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Chapter 11: Problem 54

A boat is towed with a force of 150 Ib with a rope that makes an angle of \(30^{\circ}\) to the horizontal. Find the horizontal and vertical components of the force.

Short Answer

Expert verified
Answer: The horizontal component of the force is \(75\sqrt{3} \ \text{lb}\), and the vertical component of the force is \(75 \ \text{lb}\).

Step by step solution

01

Identify the given information

We know that the force applied on the boat is 150 lb and the angle of the rope is \(30^{\circ}\) with the horizontal.
02

Calculate the horizontal component of the force

To find the horizontal component of the force, we will use the cosine function as follows: Horizontal Component \(= (150 \ \text{lb})\times \cos(30^{\circ})\). We know that \(\cos(30^{\circ}) = \frac{\sqrt{3}}{2}\). So, the horizontal component of the force is: Horizontal Component \(= (150 \ \text{lb})\times \frac{\sqrt{3}}{2} = 75\sqrt{3} \ \text{lb}\).
03

Calculate the vertical component of the force

To find the vertical component of the force, we will use the sine function as follows: Vertical Component \(= (150 \ \text{lb})\times \sin(30^{\circ})\). We know that \(\sin(30^{\circ}) = \frac{1}{2}\). So, the vertical component of the force is: Vertical Component \(= (150 \ \text{lb})\times \frac{1}{2} = 75 \ \text{lb}\).
04

Summarize the results

In conclusion, the horizontal component of the force is \(75\sqrt{3} \ \text{lb}\), and the vertical component of the force is \(75 \ \text{lb}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Trigonometric Functions
Trigonometric functions are mathematical tools that relate angles to the sides of a right triangle. The most commonly used functions are sine, cosine, and tangent. These functions are essential in various calculations, including finding vector components.
In our exercise, when the rope creates an angle of \(30^{\circ}\) with the horizontal, we use trigonometric functions to decompose the force into horizontal and vertical parts. For these kinds of problems, it's crucial to remember:
  • Cosine (\(\cos\)) is used to find the component of a vector along the adjacent side of an angle (horizontal in our case).
  • Sine (\(\sin\)) is used to find the component of a vector along the opposite side of an angle (vertical in our exercise).
By understanding these functions, you can effectively split any vectors into their horizontal and vertical components in problem-solving scenarios.
Basics of Force Decomposition
Force decomposition involves breaking down a vector, such as force, into two or more components. These components can help in analyzing and solving problems involving motion or balancing forces.
In the boat towing problem, we decompose the total force of 150 lb applied at an angle into horizontal and vertical components. This process is helpful because it simplifies the understanding and calculation of the effects of the force in vector form.
  • The horizontal component indicates how much of the force is moving the boat in the horizontal direction, which we calculate using cosine.
  • The vertical component indicates how much lift or downward force is applied, which is found using the sine function.
Force decomposition allows us to work with simpler, single-direction forces, making it easier to understand the interaction and movement dynamics in physical systems.
Angle Calculation in Vector Analysis
Angles play a critical role in vector analysis. They determine how forces and vectors are oriented in a given space. Knowing the angle between a force vector and a reference line (like the horizontal) is key to breaking down the vector into its components.
In the given problem, the angle is \(30^{\circ}\), which is used with trigonometric functions to find force components. Calculating these components involves understanding the basic trigonometric properties:
  • The cosine of \(30^{\circ}\) gives the horizontal portion, showing how much force affects movement parallel to the ground.
  • The sine of \(30^{\circ}\) gives the vertical portion, showing the lift or drop effect due to the force.
Without accurate angle calculation, it’s impossible to correctly resolve vectors, underscoring the angle’s pivotal role in vector decomposition and applied physics.

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