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.

Question

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)

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1 answer
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Answer 1

Explain Bot answered
8 months ago

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.