Three forces act on a moving object. One force has a magnitude of 76.5 N and is directed due north. Another has a magnitude of 46.3 N and is directed due west. What must be (a) the magnitude and (b) the direction of the third force, such that the object continues to move with a constant velocity? Express your answer as a positive angle south of east.
Find the resultant of the two perpendicular forces
|F| = sqrt (76.5^2+46.3^2)
tan A = 76.5/46.3
where A is angle north of west
our force is equal and opposite
same |F|
same angle but south of east
To find the magnitude and direction of the third force, we need to consider the net force acting on the object.
Since the object is moving with a constant velocity, the net force acting on it must be zero.
The northward force has a magnitude of 76.5 N and is directed due north.
The westward force has a magnitude of 46.3 N and is directed due west.
To find the magnitude of the third force, we need to find the sum of the magnitudes of the northward and westward forces:
Magnitude of the third force = √(76.5^2 + 46.3^2) N
Now let's find the direction of the third force.
The angle south of east can be calculated using trigonometry.
Angle south of east = tan^(-1)(46.3 / 76.5)
Substitute these values into a calculator to find:
Magnitude of the third force ≈ 89.2 N
Angle south of east ≈ 32.8 degrees
Therefore, the magnitude of the third force is approximately 89.2 N, and the direction is approximately 32.8 degrees south of east.
To determine the magnitude and direction of the third force, we need to consider the net force acting on the object. If the object is moving with a constant velocity, it means that the net force acting on it is zero.
To find the net force, we can use vector addition to combine the two given forces. Since the forces are acting at right angles to each other, we can use the Pythagorean theorem to find the magnitude of the net force:
Net force (F_net) = √(Force1^2 + Force2^2)
F_net = √(76.5^2 + 46.3^2)
F_net ≈ 89.5 N
Now that we know the magnitude of the net force, we can find the direction. To find the angle of the net force south of east, we can use trigonometry. We can use the inverse tangent function to find the angle:
tan(theta) = Force2 / Force1
theta = atan(Force2 / Force1)
theta = atan(46.3 / 76.5)
theta ≈ 30.9 degrees
Since the angle is south of east, we subtract the result from 180 degrees:
Angle = 180 - theta
Angle = 180 - 30.9
Angle ≈ 149.1 degrees
Therefore, the magnitude of the third force must be approximately 89.5 N, and the direction should be a positive angle of approximately 149.1 degrees south of east.