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)
Surely you do not expect anyone to understand this without a figure.
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To find the magnitude and direction of the resultant force on the balloon, we need to add the two forces vectorially. We can use the Pythagorean theorem to find the magnitude and trigonometry to find the direction.
1. Start by drawing a diagram of the forces acting on the balloon. Label F1 as 1660 N and F2 as 5040 N.
2. Add the vectors by placing the tail of F2 at the head of F1. The line connecting the tail of F1 to the head of F2 represents the resultant vector R.
3. Use the Pythagorean theorem to find the magnitude of the resultant force R. The formula is:
R = √(F1^2 + F2^2)
= √((1660 N)^2 + (5040 N)^2)
≈ 5283.25 N (rounded to two decimal places)
4. Next, use trigonometry to find the direction of the resultant force R. The formula is:
θ = tan^(-1)(F1/F2)
= tan^(-1)((1660 N)/(5040 N))
≈ 18.21° (rounded to two decimal places)
5. The direction is counterclockwise from the horizontal, so the final answer is 18.21° counterclockwise from the horizontal.
Therefore, the magnitude of the resultant force on the balloon is approximately 5283.25 N, and its direction is 18.21° counterclockwise from the horizontal.