Four forces act on a hot-air balloon, shown from the side in the figure below. Find the magnitude and direction of the resultant force on the balloon if F1 = 5050 N, F2 = 1450 N, F3 = 950 N, and F4 = 4200 N.
_______ N at ________ ° clockwise from F2
To find the magnitude and direction of the resultant force on the balloon, you need to calculate the vector sum of the four forces.
First, let's calculate the horizontal and vertical components of each force.
F1: Fx1 = 0 (This force acts vertically)
Fy1 = -5050 N (Negative because it acts downward)
F2: Fx2 = 1450 N (Positive because it acts to the right)
Fy2 = 0 (This force acts horizontally)
F3: Fx3 = 950 N * cos(180°) = -950 N (Negative because it acts to the left)
Fy3 = 950 N * sin(180°) = 0 (This force acts vertically)
F4: Fx4 = 4200 N * cos(90°) = 0 (This force acts horizontally)
Fy4 = 4200 N * sin(90°) = 4200 N (Positive because it acts upward)
Now, let's calculate the resultant force components:
Rx = Fx1 + Fx2 + Fx3 + Fx4
= 0 + 1450 N - 950 N + 0
= 500 N
Ry = Fy1 + Fy2 + Fy3 + Fy4
= -5050 N + 0 + 0 + 4200 N
= -850 N
The magnitude of the resultant force (R) can be calculated using the Pythagorean theorem:
R = sqrt(Rx^2 + Ry^2)
= sqrt((500 N)^2 + (-850 N)^2)
≈ 990.37 N
To find the direction of the resultant force, we can use the inverse tangent function:
θ = atan(Ry / Rx)
= atan((-850 N) / 500 N)
≈ -58.66°
Since the question asks for the direction clockwise from F2, we need to calculate the angle between the resultant force and F2:
Θ = 180° - θ
= 180° - (-58.66°)
≈ 238.66°
Therefore, the magnitude of the resultant force is approximately 990.37 N at 238.66° clockwise from F2.