Problem 42 - Trigonometry

Question

Find the work performed when the given force \(\mathbf{F}\) is applied to an object, whose resulting motion is represented by the displacement vector \(d\). Assume the force is in pounds and the displacement is measured in feet. \(\mathbf{F}=-67 \mathbf{i}+59 \mathbf{j}, \mathbf{d}=-96 \mathbf{i}-28 \mathbf{j}\)

Step-by-Step Solution

Answer

The work performed is 4780 foot-pounds.

Step 1: Understanding the Dot Product

The work done by a force on an object along a displacement vector is calculated using the dot product of the force vector \(\mathbf{F}\) and the displacement vector \(\mathbf{d}\). The dot product is a measure that combines both vectors and can be calculated when vectors are represented by their components.

Step 2: Calculate the Dot Product

To calculate the work, we need to find the dot product of \(\mathbf{F}\) and \(\mathbf{d}\). The formula for the dot product \(\mathbf{F} \cdot \mathbf{d}\) is: \[ (-67i + 59j) \cdot (-96i - 28j) = (-67)(-96) + (59)(-28) \]Calculating these gives: \[ 6432 - 1652 = 4780 \]

Key Concepts

Understanding the Dot Product

The dot product is a mathematical operation that takes two equal-length sequences of numbers, usually vectors, and returns a single number. When working with physical concepts like work and energy, the dot product becomes incredibly practical. The dot product of two vectors is calculated by multiplying their corresponding components and then summing these products.

In the context of physics, the dot product is especially useful because it calculates how much of one vector (like force) "fits" onto another vector (like displacement). This alignment occurs along their path of interaction, and the resulting scalar will determine how work is performed.

• Formula: 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: \( \mathbf{a} \cdot \mathbf{b} = a_1 b_1 + a_2 b_2 \).
• Properties: The dot product is commutative, meaning \( \mathbf{a} \cdot \mathbf{b} = \mathbf{b} \cdot \mathbf{a} \).

By understanding the dot product, we can effectively calculate physical phenomena such as work, where both force and displacement are key factors.

The Force Vector

A force vector represents both the magnitude and direction of a force acting on an object. In physics, force is a vector quantity because it affects both the movement and direction of the object.

A force vector can be visualized as an arrow where the direction of the arrow indicates the direction of the force, and its length represents the magnitude.

• Notation: Often represented as \( \mathbf{F} = F_x \mathbf{i} + F_y \mathbf{j} \), where \( F_x \) and \( F_y \) are the components in the respective directions.
• Importance: Understanding the force vector helps in analyzing the action applied on an object which causes motion, deformation, or change in its state of motion.

In the exercise, the given force vector \( \mathbf{F} = -67 \mathbf{i} + 59 \mathbf{j} \) specifies that the force has components in both the \(i\) and \(j\) directions, combining to create the total applied force. This aids in calculating the work when the object is displaced by another vector.

The Displacement Vector

The displacement vector is a critical component in calculating work performed by a force. Displacement refers to the change in position of an object. It is a vector quantity, which means it has both magnitude and direction.

Displacement vectors help represent the path taken by an object and estimate the efficacy of the force applied.

• Definition: A displacement vector is often expressed as \( \mathbf{d} = d_x \mathbf{i} + d_y \mathbf{j} \), where \( d_x \) and \( d_y \) are the changes in position along the respective axes.
• Use in Work Calculations: It plays a crucial role in calculating work since work is defined as the dot product of the force and displacement vectors. A well-aligned displacement vector can maximize the work done.

In our exercise, the displacement vector \( \mathbf{d} = -96 \mathbf{i} - 28 \mathbf{j} \) indicates a movement in both the negative \(i\) and \(j\) directions, crucial in understanding how the force causes work or energy transfer.