Problem 36 - Algebra and Trigonometry

Question

Find the work done by the force \(\mathbf{F}\) in moving an object from \(P\) to \(Q .\) $$ \mathbf{F}=400 \mathbf{i}+50 \mathbf{j} ; \quad P(-1,1), Q(200,1) $$

Step-by-Step Solution

Answer

The work done is 80400 units.

Step 1: Understand the Force Vector

The given force vector is \( \mathbf{F} = 400 \mathbf{i} + 50 \mathbf{j} \). This means the force has a magnitude of 400 in the direction of the x-axis and 50 in the direction of the y-axis.

Step 2: Determine the Displacement Vector

Find the displacement vector \( \mathbf{d} \) by subtracting the coordinates of point \( P(-1, 1) \) from point \( Q(200, 1) \). This results in the displacement vector \( \mathbf{d} = (200 - (-1))\mathbf{i} + (1 - 1)\mathbf{j} = 201 \mathbf{i} \).

Step 3: Compute the Dot Product

Calculate the dot product of the force vector \( \mathbf{F} \) and the displacement vector \( \mathbf{d} \). The dot product is given by \( \mathbf{F} \cdot \mathbf{d} = (400 \cdot 201) + (50 \cdot 0) = 80400 \).

Step 4: Calculate the Work Done

The work done by the force is equivalent to the dot product computed, which is \( 80400 \). Thus, the work done by the force in moving the object from point \( P \) to point \( Q \) is \( 80400 \) units.

Key Concepts

Force Vector

In physics, a force vector represents both the magnitude and direction of a force applied to an object. The force vector is crucial to understanding how objects interact in the physical world. In our exercise, the force vector \( \mathbf{F} = 400 \mathbf{i} + 50 \mathbf{j} \) indicates that there is a much larger force component along the x-axis compared to the y-axis.
The components of a force vector are:

• Magnitude: Given by the vector's components. Here, a magnitude of 400 in the x-direction and 50 in the y-direction.
• Direction: It tells us in which direction the force is applied, consisting of both horizontal (x-axis) and vertical (y-axis) parts.

Representing forces as vectors allows us to analyze their combined effect using vector addition and other mathematical operations, such as the dot product, which are essential for solving many physics problems.

Displacement Vector

The displacement vector describes the change in position of an object. It is calculated by subtracting the initial position vector from the final position vector. In our problem, we are given the points \( P(-1, 1) \) and \( Q(200, 1) \). We compute the displacement vector \( \mathbf{d} \) as follows:

• Subtract the coordinates of point \( P \) from point \( Q \).
• Compute as \( (200 - (-1))\mathbf{i} + (1 - 1)\mathbf{j} \).
• This results in \( 201 \mathbf{i} \), indicating movement along the x-axis only.

Understanding displacement vectors helps us determine how an object moves from one place to another, and this is directly used to find work done in conjunction with the force vector.

Dot Product

The dot product, also known as the scalar product, is a mathematical operation that combines two vectors and results in a scalar quantity. It's quite handy for calculating work done when a force moves an object. For vectors \( \mathbf{A} = a_1 \mathbf{i} + a_2 \mathbf{j} \) and \( \mathbf{B} = b_1 \mathbf{i} + b_2 \mathbf{j} \), the dot product is calculated by:
\[ \mathbf{A} \cdot \mathbf{B} = a_1 b_1 + a_2 b_2 \] In our context:

• Force vector \( \mathbf{F} \) and displacement vector \( \mathbf{d} \) are used.
• The dot product \( \mathbf{F} \cdot \mathbf{d} = (400 \times 201) + (50 \times 0) = 80400 \).

The result tells us the component of force contributing to the movement along the displacement direction, yielding the amount of work done.

Physics Problem Solving

Physics problem solving involves breaking down complex phenomena into manageable concepts. Our task of calculating work done by a force involves multiple components. Here’s how to approach it:

• Understand the problem: Break the given scenario into known variables, such as force and displacement.
• Identify the vectors: Write down the force and displacement components separately to visualize the problem.
• Perform calculations step by step: Find the displacement vector, compute the dot product, and then use this to find the work done.
• Interpret the results: Once the calculations are completed, interpret the scalar values to understand their physical significance.

Breaking a problem into these ordered steps ensures clarity and gives confidence when solving physics problems effectively.