Two forces of 128 lb and 253 lb act on a point. The resultant force is 320 lb. Find the angle between the forces.

검증된 단계별 안내

Identify the given quantities: two forces with magnitudes \(F_1 = 128\) lb and \(F_2 = 253\) lb, and their resultant force \(R = 320\) lb. We need to find the angle \(\theta\) between these two forces.

Recall the law of cosines for vectors, which relates the magnitudes of two vectors and their resultant: \(R^2 = F_1^2 + F_2^2 + 2 F_1 F_2 \cos(\theta)\)

Substitute the known values into the equation: \(320^2 = 128^2 + 253^2 + 2 \times 128 \times 253 \times \cos(\theta)\)

Rearrange the equation to isolate \(\cos(\theta)\): \(\cos(\theta) = \frac{320^2 - 128^2 - 253^2}{2 \times 128 \times 253}\)

Calculate the right-hand side value (without final numeric evaluation here), then use the inverse cosine function to find the angle: \(\theta = \cos^{-1}(\text{value from previous step})\)

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이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

주요 개념

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

Resultant of Two Forces

The resultant force is the single force that has the same effect as the two given forces acting together. It can be found using vector addition, considering both magnitude and direction of the forces.

Law of Cosines in Vector Addition

The law of cosines relates the magnitudes of two vectors and the angle between them to the magnitude of their resultant. It is expressed as R² = A² + B² - 2AB cos(θ), where θ is the angle between the forces.

Solving for the Angle Between Forces

By rearranging the law of cosines formula, the angle between two forces can be found using θ = cos⁻¹((A² + B² - R²) / (2AB)). This allows determination of the angle when the magnitudes and resultant are known.