## Mastering Physics Solutions: Two Forces Acting at a Point

Two Forces Acting at a Point

Solutions Below:

Two forces, F1 and F2, act at a point, as shown in the picture. F1 has a magnitude of 8.40N and is directed at an angle of α = 60.0° above the negative x axis in the second quadrant. F2 has a magnitude of 6.20 N and is directed at an angle of β = 52.7° below the negative x axis in the third quadrant.

Part A

What is the x component Fx of the resultant force?

Since both forces are in the same overall, x-direction (negative), just solve by (remember to include the negative signs):

Fx = -cos(60)F1 + -cos(52.7)F2
Fx = -0.5(8.40N) + -0.606(6.20N)

Fx = -7.96N

Part B

What is the y component Fy of the resultant force?

Same concept as Part A. Except that the first force is going up, the second is going down. Solve by:

Fy = sin(60)F1 – sin(52.7)F2
Fy = 7.27 – 4.93

Fy = 2.34N

Part C

What is the magnitude F of the resultant force?

Since we already found the resultant x and y components, we can just use these to solve for the resultant magnitude:

F = sqrt(F2x + F2y)
F = sqrt(-7.962 + 2.342)
F = sqrt(68.84)

F = 8.29N

Part D

What is the angle γ that the resultant force forms with the negative x axis? In this problem, assume that positive angles are measured clockwise from the negative x axis.

Since we have all 3 sides to a triangle (x and y components, plus the magnitude/hypotenuse), we can easily solve using trigonometric ratios. For example, using tangent (opposite / adjacent):

tan(γ) = Fy / Fx
tan(γ) = 2.34 / -7.96
tan(γ) = -0.294
γ = tan-1(-0.294)

γ = 16.4°

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