Two forces act concurrently on a point P. One force is 60nt due east. The second force is 80nt due north. (a) Find the magnitude and direction of their resultant. (b) What is the magnitude and direction of the equilibriant?
To find the magnitude and direction of the resultant force, we can use vector addition. We can calculate the magnitudes of the forces using the Pythagorean theorem and find the direction using trigonometric functions.
(a) Magnitude and direction of the resultant force:
Let's denote the 60 N force as F1 and the 80 N force as F2.
1. Determine the magnitude of the resultant force:
To find the magnitude of the resultant force, we use the Pythagorean theorem:
Resultant magnitude = √(F1^2 + F2^2)
= √(60^2 + 80^2)
= √(3600 + 6400)
= √10000
= 100 N.
2. Determine the direction of the resultant force:
To find the direction, we can use the inverse tangent function (tan^(-1)):
Resultant direction = tan^(-1)(F2 / F1)
= tan^(-1)(80 / 60)
≈ 53.13 degrees north of east.
Therefore, the magnitude of the resultant force is 100 N and its direction is approximately 53.13 degrees north of east.
(b) Magnitude and direction of the equilibrant force:
The equilibrant force is equal in magnitude but opposite in direction to the resultant force.
Magnitude of the equilibrant force = 100 N
Direction of the equilibrant force = Approximately 53.13 degrees south of west.
To find the magnitude and direction of the resultant force, we can use the Pythagorean theorem and trigonometry.
(a) Finding the Resultant Force:
1. Draw a diagram to visualize the forces. Place the 60N force in the east direction (right) and the 80N force in the north direction (up).
2. Create a right triangle using the two forces, where the hypotenuse represents the resultant force.
3. Use the Pythagorean theorem to find the magnitude of the resultant force:
resultant force = sqrt((60N)^2 + (80N)^2)
resultant force = sqrt(3600N^2 + 6400N^2)
resultant force = sqrt(10000N^2)
resultant force = 100N
So, the magnitude of the resultant force is 100N.
4. To find the direction of the resultant force, we can use trigonometry. Take the inverse tangent (arctan) of the opposite side (80N) divided by the adjacent side (60N):
direction = arctan(80N / 60N)
direction = arctan(4/3)
direction ≈ 53.13°
The direction of the resultant force is approximately 53.13° north of east.
(b) Finding the Equilibrant Force:
The equilibrant force is a force that can balance out the combined effect of the given forces. It is equal in magnitude, but opposite in direction to the resultant force.
1. Since the resultant force has a magnitude of 100N and a direction of 53.13° north of east, the equilibrant force will have the same magnitude (100N) but will be 180° opposite in direction.
Therefore, the direction of the equilibrant force is approximately 53.13° south of west.
Hence, the magnitude and direction of the resultant force are 100N and approximately 53.13° north of east, respectively. The magnitude and direction of the equilibrant force are also 100N and approximately 53.13° south of west, respectively.