (b) Dot with \((-4, 1)\): \(4(-4) + 1(1) = -16 + 1 = -15\)- (c) Dot with \((-1, 4)\): \(4(-1) + 1(4) = -4 + 4 = 0\)- (d) Dot with \((1, 1)\): \(4(1) + 1(1) = 4 + 1 = 5\)

Step 5: Identify When Dot Product is Zero

The only line for which the dot product is zero is option (c), where the direction vector \((-1, 4)\) results in a dot product of zero with the field vector.

Key Concepts

Work Done

The concept of work done is fundamental in understanding how forces affect the motion of objects. In physics, work done by a force is defined as the product of the force applied and the displacement in the direction of the force. It is mathematically given by the equation:\[W = \mathbf{F} \, . \, \mathbf{d}\]where \mathbf{F} is the force vector and \mathbf{d} is the displacement vector.
This means that only the component of the force that acts in the direction of movement does work. Importantly, if a force is perpendicular to the displacement, no work is done by that force, because the cosine of 90 degrees is zero. In the context of a gravitational field, when the field vector is perpendicular to the path along which a particle moves, the work done is zero.
This concept is pivotal in problems where we need to determine paths that correlate with zero work done, as such paths align perpendicularly to the direction of gravitational field lines.

Dot Product

The dot product, also known as the scalar product, is a key mathematical operation that helps in determining the work done by a force. It is a way to multiply two vectors that results in a scalar. The formula for the dot product of two vectors \( \mathbf{A} = (a_1, a_2) \) and \( \mathbf{B} = (b_1, b_2) \) is:\[\mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2\]This operation takes the sum of the products of the corresponding components of the vectors. The dot product is especially significant in physics because it reveals whether two vectors are perpendicular.
If the dot product equals zero, the vectors are orthogonal (i.e., they form a 90-degree angle). In our exercise, we used the dot product to determine along which path the gravitational field does zero work. By checking the dot product of the gravitational field vector and the direction vectors of the possible paths, we found that the dot product was zero in option (c), which indicates no work done along that path.

Direction Vector

A direction vector provides critical information about the orientation and direction of a line in a coordinate plane. For a straight line given by the equation \( ax + by = c \), the direction vector is represented by \((-b, a)\). This vector tells us which way the line points by describing the horizontal and vertical changes necessary to create the path of the line.
Understanding how to determine the direction vector is essential, especially when solving problems involving perpendicularity and parallelism in vectors. In our problem, each given line equation is analyzed to extract its direction vector, which is then used in calculating the dot product with the gravitational field vector.
This process helps in identifying when the orientation of the line causes it to be perpendicular to the field vector, leading to zero work done. Remember, lines with direction vectors perpendicular to a force field indicate that the field is not doing any work while moving along them.