Two forces with magnitude of 2,000 pounds and 900 pounds act on a machine part at 105° and 30°, respectively. Find the direction and magnitude of the resultant of these two forces.
A),900 pounds at 83.7°
B)2,900 pounds at 135°
C)1,100 pounds at 75°
D)2,396.2 pounds at 83.7°

Question

Quizplus · Accepted Answer

To find the resultant of two forces, we can use the law of cosines.
Let F1 be the force of magnitude 2,000 pounds at an angle of 105°, and let F2 be the force of magnitude 900 pounds at an angle of 30°.

To find the resultant R, we have:
R^2 = F1^2 + F2^2 + 2(F1)(F2)(cos(theta))
where theta is the angle between the two forces.
theta = 105° - 30° = 75°
R^2 = (2000)^2 + (900)^2 + 2(2000)(900)(cos(75°))
R ≈ 2,396.2 pounds

To find the direction of the resultant, we can use the law of sines:
sin(alpha) / F1 = sin(theta) / R
alpha = arcsin((sin(theta) / R) * F1)
alpha ≈ 83.7°

Therefore, the magnitude and direction of the resultant are approximately 2,396.2 pounds at 83.7°, which corresponds to choice D.

Two Forces with Magnitude of 2,000 Pounds and 900 Pounds