Problem 22 - Master Resource Book in JEE Main Physics

Question

A body constrained to move in \(y\)-direction, is subjected to a force given by \(\mathbf{F}=(-2 \hat{\mathbf{i}}+15 \hat{\mathbf{j}}+6 \hat{\mathbf{k}})\) done by this force in moving the body through a distance of \(10 \mathrm{~m}\) along \(y\)-axis? (a) \(190 \mathrm{~J}\) (b) \(160 \mathrm{~J}\) (c) \(150 \mathrm{~J}\) (d) \(20 \mathrm{~J}\)

Step-by-Step Solution

Answer

The work done is 150 J, corresponding to option (c).

Step 1: Identify relevant components

The force vector is given as \( \mathbf{F} = (-2 \hat{\mathbf{i}} + 15 \hat{\mathbf{j}} + 6 \hat{\mathbf{k}}) \). Since the motion is only in the \( y \)-direction, we only consider the \( j \)-component of the force, which is \( 15 \hat{\mathbf{j}} \).

Step 2: Use the work formula

Work done \( W \) is calculated using the dot product of force \( \mathbf{F} \) and displacement \( \mathbf{d} \). Since displacement is only along the \( y \)-axis, the displacement vector is \( \mathbf{d} = 10 \hat{\mathbf{j}} \). Therefore, the work done is: \[ W = \mathbf{F} \cdot \mathbf{d} = 15 \hat{\mathbf{j}} \cdot 10 \hat{\mathbf{j}} \]

Step 3: Calculate the dot product

Perform the dot product calculation: \[ W = 15 \times 10 = 150 \text{ J} \].

Step 4: Compare with given options

The calculated work done is \( 150 \text{ J} \), which corresponds to option (c) in the given choices.

Key Concepts

Dot product

The dot product, also known as the scalar product, is a fundamental operation to understand in physics and mathematics, especially when dealing with vectors. When you take the dot product of two vectors, you multiply their magnitudes (lengths) and the cosine of the angle between them. This ultimately results in a scalar quantity (a number without direction).
Here’s how it works:

• If you have two vectors, let's call them \( \mathbf{A} = a_1 \hat{\mathbf{i}} + a_2 \hat{\mathbf{j}} + a_3 \hat{\mathbf{k}} \) and \( \mathbf{B} = b_1 \hat{\mathbf{i}} + b_2 \hat{\mathbf{j}} + b_3 \hat{\mathbf{k}} \), their dot product is calculated as \( \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3 \).
• This operation essentially measures the extent to which two vectors are pointing in the same direction.

In our scenario, the dot product helps us calculate work done."
Work is the product of the force component in the direction of movement and the displacement. Hence, using the dot product simplifies work calculations when each vector is along an axis.

Vector components

Vectors are quantities that have both magnitude and direction, and they can be broken down into components along predefined axes (usually x, y, and z in three dimensions). When solving physics problems, understanding vector components is crucial. This is especially true when movements or forces are constrained to certain directions, as in the current problem where the motion is only in the y-direction.

• The vector \( \mathbf{F} = (-2 \hat{\mathbf{i}} + 15 \hat{\mathbf{j}} + 6 \hat{\mathbf{k}}) \) is given, which is comprised of three components that each describe the force along the respective i, j, and k directions.
• In problems restricted to movement along a single axis (here, the y-axis), we can ignore components of the force that act in other directions (the i and k components in this case).
• This leaves us with the relevant vector part to consider: \( 15 \hat{\mathbf{j}} \).

By focusing on this component, we efficiently determine how much force contributes to the work done.

Physics problem-solving

Physics is the study of the natural world, and solving problems in physics often involves breaking down complex scenarios into more manageable parts. The main goal is to apply physical laws and principles efficiently. Let’s explore this approach through the example problem.

• First, identify what you are given and what you need to find. In this case, a force vector and a specific direction of movement.
• Decompose the problem by concentrating only on the components essential to the solution. Here, we only need the j-component since the motion is in the y-direction.
• Apply the correct formula, here, the work done formula \( W = \mathbf{F} \cdot \mathbf{d} \), integrating necessary vector operations like the dot product.
• Conduct calculations carefully to ensure accuracy, verifying results against possible multiple-choice answers when provided.

Successful physics problem-solving involves identifying key concepts and tools, like vector components and the dot product, and combining them methodically to find solutions.