Problem 44 - 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. \(\mathbf{F}=13 \mathbf{j}, \mathrm{d}=44 \mathrm{i}\)

Step-by-Step Solution

Answer

Work = 0, as the force and displacement are perpendicular.

Step 1: Understand the Concept of Work

Work is a measure of energy transfer when a force is applied to an object, resulting in a displacement. In physics, it is calculated by taking the dot product of the force \(\mathbf{F}\) and displacement \(\mathbf{d}\) vectors, as given by \[\text{Work} = \mathbf{F} \cdot \mathbf{d}.\]

Step 2: Identify the Force and Displacement Vectors

In this problem, the force vector is \(\mathbf{F} = 13\mathbf{j}\) and the displacement vector is \(\mathrm{d} = 44\mathrm{i}\). The force is applied in the \(y\)-direction, and the displacement is along the \(x\)-direction.

Step 3: Calculate the Dot Product

Since the force and displacement vectors are perpendicular (one is along \(\mathbf{j}\) and the other is along \(\mathbf{i}\)), the dot product is zero. The formula for the dot product is \[\mathbf{F} \cdot \mathbf{d} = F_{x}d_{x} + F_{y}d_{y},\] where both \(F_{x}\) of force and \(d_{y}\) of displacement are zero. Thus, \(\text{Work} = 0\).

Key Concepts

Dot Product

The dot product is a fundamental concept in physics and vector mathematics. It is a way to multiply two vectors which result in a scalar (a single number). The dot product is especially useful for finding out how much one vector induces an effect along another vector.

Mathematically, the dot product of two vectors \(\mathbf{A}\) and \(\mathbf{B}\) can be expressed as:

• \(\mathbf{A} \cdot \mathbf{B} = A_xB_x + A_yB_y + A_zB_z\)

where \(A_x\), \(A_y\), and \(A_z\) are the components of vector \(\mathbf{A}\), and \(B_x\), \(B_y\), and \(B_z\) are the components of vector \(\mathbf{B}\).

A key property of dot product is that it incorporates the cosine of the angle between the two vectors, denoted as \(\theta\):

• \(\mathbf{A} \cdot \mathbf{B} = |\mathbf{A}| |\mathbf{B}||\cos \theta|\)

This equation shows that when two vectors are perpendicular (i.e., \(\theta = 90^\circ\)), the dot product is zero, reflecting no influence of one vector on the direction of the other.

Force Vector

A force vector is a vector that represents the magnitude and direction of a force acting on an object. Force is a vector quantity, meaning it has both size and direction, and is essential for understanding how objects move and interact with energy.

Consider the force vector given as \(\mathbf{F} = 13\mathbf{j}\). In this representation:

• \(13\mathbf{j}\): The force has a magnitude of 13 units in the \(y\)-direction.

Force vectors are crucial when calculating work or analyzing motion because they tell you the exact line of action of the force. Knowing the direction of a force can help determine the resulting motion of the object and how forces might add up or cancel out when multiple forces act simultaneously.

Displacement Vector

A displacement vector is used to describe how far out of position an object is from its starting point, considering not only distance but also the direction of movement. It is a vector because it includes both magnitude and direction.

The displacement vector in our problem is \(d = 44\mathbf{i}\). Here, it indicates:

• 44\(\mathbf{i}\): The object is displaced 44 units in the \(x\)-direction.

Displacement vectors are essential in physics as they showcase not just the distance an object has moved but its direct path from the original point.

Understanding displacement vectors helps analyze the motion of objects and their paths, and plays a vital role in interpreting how forces applied in a specific direction cause movement along certain paths. This is critical in calculating work, determining energies, and analyzing reaction forces in various applications.