Problem 52 - Master Resource Book in JEE Main Physics

Question

A force \((4 \hat{\mathbf{i}}+\hat{\mathbf{j}}-2 \hat{\mathbf{k}}) \mathrm{N}\) acting on a body maintains its velocity at \((2 \hat{\mathbf{i}}+2 \hat{\mathbf{j}}+3 \hat{\mathbf{k}}) \mathrm{ms}^{-1}\). The power exerted is [Kerala CET 2010] (a) \(4 \mathrm{~W}\) (b) \(5 \mathrm{~W}\) (c) \(2 \mathrm{~W}\) (d) \(8 \mathrm{~W}\)

Step-by-Step Solution

Answer

The power exerted is 4 W, corresponding to option (a).

Step 1: Understanding the Problem

We are given a force \( \mathbf{F} = (4 \hat{\mathbf{i}} + \hat{\mathbf{j}} - 2 \hat{\mathbf{k}}) \mathrm{N} \) and a velocity \( \mathbf{v} = (2 \hat{\mathbf{i}} + 2 \hat{\mathbf{j}} + 3 \hat{\mathbf{k}}) \mathrm{ms}^{-1} \). We need to find the power exerted by this force.

Step 2: Using Power Formula

The formula for power \( P \) due to a force is given by the dot product of the force vector \( \mathbf{F} \) and the velocity vector \( \mathbf{v} \): \[ P = \mathbf{F} \cdot \mathbf{v} \]

Step 3: Applying the Dot Product

Calculate the dot product: \[\mathbf{F} \cdot \mathbf{v} = (4 \hat{\mathbf{i}} + \hat{\mathbf{j}} - 2 \hat{\mathbf{k}}) \cdot (2 \hat{\mathbf{i}} + 2 \hat{\mathbf{j}} + 3 \hat{\mathbf{k}})\]

Step 4: Calculating Each Component's Contribution

Calculate each component's contribution to the dot product:- \((4 \hat{\mathbf{i}}) \cdot (2 \hat{\mathbf{i}}) = 8\)- \((\hat{\mathbf{j}}) \cdot (2 \hat{\mathbf{j}}) = 2\)- \((-2 \hat{\mathbf{k}}) \cdot (3 \hat{\mathbf{k}}) = -6\)

Step 5: Summing the Components

Sum all the contributions:\[ 8 + 2 - 6 = 4\]

Step 6: Selecting the Correct Answer

The power exerted is 4 watts, which corresponds to option (a).

Key Concepts

Dot Product

The dot product, also known as the scalar product, is a way to multiply two vectors and get a scalar (a single number) as the result. This operation is particularly useful in physics when you need to calculate the power exerted by a force on an object moving with a certain velocity.

The mathematical operation involves multiplying corresponding components of two vectors and then summing the results. For vectors expressed as \( \mathbf{A} = a_1 \hat{\mathbf{i}} + a_2 \hat{\mathbf{j}} + a_3 \hat{\mathbf{k}} \) and \( \mathbf{B} = b_1 \hat{\mathbf{i}} + b_2 \hat{\mathbf{j}} + b_3 \hat{\mathbf{k}} \),the dot product is computed as follows:

• Multiply the i-components: \( a_1 \cdot b_1 \)
• Multiply the j-components: \( a_2 \cdot b_2 \)
• Multiply the k-components: \( a_3 \cdot b_3 \)
• Sum these products: \( a_1b_1 + a_2b_2 + a_3b_3 \)

The result of the dot product is a scalar value indicating the magnitude of the projection of one vector onto the other.

When applying this to power calculations, if \( \mathbf{F} \) is a force vector and \( \mathbf{v} \) is a velocity vector, then the power \( P \) is:\[ P = \mathbf{F} \cdot \mathbf{v} \]

Vectors in Physics

Vectors are fundamental in physics as they provide a way to represent quantities that have both magnitude and direction. Forces and velocities are typical examples of vector quantities, and understanding how to work with them is crucial for solving many physics problems.

A vector is usually represented in a three-dimensional space with components along the i, j, and k axes. For example, a vector \( \mathbf{v} = v_1 \hat{\mathbf{i}} + v_2 \hat{\mathbf{j}} + v_3 \hat{\mathbf{k}} \) describes its direction and magnitude in terms of unit vectors, \( \hat{\mathbf{i}}, \hat{\mathbf{j}}, \hat{\mathbf{k}} \), that point along the x, y, and z axes, respectively.

Working with vectors might include:

• Calculating magnitudes: \( |\mathbf{v}| = \sqrt{v_1^2 + v_2^2 + v_3^2} \)
• Adding or subtracting vectors
• Finding projections
• Understanding vector components interactions, like in dot products

Knowing how vectors operate is key when calculating real-world scenarios, enabling us to determine directions of motion, forces, and velocities.

The understanding of vectors as physical representations helps in setting up equations that faithfully model what happens in physical systems, hence facilitating solutions to complex physics problems.

Force and Velocity

Force and velocity are two primary concepts used in physics to analyze motion. Force, often represented by \( \mathbf{F} \), is the push or pull exerted on an object that can cause changes in its velocity. It is a vector quantity, meaning it has both magnitude and direction.

Velocity, denoted as \( \mathbf{v} \), measures how fast an object changes its position. Like force, velocity is also a vector, describing both the speed and the direction of motion.

When analyzing scenarios involving force and velocity, it is crucial to consider their vector nature:

• Force can be calculated using Newton's second law: \( \mathbf{F} = m \cdot \mathbf{a} \), where \( m \) is mass and \( \mathbf{a} \) is acceleration.
• Velocity ties directly to an object's motion and is defined as the rate of change of displacement.

In the context of power, understanding these vectors is essential as the power exerted by a force maintaining velocity is derived from their dot product. This represents the effective amount of force applied along the direction of motion. Calculating power helps us understand how energy is transferred in mechanical systems, and is fundamental in designing engines, motors, and various mechanical devices. Together, these concepts enable us to predict motion and energy transformations in the physical world.