Four forces act on a hot-air balloon with F1 = 1660 N and F2 = 5040 N , shown from the side in Figure 4-29. Find the magnitude and direction of the resultant force on the balloon.
N
° (counterclockwise from the horizontal)
990
To find the magnitude and direction of the resultant force on the balloon, we need to first determine the horizontal and vertical components of the forces.
Let's assume that the forces F1 and F2 are acting in the positive x-direction. Since the forces are shown from the side, we can assume that there are no vertical forces acting on the balloon. Therefore, the vertical component of the forces will be zero.
Now, let's find the horizontal components of the forces:
Horizontal component of F1 (F1x) = F1 * cos(0°) = 1660 N * cos(0°) = 1660 N
Horizontal component of F2 (F2x) = F2 * cos(0°) = 5040 N * cos(0°) = 5040 N
Now, let's find the vertical components of the forces:
Vertical component of F1 (F1y) = F1 * sin(0°) = 1660 N * sin(0°) = 0 N
Vertical component of F2 (F2y) = F2 * sin(0°) = 5040 N * sin(0°) = 0 N
Since the vertical components of both forces are zero, the net vertical force on the balloon is also zero.
Now, let's find the horizontal net force by adding the horizontal components of the forces:
Horizontal net force (Fx) = F1x + F2x = 1660 N + 5040 N = 6700 N
The magnitude of the net force is given by the Pythagorean theorem:
Magnitude of the net force (Fnet) = sqrt((Fx)^2 + (Fy)^2) = sqrt((6700 N)^2 + (0 N)^2) = sqrt(44,890,000 N^2) = 6700 N
So, the magnitude of the resultant force on the balloon is 6700 N.
Since the vertical component is zero and the horizontal component is positive, the direction of the resultant force is 0° (counterclockwise from the horizontal).
To find the resultant force on the hot-air balloon, we need to sum up the individual forces acting on it. In this case, we have two forces, F1 and F2.
To find the magnitude of the resultant force, we use the Pythagorean theorem. The Pythagorean theorem states that the square of the hypotenuse of a right triangle is equal to the sum of the squares of the other two sides.
So, let's calculate the magnitude of the resultant force using the Pythagorean theorem:
Magnitude of the Resultant force = sqrt(F1^2 + F2^2)
Magnitude of the Resultant force = sqrt(1660^2 + 5040^2)
After calculating this, we get the magnitude of the resultant force.
To determine the direction of the resultant force, we can use trigonometry. We can calculate the angle between the resultant force and the horizontal axis by using the inverse tangent (arctan) function.
Direction of the Resultant force = arctan(F2 / F1)
After calculating this, we get the direction of the resultant force.
Now you can plug in the values and calculate the magnitude and direction of the resultant force using the given values for F1 and F2.