Q: Which equation correctly gives the direction angle $$\theta of $$the resultant force $$\vec{R}$$ measured counterclockwise from the x-axis?

$$\theta = \arctan \left( \dfrac{R_y}{R_x} \right)$$

Q: If the resultant force $$\vec{R}$$ has components $$R_x = 500\ \mathrm{lb}$$ and $$R_y = -100\ \mathrm{lb}$$, what is the direction angle $$\theta$$ measured from the x-axis?

$$\theta = \arctan \left( \dfrac{-100}{500} \right)$$

Question 1

Given the forces $$F_1 = 250$$\ \mathrm{lb}$$ at 30$$^\circ$$ above the x-axis and F_2$ = 375\ \mathrm{lb} at$$ $$45^\circ$$ below the x-axis, what is the x-component of the resultant force $$\vec{R}$$?

Accepted Answer

$$R_x = 250 \cos 30^\circ + 375 \cos 45^\circ$$

Question 2

What is the y-component of the resultant force $$\vec{R}$$ for the given system?

Accepted Answer

$$R_y = 250 \sin 30^\circ - 375 \sin 45^\circ$$

Question 3

Calculate the magnitude of the resultant force $$\vec{R}$$ using its components $$R_x$$ and $$R_y.$$

Accepted Answer

$$|\vec{R}| = \sqrt{R_x^2$ + R_y^2$}$$

Question 4

Which equation correctly gives the direction angle $$	heta of $$the resultant force $$\vec{R}$$ measured counterclockwise from the x-axis?

Accepted Answer

$$	heta = \arctan \left( \dfrac{R_y}{R_x} ight)$$

Question 5

If $$F_1$$ acts at $$30^\circ$$ above the x-axis and $$F_2 at$$ $$45^\circ$$ below the x-axis, which of the following best describes the direction of the resultant force $$\vec{R}$$?

Accepted Answer

The resultant will be closer to the x-axis than either $$F_1 or$$ $$F_2.$$

Question 6

What is the correct unit vector $$\hat{u}_R in $$the direction of the resultant force $$\vec{R}$$?

Accepted Answer

$$\hat{u}_R = \dfrac{\vec{R}}{|\vec{R}|}$$

Question 7

Suppose $$F_1$$ and $$F_2$$ are replaced by a single force $$\vec{R} at $$the origin. Which property must $$\vec{R}$$ have to be equivalent to the original system?

Accepted Answer

$$\vec{R}$$ must have the same magnitude and direction as the vector sum of $$F_1$$ and $$F_2.$$

Question 8

If the angle of $$F_1 is $$increased from $$30^\circ to$$ $$60^\circ ($$keeping its magnitude the same), what happens to the x-component of $$F_1$$?

Accepted Answer

It decreases.

Question 9

Which of the following best explains why vector addition is used to find the resultant force in this problem?

Accepted Answer

Because forces are vector quantities and must be added using both magnitude and direction.

Question 10

If the resultant force $$\vec{R}$$ has components $$R_x = 500\ \mathrm{lb}$$ and $$R_y = -100\ \mathrm{lb}$$, what is the direction angle $$	heta$$ measured from the x-axis?

Accepted Answer

$$	heta = \arctan \left( \dfrac{-100}{500} ight)$$

Question 11

A construction worker pulls on a cable with a force of $$250\ \mathrm{lb} at$$ $$30^\circ$$ above the x-axis. What is the vertical (y) component of this force?

Accepted Answer

$$250 \sin 30^\circ$$

Question 12

If $$F_2 is $$increased to $$500\ \mathrm{lb}$$ while keeping its angle the same, what effect does this have on the magnitude of the resultant force $$\vec{R}$$?

Accepted Answer

The magnitude of $$\vec{R}$$ increases.

Question 13

Which of the following is the correct method to find the resultant of two non-perpendicular forces acting at a point?

Accepted Answer

Resolve each force into its x and y components, sum the components, and then recombine.

Question 14

If the resultant force $$\vec{R} is $$found to be $$600\ \mathrm{lb} at$$ $$10^\circ$$ below the x-axis, what is the unit vector in the direction of $$\vec{R}$$?

Accepted Answer

$$\cos(-10^\circ)\hat{i} + \sin(-10^\circ)\hat{j}$$

Question 15

A force $$F$$ acts at an angle $$	heta$$ above the x-axis. Which formula gives the y-component of this force?

Accepted Answer

$$F_y = F \sin 	heta$$

Resultant Force: Magnitude, Direction, and Unit Vector

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