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?

검증된 단계별 안내

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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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.

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.