Problem 40 - Essential Calculus Early Transcendentals

Question

A boat sails south with the help of a wind blowing in the direction \(S36^{\circ} \mathrm{E}\) with magnitude 400 \(\mathrm{lb}\) . Find the work done by the wind as the boat moves 120 \(\mathrm{ft.}\)

Step-by-Step Solution

Answer

The work done by the wind is approximately 38,832 ft-lb.

Step 1: Understand the Problem

The boat is moving south, and the wind is blowing in the direction \( S36^{\circ} \text{E} \). We need to determine how much work the wind does as the boat moves 120 feet.

Step 2: Define the Relevant Work Formula

Work is defined as the dot product of force and displacement vectors. The formula for work is \( W = F \cdot d = |F| \cdot |d| \cdot \cos \theta \), where \( \theta \) is the angle between the force and displacement vectors.

Step 3: Determine the Angle Between Directions

The direction \( S36^{\circ} \text{E} \) means 36 degrees east of south. Since the boat is moving directly south, the angle \( \theta \) between the force (wind) direction and the displacement (south) direction is 36 degrees.

Step 4: Calculate the Work

We have \( |F| = 400 \) lb, \( |d| = 120 \) ft, and \( \theta = 36^{\circ} \). Calculate the work using the formula: \[ W = 400 \cdot 120 \cdot \cos(36^{\circ}) \] First, calculate \( \cos(36^{\circ}) \), which is approximately 0.809. Therefore, \[ W = 400 \cdot 120 \cdot 0.809 \approx 38,832 \text{ ft-lb} \].

Key Concepts

Vector Dot Product

The vector dot product is a fundamental concept in physics, especially when discussing work. It allows us to find out how much of one vector extends in the direction of another. Imagine two vectors, the first representing the force and the second representing the displacement. To find the work, we multiply these two vectors using the dot product.
The mathematical formula for the dot product of two vectors, \( \mathbf{F} \) and \( \mathbf{d} \), is given by:

• \( \mathbf{F} \cdot \mathbf{d} = |\mathbf{F}| \cdot |\mathbf{d}| \cdot \cos \theta \)

Here, \( |\mathbf{F}| \) and \( |\mathbf{d}| \) represent the magnitudes of the force and displacement vectors. Meanwhile, \( \theta \) is the angle between them. This formula tells us that only the component of the force that acts in the direction of the displacement contributes to work.
Overall, the vector dot product helps us understand physical phenomena where direction plays an essential role.

Force and Displacement

Force and displacement are core physical concepts that together help us measure work done by or on an object. In the context of work, force is the push or pull applied to an object. It's what gets things moving or keeps them still.
Displacement represents the change in an object's position. It’s a vector, which means it has both a direction and a magnitude, like moving 120 feet directly south.
When calculating work done, both force and displacement play crucial roles. Only the component of the force that acts in the direction of the displacement will work on the object. If the force acts perpendicular to the displacement, no work is done.

• Force: A vector with both magnitude and direction, measured in pounds (lb) or newtons (N).
• Displacement: A vector representing movement, measured in feet (ft) or meters (m).

Angle Between Vectors

The angle between vectors, \( \theta \), is a critical element in calculating work through the dot product. It explicitly tells us how aligned the two vectors are, influencing how much of the force contributes to the displacement in a useful way.
In our exercise, the wind blows at an angle of 36 degrees east of south. The angle between this force vector and the straight southward displacement vector is exactly 36 degrees. Knowing this angle is fundamental because:

• If the angle is 0 degrees, the vectors are perfectly aligned, and all of the force contributes to the work.
• At 90 degrees, no work is done as the force does not contribute to displacement.
• An angle in between uses cosine to determine how much of the force actually works to move the object.

Understanding angles helps us harness real-world scenarios, like wind aiding a moving boat. The calculation of work using \( \cos(36^{\circ}) \) demonstrates the partial impact of force based on direction.