Question

100 POINTS!!!

The magnitude and direction of two forces acting on an object are 100 pounds, S80°E, and 90 pounds, N55°E,
respectively. Find the magnitude, to the nearest hundredth of a pound, and the direction angle, to the nearest tenth
of a degree, of the resultant force.

Answer 1

• 175.58 pounds
• N 78.7° E

Explanation:

You want the resultant force equivalent to the two forces ...

• 100 pounds S 80° E
• 90 pounds N 55° E

Resultant

There are a number of ways the resultant force can be found. One of them is to add the components of the two forces. Another makes use of the law of cosines:

r² = u² +v² -2uv·cos(θ)

where θ is the supplement of the angle between the two forces.

r² = 100² +90² -2·100·90·cos(135°) ≈ 30827.922061

r ≈ 175.579 ≈ 175.58

The resultant force is about 175.58 pounds.

Direction

The angle added to that of v can be found using the law of sines:

sin(φ)/100 = sin(135°)/175.58

φ = arcsin(sin(135°)·100/175.58) ≈ 23.749°

The angle East of North is then ...

55° +23.7° = 78.7°

The direction of the resultant is N 78.7° E.

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Additional comment

The angles used in the second attachment are bearing angles measured clockwise from north. The rectangular coordinates shown as the sum of the forces represent (north) + (east)i.

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