Ch 03: Vectors and Coordinate Systems

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 03: Vectors and Coordinate Systems

Problem 39

Chapter 3, Problem 39

Your neighbor Paul has rented a truck with a loading ramp. The ramp is tilted upward at 25°, and Paul is pulling a large crate up the ramp with a rope that angles 10° above the ramp. If Paul pulls with a force of 550 N, what are the horizontal and vertical components of his force? (Force is measured in newtons, abbreviated N.)

Verified step by step guidance

Step 1: Understand the problem. Paul is pulling a crate up a ramp with a force of 550 N. The rope is angled 10° above the ramp, and the ramp itself is tilted at 25°. We need to find the horizontal and vertical components of Paul's pulling force.

Step 2: Define the coordinate system. The horizontal direction is parallel to the ground, and the vertical direction is perpendicular to the ground. To find the components of the force, we need to resolve the force vector into these two directions.

Step 3: Calculate the total angle of the force relative to the horizontal. The ramp is tilted at 25°, and the rope is angled 10° above the ramp. Therefore, the total angle θ of the force relative to the horizontal is θ = 25° + 10° = 35°.

Step 4: Use trigonometric functions to resolve the force into components. The horizontal component of the force (F_x) is given by F_x = F * cos(θ), and the vertical component of the force (F_y) is given by F_y = F * sin(θ), where F = 550 N and θ = 35°.

Step 5: Substitute the values into the equations. For F_x, use F_x = 550 * cos(35°). For F_y, use F_y = 550 * sin(35°). These expressions give the horizontal and vertical components of Paul's pulling force.

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Key Concepts

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

Force Components

In physics, any force can be broken down into its horizontal and vertical components using trigonometric functions. The horizontal component is found by multiplying the force by the cosine of the angle, while the vertical component is determined by multiplying the force by the sine of the angle. This decomposition is essential for analyzing forces acting at angles, especially in problems involving inclined planes.

Inclined Plane

An inclined plane is a flat surface that is tilted at an angle to the horizontal. It is a simple machine that allows for the easier movement of objects by reducing the amount of force needed to lift them vertically. The angle of inclination affects the gravitational force acting on the object and the required force to move it up the ramp, making it crucial to consider in problems involving ramps.

Trigonometric Functions

Trigonometric functions, such as sine and cosine, relate the angles of a triangle to the ratios of its sides. In the context of forces, these functions are used to resolve forces into their components based on the angle of application. Understanding how to apply these functions is vital for calculating the effects of forces acting at angles, particularly in scenarios involving ramps and inclined surfaces.

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