Problem 35 - Precalculus Mathematics for Calculus

Question

Find the work done by the force \(\mathbf{F}\) in moving an object from \(P\) to \(Q\). $$\mathbf{F}=4 \mathbf{i}-5 \mathbf{j}: \quad P(0,0), Q(3,8)$$

Step-by-Step Solution

Answer

The work done is -28 units.

Step 1: Understand the Problem

We need to find the work done by the force vector \(\mathbf{F} = 4\mathbf{i} - 5\mathbf{j}\) in moving an object from point \(P(0,0)\) to point \(Q(3,8)\). Work is calculated using the dot product of the force vector and the displacement vector.

Step 2: Calculate the Displacement Vector

The displacement vector \(\mathbf{d}\) is the vector from point \(P\) to point \(Q\). Calculate this as \(\mathbf{d} = Q - P = (3-0)\mathbf{i} + (8-0)\mathbf{j} = 3\mathbf{i} + 8\mathbf{j}\).

Step 3: Use the Dot Product Formula

The work done \(W\) is found using the formula for the dot product: \(W = \mathbf{F} \cdot \mathbf{d}\). Substitute the values to get \(W = (4\mathbf{i} - 5\mathbf{j}) \cdot (3\mathbf{i} + 8\mathbf{j})\).

Step 4: Calculate the Dot Product

Calculate the dot product as follows: \((4)(3) + (-5)(8) = 12 - 40 = -28\).

Step 5: Conclude the Calculation

The work done by the force \(\mathbf{F}\) is \(-28\) units. Negative work indicates that the force is acting opposite to the direction of displacement.

Key Concepts

Displacement Vector

In physics, the concept of a displacement vector is crucial when calculating work done by forces. Displacement is a vector quantity because it has both a magnitude and a direction. It is not the same as distance; rather, it is the shortest path between two points in a specific direction.
For this exercise, the displacement vector \( \mathbf{d} \) is calculated from point \( P(0,0) \) to point \( Q(3,8) \).

• The displacement vector is found by subtracting the vector coordinates of the starting point \( P \) from the coordinates of the ending point \( Q \).
• Mathematically, it is given by: \( \mathbf{d} = Q - P = (3-0)\mathbf{i} + (8-0)\mathbf{j} = 3\mathbf{i} + 8\mathbf{j} \).

When performing calculations involving vectors, it’s essential to carry over both direction and size, which results in the displacement vector defining the path of movement precisely.

Dot Product

The dot product, also known as scalar product, is a fundamental operation in vector algebra. It takes two vectors and returns a scalar, a single number, which can provide insight into the relationship between the two vectors.

• This operation is crucial for finding the work done by a force because it considers both the magnitude of the vectors and the angle between them.
• The work \( W \) is calculated using the dot product formula: \( W = \mathbf{F} \cdot \mathbf{d} \).
• For vectors \( \mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j} \) and \( \mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j} \), their dot product is calculated as \( a_1b_1 + a_2b_2 \).

This formula simplifies the work calculation to examining how much of the force is applied in the direction of the displacement, which is the defining criterion for determining work.

Force Vector

The force vector \( \mathbf{F} \) is a vector that describes the magnitude and direction of force applied to an object. Just like other vectors, it has components along the axes of a coordinate system.
In this example, the force vector is given as \( \mathbf{F} = 4\mathbf{i} - 5\mathbf{j} \).

• The \( 4\mathbf{i} \) component indicates a force acting in the positive x-direction, while the \( -5\mathbf{j} \) component represents a force in the negative y-direction.

Force vectors are incredibly important in physics because they describe exactly how forces influence the movement of objects. Calculating the work done involves determining how this force vector interacts with the displacement vector.

Negative Work

Negative work is a concept that arises when the direction of force applied is opposite to the direction of displacement, leading to a slowing down or stopping of the object.
In this problem, after calculating the dot product, we determined the work done as \(-28\) units.

• The negative sign in work reflects that there is a force component acting against the displacement from point \( P \) to point \( Q \).
• In practical terms, negative work indicates that the energy is being drained from the moving object into the force that opposes the movement, such as friction or opposing force.

It’s essential to grasp this concept, as negative work often occurs in everyday scenarios where forces like friction play a significant role, making it crucial for accurate problem-solving in physics.