To extricate an SUV stuck in the mud, workmen use three horizontal ropes, producing the force vectors shown in Fig. E4.24.24.2. Use the components to find the magnitude and direction of the resultant of the three pulls.

Ch 04: Newton's Laws of Motion

Young & Freedman Calc - University Physics 14th Edition

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Young & Freedman Calc 14th Edition

Ch 04: Newton's Laws of Motion

Problem 2b

Chapter 4, Problem 2b

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.

Verified step by step guidance

Step 1: Break each force vector into its x and y components using trigonometric functions. For a force F at an angle θ, the x-component is F*cos(θ) and the y-component is F*sin(θ).

Step 2: Calculate the x and y components for each force vector. For example, for the 985 N force at 31°: x-component = 985*cos(31°), y-component = 985*sin(31°). Repeat this for the other forces (788 N at 32° and 411 N at 53°).

Step 3: Sum up all the x-components to find the total x-component of the resultant force. Similarly, sum up all the y-components to find the total y-component of the resultant force.

Step 4: Use the Pythagorean theorem to calculate the magnitude of the resultant force. The magnitude is given by √(total_x^2 + total_y^2).

Step 5: Determine the direction of the resultant force using the inverse tangent function. The direction angle θ is given by tan⁻¹(total_y / total_x). Ensure the angle is measured relative to the positive x-axis.

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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. Each vector can be broken down into its components along the x and y axes, allowing for easier calculations of net forces. In this scenario, the forces exerted by the ropes on the SUV are represented as vectors, which must be analyzed to determine the resultant force.

Vector Addition

Vector addition is the process of combining multiple vectors to find a resultant vector. This involves adding the corresponding components of each vector. For the SUV problem, the horizontal and vertical components of the three force vectors must be summed separately to find the overall force acting on the SUV, which will help in determining its movement.

Resultant Force

The resultant force is the single force that represents the combined effect of all individual forces acting on an object. It is calculated by vector addition of all forces. In this case, finding the magnitude and direction of the resultant force will indicate how effectively the combined pulls from the ropes can extricate the SUV from the mud.

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Related Practice

Textbook Question

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

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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

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 $\overrightarrow{F}$ whose direction makes an angle of $30.0$ ° with the ramp (Fig. E $4.4$ ). How large will the component $F_{y}$ perpendicular to the ramp be then?

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Textbook Question

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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