Problem 56 - Master Resource Book in JEE Main Physics

Question

A force of \((7 \hat{i}+6 \hat{k}) \mathrm{N}\) makes a body move on a rough plane with a velocity of \((3 \hat{\mathbf{j}}+4 \hat{\mathrm{k}}) \mathrm{ms}^{-1}\). Calculate the power in watt (a) 24 (b) 34 (c) 21 (d) 45

Step-by-Step Solution

Answer

The power is 24 watts, option (a).

Step 1: Understand the Concept of Power

Power is the rate at which work is done. Mathematically, it is expressed as the dot product of force and velocity vectors: \[ P = \mathbf{F} \cdot \mathbf{v} \] where \( \mathbf{F} \) is the force vector and \( \mathbf{v} \) is the velocity vector.

Step 2: Express Force and Velocity Vectors

The given force vector is \( \mathbf{F} = 7 \hat{i} + 6 \hat{k} \) N.The given velocity vector is \( \mathbf{v} = 3 \hat{j} + 4 \hat{k} \) m/s.

Step 3: Identify Non-zero Components for Dot Product

For the dot product, identify the matching components in both vectors. Here, \( \hat{i} \) and \( \hat{j} \) are not present in both vectors; only the \( \hat{k} \) component is present in both vectors.

Step 4: Calculate the Dot Product

Calculate the dot product where components match. Since only the \( \hat{k} \) component is common:\[ P = (6) \times (4) = 24 \]

Step 5: Conclude with the Value of Power

The calculated power is 24 watts. Therefore, the correct option is (a) 24.

Key Concepts

Dot Product

In physics, the dot product is a mathematical operation that can be used with vectors. It helps us find how much one vector goes in the same direction as another. You often encounter this when dealing with problems related to work and power. To calculate it, you multiply corresponding components of two vectors and find the sum of those products.
A simple formula for the dot product between two vectors \( \mathbf{A} = a_1 \hat{i} + a_2 \hat{j} + a_3 \hat{k} \) and \( \mathbf{B} = b_1 \hat{i} + b_2 \hat{j} + b_3 \hat{k} \) is:

• \( \mathbf{A} \cdot \mathbf{B} = a_1b_1 + a_2b_2 + a_3b_3 \)

Note that the dot product will result in a scalar, which means just a number, not another vector. This scalar represents how much of one vector's force or strength goes in the direction of another. To see this concept in action, look at the vectors from the problem. Both the force and velocity vectors have a \( \hat{k} \) component. By multiplying these components:

• \( 6 \times 4 = 24 \)

We see that the power rating in watts is derived from this dot product calculation.

Force and Velocity

Force and velocity are two key concepts in understanding motion and dynamics in physics. They are often expressed as vectors, which means they have both magnitude and direction. When analyzing movement, especially in calculating power, it’s important to understand how these two relate. **Force Vector:**A force vector indicates a push or pull on an object. In this case, the force \( \mathbf{F} = 7 \hat{i} + 6 \hat{k} \) N shows that there's no component in the \( \hat{j} \) direction. This means the force pushes only along the x-axis and z-axis.
**Velocity Vector:**On the other hand, velocity \( \mathbf{v} = 3 \hat{j} + 4 \hat{k} \) m/s indicates how fast something moves and in what direction. It shows motion in the y-axis and z-axis.To compute power using these vectors, the only relevant part is where these directions overlap; this is where the dot product comes in. Here, both forces and velocity share a \( \hat{k} \) direction, which contributes to the power calculation.
Power is directly calculated as the dot product of these vectors:

• Power = Force \( \cdot \) Velocity = (6) \( \cdot \) (4) = 24 Watts

Vector Components

To solve problems involving vectors, it's crucial to understand vector components. A component in a vector breaks down its influence in each direction—horizontal, vertical, or any axis, depending on the dimensional space.Consider the standard 3D vector format, \( \, \hat{i}, \hat{j}, \hat{k} \, \).

• \( \hat{i} \) is the x-component.
• \( \hat{j} \) is the y-component.
• \( \hat{k} \) is the z-component.

Vectors are combined using these components, and operations like the dot product depend on matching these components across vectors.For example, in our exercise:

• Force vector: \( 7 \hat{i} + 6 \hat{k} \) has no \( \hat{j} \) component.
• Velocity vector: \( 3 \hat{j} + 4 \hat{k} \) has no \( \hat{i} \) component.

When calculating the dot product, only the \( \hat{k} \) components contribute, as they are the only matching components; hence,

• \( 6 \times 4 = 24 \).

By focusing on the matching parts of each vector, you simplify problems and understand how different vector directions affect calculations like power.