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Ch. 7 - Applications of Trigonometry and Vectors
Lial - Trigonometry 12th Edition
Lial12th EditionTrigonometryISBN: 9780136552161Not the one you use?Change textbook
All textbooksLial 12th EditionCh. 7 - Applications of Trigonometry and VectorsProblem 41
Chapter 8, Problem 41

A force of 18.0 lb is required to hold a 60.0-lb stump grinder on an incline. What angle does the incline make with the horizontal?

Verified step by step guidance
1
Identify the forces acting on the stump grinder on the incline: the weight (60.0 lb) acting vertically downward and the force (18.0 lb) required to hold it in place along the incline.
Recognize that the force holding the grinder is balancing the component of the weight parallel to the incline. The weight can be resolved into two components: one parallel to the incline and one perpendicular to it.
Use the relationship between the force parallel to the incline and the angle \( \theta \) of the incline: the parallel component of the weight is given by \( W \sin(\theta) \), where \( W = 60.0 \) lb.
Set up the equation equating the force required to hold the grinder to the parallel component of the weight: \( 18.0 = 60.0 \sin(\theta) \).
Solve for the angle \( \theta \) by isolating \( \sin(\theta) \): \( \sin(\theta) = \frac{18.0}{60.0} \), then find \( \theta = \arcsin\left(\frac{18.0}{60.0}\right) \).

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Key Concepts

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

Forces on an Inclined Plane

When an object rests on an inclined plane, its weight can be resolved into components parallel and perpendicular to the plane. The parallel component causes the object to slide down, while the perpendicular component presses it against the surface. Understanding these components is essential to analyze forces acting on the object.
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Example 2

Trigonometric Resolution of Forces

Trigonometry allows us to relate the angle of the incline to the force components using sine and cosine functions. Specifically, the component of weight parallel to the incline is given by weight times sine of the angle, and the perpendicular component by weight times cosine of the angle.
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Fundamental Trigonometric Identities

Equilibrium and Force Balance

For the stump grinder to be held stationary, the applied force must balance the component of weight pulling it down the incline. Setting the applied force equal to the parallel component of weight allows solving for the incline angle, ensuring the system is in equilibrium.
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45-45-90 Triangles
Related Practice
Textbook Question

To build the pyramids in Egypt, it is believed that giant causeways were constructed to transport the building materials to the site. One such causeway is said to have been 3000 ft long, with a slope of about 2.3°. How much force would be required to hold a 60-ton monolith on this causeway?


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Textbook Question

Use the law of sines to prove that each statement is true for any triangle ABC, with corresponding sides a, b, and c.


(a - b)/(a + b) = (sin A - sin B)/(sin A + sin B)

Textbook Question

A balloonist is directly above a straight road 1.5 mi long that joins two villages. She finds that the town closer to her is at an angle of depression of 35°, and the farther town is at an angle of depression of 31°. How high above the ground is the balloon? 


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Textbook Question

Given vectors u and v, find: 2u.

u = 2i, v = i + j

Textbook Question

Given vectors u and v, find: 2u + 3v.

u = 2i, v = i + j

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Textbook Question

Given vectors u and v, find: v - 3u. 

u = 2i, v = i + j