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 40

Chapter 3, Problem 40

Tom is climbing a 3.0-m-long ladder that leans against a vertical wall, contacting the wall 2.5 m above the ground. His weight of 680 N is a vector pointing vertically downward. (Weight is measured in newtons, abbreviated N.) What are the components of Tom's weight parallel and perpendicular to the ladder?

Verified step by step guidance

Determine the angle of the ladder with respect to the ground using trigonometry. The ladder forms a right triangle with the wall and the ground. Use the equation \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \), where the opposite side is 2.5 m (height of the ladder against the wall) and the hypotenuse is 3.0 m (length of the ladder). Solve for \( \theta \).

Once \( \theta \) is determined, calculate the component of Tom's weight parallel to the ladder using the formula \( W_{\parallel} = W \sin(\theta) \), where \( W \) is Tom's weight (680 N) and \( \sin(\theta) \) is the sine of the angle found in step 1.

Calculate the component of Tom's weight perpendicular to the ladder using the formula \( W_{\perp} = W \cos(\theta) \), where \( W \) is Tom's weight (680 N) and \( \cos(\theta) \) is the cosine of the angle found in step 1.

Verify the results by checking that the vector sum of the parallel and perpendicular components equals the total weight. This can be done using the Pythagorean theorem: \( W = \sqrt{W_{\parallel}^2 + W_{\perp}^2} \).

Interpret the results: The parallel component represents the force along the ladder's direction, while the perpendicular component represents the force pressing the ladder against the wall and ground. These components are crucial for understanding the forces acting on the ladder and ensuring its stability.

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Key Concepts

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

Weight as a Vector

Weight is a force that acts downward due to gravity, represented as a vector with both magnitude and direction. In this scenario, Tom's weight of 680 N acts vertically downward, and understanding its vector nature is crucial for resolving it into components relative to the inclined ladder.

Components of Forces

Forces can be resolved into components that act along specific directions, typically parallel and perpendicular to a surface. In this case, Tom's weight can be broken down into two components: one that acts parallel to the ladder and another that acts perpendicular to it, which helps in analyzing the forces acting on the ladder.

Trigonometric Relationships in Right Triangles

When dealing with inclined surfaces, trigonometric functions such as sine and cosine are used to relate the angles and sides of right triangles. By applying these functions to the angle formed by the ladder with the ground, we can calculate the parallel and perpendicular components of Tom's weight based on the ladder's angle and length.

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