Four forces act on a hot-air balloon with F1 = 1660 N and F2 = 5080 N , shown from the side in Figure 4-29. Find the magnitude and direction of the resultant force on the balloon.
a)N
b)° (counterclockwise from the horizontal)
To find the magnitude and direction of the resultant force on the balloon, we first need to calculate the horizontal and vertical components of the forces.
Given:
F1 = 1660 N (force 1)
F2 = 5080 N (force 2)
1. Calculate the horizontal components:
F1x = F1 * cos(θ1)
F2x = F2 * cos(θ2)
2. Calculate the vertical components:
F1y = F1 * sin(θ1)
F2y = F2 * sin(θ2)
3. Calculate the total horizontal and vertical components:
Fx = F1x + F2x
Fy = F1y + F2y
4. Calculate the magnitude of the resultant force using Pythagoras' theorem:
R = √(Fx^2 + Fy^2)
5. Calculate the direction of the resultant force using trigonometry:
θ = tan^(-1)(Fy / Fx)
Now let's calculate the magnitude and direction of the resultant force step by step:
Step 1: Calculate the horizontal components:
θ1 = 0° (assuming force 1 is horizontal)
θ2 = 180° (assuming force 2 is horizontal in the opposite direction)
F1x = F1 * cos(0°) = 1660 * cos(0°) = 1660 * 1 = 1660 N (horizontal component of force 1)
F2x = F2 * cos(180°) = 5080 * cos(180°) = 5080 * (-1) = -5080 N (horizontal component of force 2, negative because it acts in the opposite direction)
Step 2: Calculate the vertical components:
θ1 = 0° (assuming force 1 is vertical)
θ2 = 180° (assuming force 2 is vertical in the opposite direction)
F1y = F1 * sin(0°) = 1660 * sin(0°) = 1660 * 0 = 0 N (vertical component of force 1)
F2y = F2 * sin(180°) = 5080 * sin(180°) = 5080 * 0 = 0 N (vertical component of force 2)
Step 3: Calculate the total horizontal and vertical components:
Fx = F1x + F2x = 1660 N + (-5080 N) = -3420 N (total horizontal component)
Fy = F1y + F2y = 0 N + 0 N = 0 N (total vertical component)
Step 4: Calculate the magnitude of the resultant force:
R = √(Fx^2 + Fy^2) = √((-3420 N)^2 + (0 N)^2) = √(11696400 N^2) = 3417.9 N (rounded to the nearest tenth)
Step 5: Calculate the direction of the resultant force:
θ = tan^(-1)(Fy / Fx) = tan^(-1)(0 N / -3420 N) = tan^(-1)(0) = 0°
Therefore, the magnitude of the resultant force on the balloon is approximately 3417.9 N and the direction is 0° counterclockwise from the horizontal.