Problem 45 - 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. \(\mathbf{F}=39 \mathbf{j}, \mathrm{d}=72 \mathrm{i}\)

Step-by-Step Solution

Answer

The work done is 0 foot-pounds.

Step 1: Understand the Formula for Work

To find the work done by a force on an object through a displacement, we use the formula for work:\[W = \mathbf{F} \cdot \mathbf{d}\]where \(W\) is the work, \(\mathbf{F}\) is the force vector, and \(\mathbf{d}\) is the displacement vector. For force in pounds and displacement in feet, the result will be in foot-pounds.

Step 2: Perform the Dot Product

The dot product of two vectors \(\mathbf{F} = a\mathbf{i} + b\mathbf{j}\) and \(\mathbf{d} = c\mathbf{i} + d\mathbf{j}\) is given by:\[\mathbf{F} \cdot \mathbf{d} = ac + bd\]Given \(\mathbf{F} = 39 \mathbf{j}\) and \(\mathbf{d} = 72 \mathbf{i}\), we find:\[\mathbf{F} \cdot \mathbf{d} = (0)(72) + (39)(0) = 0\]

Step 3: State the Result

Since the dot product is zero, the work done by the force is 0 foot-pounds. This is because the force and displacement are perpendicular to each other, resulting in no work being done.

Key Concepts

Work

In the realm of physics, work is a pivotal concept that measures how much energy is transferred by a force acting along a distance. When discussing work in the context of vector mathematics, the dot product plays an essential role.
Work is calculated by taking the dot product of two vectors: the force vector and the displacement vector. Thus, the formula for work is:

• \[ W = \mathbf{F} \cdot \mathbf{d} \]

The result of the dot product in the formula gives us the amount of work performed by the force on the object.
It's informative to note that the work is measured in foot-pounds when force is in pounds and displacement is in feet.
This is particularly useful in practical applications where understanding how much work is done can lead to insights into efficiency and energy use.When vectors are perpendicular, as in our problem, no work is done, and the value is zero.
This is because the direction of the force does not contribute towards the displacement, illustrating how the orientation of vectors affects work.

Vector Operations

Vector operations, such as the dot product, are crucial in understanding the relationships between different vectors. A vector is a quantity that has both magnitude and direction.
In our problem, vectors represent physical quantities: the force vector \[ \mathbf{F} = 39 \mathbf{j} \]and the displacement vector\[ \mathbf{d} = 72 \mathbf{i} \].To find the work done, it's essential to understand the concept of the dot product, which is an algebraic operation that takes two equal-length sequences of numbers and returns a single number.
The dot product of vectors \(\mathbf{a} = a_1\mathbf{i} + a_2\mathbf{j}\) and \(\mathbf{b} = b_1\mathbf{i} + b_2\mathbf{j}\) is calculated as:

• \[ \mathbf{a} \cdot \mathbf{b} = a_1b_1 + a_2b_2 \]

Understanding and applying the dot product allows one to determine the extent to which one vector aligns with another.
This is especially important in physics and engineering, where such operations often denote the interaction between different forces and movements.

Force and Displacement

In the scenario of calculating work, both force and displacement are described as vectors. Each vector has direction and magnitude, which are crucial for determining the work done.
The force vector, \(\mathbf{F} = 39 \mathbf{j}\), indicates that the force is acting in the vertical direction along the y-axis.
Meanwhile, the displacement vector \(\mathbf{d} = 72 \mathbf{i}\) shows how the object is moving exclusively in the horizontal direction along the x-axis.
When these two vectors are perpendicular, meaning they do not share any common directional component, no work is done as there is no component of force in the direction of displacement. This is evident from the formula:

• Work is zero when \(\mathbf{F} \cdot \mathbf{d} = 0\).

For practically applying this concept, consider situations where you push a book across a table: if you push it straight (parallel to the table), you do work.
However, lifting it straight up or down would not change its horizontal position, thus no horizontal work in relation to the horizontal displacement.