Problem 43 - Master Resource Book in JEE Main Physics

Question

A force of \((10 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}}) \mathbf{N}\) acts on a body of mass \(100 \mathrm{~g}\) and displaces it from \((6 \hat{\mathbf{i}}+5 \hat{\mathbf{j}}-3 \hat{\mathbf{k}}) \mathrm{m}\) to \((10 \hat{\mathbf{i}}-2 \hat{\mathbf{j}}+7 \hat{\mathbf{k}}) \mathrm{m}\). The work done is (a) \(21 \mathrm{~J}\) (b) \(121 \mathrm{~J}\) (c) \(361 \mathrm{~J}\) (d) \(1000 \mathrm{~J}\)

Step-by-Step Solution

Answer

The work done is 121 J, corresponding to option (b).

Step 1: Determine the Displacement Vector

The initial position vector is \((6 \hat{\mathbf{i}} + 5 \hat{\mathbf{j}} - 3 \hat{\mathbf{k}}) \text{m}\) and the final position vector is \((10 \hat{\mathbf{i}} - 2 \hat{\mathbf{j}} + 7 \hat{\mathbf{k}}) \text{m}\). To find the displacement vector, subtract the initial position vector from the final position vector: \[\hat{\mathbf{r}}_f - \hat{\mathbf{r}}_i = (10 - 6) \hat{\mathbf{i}} + (-2 - 5) \hat{\mathbf{j}} + (7 + 3) \hat{\mathbf{k}} = 4 \hat{\mathbf{i}} - 7 \hat{\mathbf{j}} + 10 \hat{\mathbf{k}}\text{ m}.\]

Step 2: Use Work Done Formula

The work done by a force is given by the dot product of the force vector and the displacement vector. The force vector is \((10 \hat{\mathbf{i}}-3 \hat{\mathbf{j}}+6 \hat{\mathbf{k}} ) \text{N} \) and the displacement vector is found in Step 1 as \((4 \hat{\mathbf{i}} - 7 \hat{\mathbf{j}} + 10 \hat{\mathbf{k}}) \text{ m} \).

Step 3: Calculate the Dot Product

Calculate the dot product of the force vector and the displacement vector:\[\mathbf{F} \cdot \mathbf{d} = (10 \hat{\mathbf{i}} - 3 \hat{\mathbf{j}} + 6 \hat{\mathbf{k}}) \cdot (4 \hat{\mathbf{i}} - 7 \hat{\mathbf{j}} + 10 \hat{\mathbf{k}})\]\[= (10 \times 4) + (-3 \times -7) + (6 \times 10)\]\[= 40 + 21 + 60\]\[= 121 \text{ J}\]

Step 4: Verify and Arrive at the Solution

The dot product and therefore the work done is calculated to be \(121 \mathrm{~J}\). Checking with the options, \(121 \mathrm{~J}\) is option (b). Therefore, the correct answer is \(121 \mathrm{~J}\).

Key Concepts

Dot Product Calculation

In physics, the dot product is a crucial tool, especially when calculating work done by a force. It involves two vectors: one for the force and another for the displacement. The dot product tells you how much one vector goes in the direction of another. This is particularly useful in finding out the work done when a force moves an object.
The mathematical formula for the dot product is: \[\mathbf{F} \cdot \mathbf{d} = F_x \times d_x + F_y \times d_y + F_z \times d_z\]Here's how it works:

• Multiply the corresponding components of each vector.
• For example, the i-component of the force and displacement vectors are multiplied, and similarly for j- and k-components.
• Add these products together to get the dot product.

In our original problem, the calculation is: \[ (10 \times 4) + (-3 \times -7) + (6 \times 10) \] to arrive at 121 J. This intuitive operation shows the directional overlap between force and movement.

Displacement Vector

The displacement vector is a vector that shows how far and in what direction an object has moved. It's found by subtracting the initial position vector from the final position vector.
Let's understand with our scenario:

• The initial position vector was \((6 \hat{\mathbf{i}} + 5 \hat{\mathbf{j}} - 3 \hat{\mathbf{k}}) \text{m}\).
• The final position vector was \((10 \hat{\mathbf{i}} - 2 \hat{\mathbf{j}} + 7 \hat{\mathbf{k}}) \text{m}\).
• The displacement vector is just the change in position, calculated as: \((10 - 6) \hat{\mathbf{i}} + (-2 - 5) \hat{\mathbf{j}} + (7 + 3) \hat{\mathbf{k}}\).

This results in the vector \(4 \hat{\mathbf{i}} - 7 \hat{\mathbf{j}} + 10 \hat{\mathbf{k}}\), which tells us how the object moved from its start location to its final location.

Force Vector

A force vector tells us about the force acting upon an object in terms of direction and magnitude. Just like displacement, force is a vector quantity having multiple dimensions.
In our example:

• The force vector is \((10 \hat{\mathbf{i}} - 3 \hat{\mathbf{j}} + 6 \hat{\mathbf{k}}) \text{N}\).
• Each component of this vector tells us how much force is applied in the respective x, y, and z directions.

Understanding the force vector is fundamental because it determines how an object will accelerate when force is applied. It is crucial when calculating work done, as the components must align properly with the displacement components for the dot product calculation.

Physics Problem Solving

Physics problem solving involves a series of logical steps. By understanding and applying certain principles, it helps break down complex problems into understandable parts.
To solve a problem about work done, follow these steps:

• First, identify the known and unknown quantities.
• Determine all vectors involved, such as displacement and force.
• Find the dot product of the force and displacement vectors to calculate work done.
• Always check your work against common sense and verify with options provided, if applicable.

Each step is logically connected to the next to ensure clarity and accuracy in the final result. This structured approach gives you confidence while solving physics-related problems.