Ch 04: Newton's Laws of Motion

Young & Freedman Calc - University Physics 14th Edition

Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610Not the one you use?Change textbook

Young & Freedman Calc 14th Edition

Ch 04: Newton's Laws of Motion

Problem 2a

Chapter 4, Problem 2a

To extricate an SUV stuck in the mud, workmen use three horizontal ropes, producing the force vectors shown in Fig. E $4.2$ . Find the $x$ - and $y$ -components of each of the three pulls.

Verified step by step guidance

Step 1: Identify the forces and angles given in the problem. From the diagram, we have three forces: 685 N at 54° (Force A), 784 N at 44° (Force B), and 469 N at 32° (Force C). These forces are applied at specific angles relative to the x-axis.

Step 2: Recall the formulas for calculating the x- and y-components of a force vector. The x-component is given by F_x = F * cos(θ), and the y-component is given by F_y = F * sin(θ), where F is the magnitude of the force and θ is the angle relative to the x-axis.

Step 3: Calculate the x-component for each force. For Force A, use F_x = 685 * cos(54°). For Force B, use F_x = 784 * cos(44°). For Force C, use F_x = 469 * cos(32°).

Step 4: Calculate the y-component for each force. For Force A, use F_y = 685 * sin(54°). For Force B, use F_y = 784 * sin(44°). For Force C, use F_y = 469 * sin(32°).

Step 5: Organize the results into a table or list format for clarity. Each force will have its x- and y-components calculated separately, and these components can be summed later if needed for further analysis.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Force Vectors

Force vectors represent the magnitude and direction of forces acting on an object. In this scenario, the forces exerted by the ropes on the SUV are depicted as vectors, which can be broken down into their x (horizontal) and y (vertical) components. Understanding how to resolve these vectors is crucial for analyzing the net force acting on the SUV.

Vector Resolution

Vector resolution involves breaking a vector into its components along the axes of a coordinate system, typically the x and y axes. This is done using trigonometric functions: the x-component is found using the cosine of the angle, while the y-component is found using the sine. This process is essential for calculating the individual contributions of each force in the problem.

Equilibrium of Forces

The equilibrium of forces occurs when the net force acting on an object is zero, meaning all forces balance out. In this context, understanding how to calculate the x and y components of the forces allows for determining whether the combined forces can successfully extricate the SUV from the mud. Analyzing the components helps ensure that the resultant force is sufficient to overcome the resistance of the mud.

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Equilibrium

Related Practice

Textbook Question

Due to a jaw injury, a patient must wear a strap (Fig. E $4.3$ ) that produces a net upward force of $5.00$ N on his chin. The tension is the same throughout the strap. To what tension must the strap be adjusted to provide the necessary upward force?

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A man is dragging a trunk up the loading ramp of a mover's truck. The ramp has a slope angle of $20.0$ °, and the man pulls upward with a force $\vec{F}$ whose direction makes an angle of $30.0$ ° with the ramp (Fig. E $4.4$ ). How large a force $\vec{F}$ is necessary for the component $F_{x}$ parallel to the ramp to be $90.0$ N?

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Textbook Question

To extricate an SUV stuck in the mud, workmen use three horizontal ropes, producing the force vectors shown in Fig. E $4.2$ . Use the components to find the magnitude and direction of the resultant of the three pulls.

Two dogs pull horizontally on ropes attached to a post; the angle between the ropes is $60.0$ °. If Rover exerts a force of $270$ N and Fido exerts a force of $300$ N, find the magnitude of the resultant force and the angle it makes with Rover's rope.

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To extricate an SUV stuck in the mud, workmen use three horizontal ropes, producing the force vectors shown in Fig. E4.24.2. Find the xx- and yy-components of each of the three pulls.

Verified video answer for a similar problem:

Key Concepts

Force Vectors

Vector Resolution

Equilibrium of Forces

To extricate an SUV stuck in the mud, workmen use three horizontal ropes, producing the force vectors shown in Fig. E $4.2$ . Find the $x$ - and $y$ -components of each of the three pulls.