Three forces with magnitudes 75 pound, 100 lb, and 125 pound act on an object at angles of 30 degrees, -45 degrees, and 120 degrees respectively. Find the direction and magnitude of the resultant vector.
convert each to (x,y) coordinates
add them up
convert result back to speed and direction
75 @ 30° = (64.95,37.50)
...
To find the direction and magnitude of the resultant vector, we need to use vector addition.
First, we will break down each force into its horizontal and vertical components. We can use trigonometry to do this.
For the force of 75 pounds at an angle of 30 degrees:
Horizontal component = 75 * cos(30) = 64.95 pounds
Vertical component = 75 * sin(30) = 37.5 pounds
For the force of 100 pounds at an angle of -45 degrees:
Horizontal component = 100 * cos(-45) = 70.71 pounds
Vertical component = 100 * sin(-45) = -70.71 pounds
For the force of 125 pounds at an angle of 120 degrees:
Horizontal component = 125 * cos(120) = -62.5 pounds
Vertical component = 125 * sin(120) = 108.25 pounds
Now, we can add up all the horizontal and vertical components separately to get the total horizontal and vertical components:
Total horizontal component = 64.95 + 70.71 - 62.5 = 73.16 pounds
Total vertical component = 37.5 - 70.71 + 108.25 = 75.04 pounds
To find the magnitude of the resultant vector, we can use the Pythagorean theorem:
Magnitude of the resultant vector = sqrt((total horizontal component)^2 + (total vertical component)^2)
= sqrt((73.16)^2 + (75.04)^2)
= sqrt(5353.1556 + 5625.0016)
= sqrt(10978.1572)
= 104.77 pounds (approximately)
To find the direction of the resultant vector, we can use inverse trigonometric functions:
Direction of the resultant vector = atan(total vertical component / total horizontal component)
= atan(75.04 / 73.16)
= atan(1.0268)
= 47.65 degrees (approximately)
Therefore, the direction of the resultant vector is approximately 47.65 degrees, and the magnitude of the resultant vector is approximately 104.77 pounds.