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Chapter 8: Problem 58

$$\text {Evaluate the given expressions.}$$ The force \(F\) that a rope exerts on a crate is related to force \(F_{x}\) directed along the \(x\) -axis by \(F=F_{x} \sec \theta\) where \(\theta\) is the standard-position angle for \(F\). See Fig. \(8.18 .\) Find \(F\) if \(F_{x}=-365 \mathrm{N}\) and \(\theta=127.0^{\circ}\)

Short Answer

Expert verified
The force \( F \) is approximately 606.457 N.

Step by step solution

01

Understanding the Problem

We are given that the force exerted by a rope on a crate, denoted as \( F \), can be calculated using the formula \( F = F_x \sec \theta \), where \( F_x \) is the force component along the \( x \)-axis and \( \theta \) is the angle in standard position. We need to find the value of \( F \) when \( F_x = -365 \) N and \( \theta = 127.0^\circ \).
02

Converting Degrees to Radians

To ensure accurate calculations using trigonometric functions, first convert the angle from degrees to radians. Use the conversion factor: \( 1^\circ = \frac{\pi}{180} \, \text{radians} \).Therefore, \( 127^\circ = 127 \times \frac{\pi}{180} \approx 2.2166 \, \text{radians} \).
03

Calculating \( \sec \theta \)

The secant function is the reciprocal of the cosine function. That is, \( \sec \theta = \frac{1}{\cos \theta} \). Calculate \( \cos 127.0^\circ \) (or \( \cos 2.2166 \, \text{radians} \)) using a calculator.\( \cos 127.0^\circ \approx -0.6018 \)\( \sec 127.0^\circ = \frac{1}{-0.6018} \approx -1.6618 \).
04

Calculating the Force \( F \)

Substitute \( F_x = -365 \) N and \( \sec 127.0^\circ \approx -1.6618 \) into the formula:\[ F = F_x \sec \theta = (-365) \times (-1.6618) \]Calculate the product to find \( F \).
05

Finding \( F \)

Perform the multiplication to find the result:\( F \approx (-365) \times (-1.6618) = 606.457 \) N.Thus, the force \( F \) is approximately 606.457 N.

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Key Concepts

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

Force Components
In physics, a force can often be broken down into components. These force components are typically aligned with coordinate axes, such as the x-axis and y-axis. This makes it easier to analyze their effects independently.

Consider a force pulling a crate by a rope. The total force exerted by the rope, known as force \( F \), has components in different directions, such as \( F_x \), which represents the projection of the force along the x-axis.
  • This component is crucial because it directly affects how the crate moves in that direction.
  • By analyzing the force in parts, it becomes simpler to apply Newton's laws of motion and solve for unknowns.
Understanding components is vital in various physics problems, especially involving inclined planes or systems with more dimensions.
Secant Function
The secant function \( \sec \theta \) is a trigonometric function, which is the reciprocal of the cosine function. In mathematical terms, it can be expressed as:
\[ \sec \theta = \frac{1}{\cos \theta} \]
This function is useful in problems involving diagonal forces or when certain angles are given in reference calculations.
  • The secant function tends to appear when dealing with angles not directly aligned with standard axes.
  • It facilitates the calculation of the magnitude of a force when given a specific angle, much like in this exercise.
Trigonometric functions like secant allow us to relate angles and sides of a triangle in a unit circle, facilitating the handling of various physics and engineering problems.
Radians Conversion
Degrees and radians are two units for measuring angles. However, trigonometric calculations in mathematics and physics often require angles to be in radians. Converting degrees to radians is essential for accurate calculations using trigonometric functions.

Here's the formula to convert degrees \( \theta \) to radians:
\[ \text{radians} = \theta\text{ (in degrees) } \times \frac{\pi}{180} \]
This conversion ensures that angles are expressed in units that align with the mathematical definition of these functions.
  • Radians provide a natural way of measuring angles because they are based on the radius of a circle.
  • This conversion is particularly important when using calculators or software that require angles in radians.
While degrees are more intuitive, radians encapsulate the geometric properties more elegantly.
Standard-Position Angle
A standard-position angle is an angle whose vertex is at the origin, and its initial side lies along the positive x-axis. It is a common way to represent angles in mathematics, particularly for trigonometric functions.
  • This angle can provide a consistent reference point, easing the complexity of understanding the relationship between angles and their corresponding trigonometric values.
  • Standard-position angles are measured counterclockwise from the positive x-axis.
For practical problems like calculating the force a rope exerts on a crate, considering the angle \( \theta \) in standard-position aids in correctly applying formulas and solving relevant equations.

Representation in the standard position also helps to visually understand the component relationships and deduce correct values for further computations.

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