As a sailboat sails 52 m due north, a breeze exerts a constant force 𝐅1 on the boat's sails. Thi

Name: Problem 9, Chapter 6: Physics
Uploaded: 2020-04-07T14:31:50-08:00
Duration: 1 min 57 s
Channel: Adnan Gill
Description: As a sailboat sails 52 m due north, a breeze exerts a constant force 𝐅1 on the boat's sails. This force is directed at an angle west of due north. A force 𝐅2 of the same magnitude directed due north would do the same amount of work on the sailboat over a distance of just 47 m . What is the angle between the direction of the force 𝐅1 and due north?

step 1 So let's make a free body diagram for the first scenario. So we have a force F1, which acts at s

Question

As a sailboat sails 52 m due north, a breeze exerts a constant force $\overrightarrow{\mathbf{F}}_{1}$ on the boat's sails. This force is directed at an angle west of due north. A force $\overrightarrow{\mathbf{F}}_{2}$ of the same magnitude directed due north would do the same amount of work on the sailboat over a distance of just $47 \mathrm{m} .$ What is the angle between the direction of the force $\overrightarrow{\mathbf{F}}_{1}$ and due north?

   As a sailboat sails 52 m due north, a breeze exerts a constant force $\overrightarrow{\mathbf{F}}_{1}$ on the boat's sails. This force is directed at an angle west of due north. A force $\overrightarrow{\mathbf{F}}_{2}$ of the same magnitude directed due north would do the same amount of work on the sailboat over a distance of just $47 \mathrm{m} .$ What is the angle between the direction of the force $\overrightarrow{\mathbf{F}}_{1}$ and due north?

Physics

John D. Cutnell,… 11th Edition

Chapter 6, Problem 9

Instant Answer

Solved by Expert Adnan Gill

04/07/2020

Step 1

We are given that the work done by both forces is the same, so we can set up the following equation: $F_1 d_1 \cos\theta_1 = F_2 d_2 \cos\theta_2$ Since the magnitudes of the forces are the same ($F_1 = F_2$), we can simplify the equation: $d_1 \cos\theta_1 = Show more…

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As a sailboat sails 52 m due north, a breeze exerts a constant force $\overrightarrow{\mathbf{F}}_{1}$ on the boat's sails. This force is directed at an angle west of due north. A force $\overrightarrow{\mathbf{F}}_{2}$ of the same magnitude directed due north would do the same amount of work on the sailboat over a distance of just $47 \mathrm{m} .$ What is the angle between the direction of the force $\overrightarrow{\mathbf{F}}_{1}$ and due north?

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Key Concepts

Force Decomposition

Force decomposition involves resolving a force into components along specific directions. In contexts like these, it is crucial to identify the component of a force that acts in the direction of displacement, as only this component performs work.

Trigonometric Relationships in Vector Analysis

Trigonometric relationships, particularly the cosine function, are used to relate the angle between vectors to the magnitude of their components. The cosine of the angle between the force and displacement vectors determines the proportion of the force that contributes to the work done.

Work

Work is defined as the dot product of force and displacement, calculated as the product of the force magnitude, displacement, and the cosine of the angle between them. This concept is essential for determining how much energy is transferred by a force acting along a displacement.

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