Two tugboats are pulling a disabled speedboat into port with forces of 1240 lb and 1480 lb. The angle between these forces is 28.2°. Find the direction and magnitude of the equilibrant.

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Identify the two forces acting on the speedboat: \(F_1 = 1240\) lb and \(F_2 = 1480\) lb, with an angle \(\theta = 28.2^\circ\) between them.

Calculate the resultant force \(R\) of the two forces using the Law of Cosines: \[R = \sqrt{F_1^2 + F_2^2 + 2 F_1 F_2 \cos(\theta)}\]

Determine the direction (angle) \(\alpha\) of the resultant force relative to the first force \(F_1\) using the Law of Sines or Law of Cosines: \[\alpha = \arctan\left(\frac{F_2 \sin(\theta)}{F_1 + F_2 \cos(\theta)}\right)\]

Recall that the equilibrant force is equal in magnitude but opposite in direction to the resultant force. So, the magnitude of the equilibrant is the same as \(R\).

Find the direction of the equilibrant by adding \(180^\circ\) to the angle \(\alpha\) (to point opposite the resultant force). This gives the direction of the equilibrant force.

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Key Concepts

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

Vector Addition and Resultant Force

When two forces act at an angle, their combined effect is found by vector addition. The resultant force is the single force that has the same effect as the two forces acting together. It can be calculated using the law of cosines and law of sines based on the magnitudes and angle between the forces.

Equilibrant Force

The equilibrant is a force that balances out other forces acting on an object, resulting in equilibrium. It has the same magnitude as the resultant force but acts in the opposite direction, ensuring the net force is zero and the object remains at rest or moves with constant velocity.

Law of Cosines and Law of Sines

The law of cosines relates the lengths of sides of a triangle to the cosine of one angle, useful for finding the magnitude of the resultant force. The law of sines relates the ratios of sides to the sines of their opposite angles, helping to find the direction of the resultant force relative to the given forces.

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Related Practice

Textbook Question

Apply the law of sines to the following: a = √5, c = 2√5, A = 30°. What is the value of sin C? What is the measure of C? Based on its angle measures, what kind of triangle is triangle ABC?

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Textbook Question

Use the figure to find each vector: u - v. Use vector notation as in Example 4.

Textbook Question

Use the figure to find each vector: u + v. Use vector notation as in Example 4.

Textbook Question

Solve each triangle. See Examples 2 and 3.

B = 74.8°, a = 8.92 in., c = 6.43 in.

Textbook Question

Use the figure to find each vector: - v. Use vector notation as in Example 4.

Textbook Question

Use the figure to find each vector: u + v. Use vector notation as in Example 4.