A force F=8.0 N is in the direction 30* counterclockwise from +y axis. What are its X and Y components?

Question

A force F=8.0 N is in the direction 30* counterclockwise from +y axis. What are its X and Y components?

A) Fx= 8cos 30* and Fy= 8sin 30*

B) Fx= 8sin 30* and Fy= 8 cos 30*

C) Fx= -8 cos 30* and Fy=8 sin 30*

D) Fx= -8 sin 30* and Fy= 8 cos 30*

Answer 1

The answer is D

The x and y components of the force are decomposed using the sine and cosine of the angle between the positive x-axis and the force vector. If the force vector is 30 degrees CCW from the positive y-axis, then it is 90+30=120 degrees from the positive x-axis. Then the components are:

Fx = F * cos(120deg) = -0.5 * F = - F * sin(30) = -(8.0 N) * sin(30deg)

Fy = F * sin(120deg) = (sqrt(3)/2) * F = F * cos(30) = (8.0 N) * cos(30deg)

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Note that there is a somewhat simpler way to think of this by visualizing the vector decomposition, noting that F is the hypotenuse of a right triangle with governing equations SOH CAH TOA:

cos(30) = adjacent/hypotenuse = Fy / F => Fy = F cos(30)

sin(30) = opposite/hypotenuse = Fx / F => Fx = F sin(30)

Keep in mind that doing things this way means you need to account for the negative direction of x yourself.