A 48.42 kg passenger is in a cabin of a Ferris wheel rotating in a vertical circular path with a uniform speed of 9.14 m/s. If the Ferris wheel has a radius of 80 m, find the normal reaction (in Newtons) exerted on the passenger by the cabin floor at the top of the circular path.

Question

BlackTom AI · Accepted Answer

To analyze the forces on the passenger at the top of the Ferris wheel, identify the directions: the center of the circular path is directly downward at that point, so the inward (centripetal) direction is downward.
First, list the forces acting on the passenger: gravity (weight) mg acts downward, and the normal reaction N from the cabin floor acts upward (opposite to the inward direction).
For circular motion, the sum of forces toward the center (inward) must equal m v^2 / r. Here, the inward direction is downward, so the downward forces are mg, while the upward normal N does not contribute to inward force; instead, it reduces the net inward force.
Thus, the equation for the inward (downward) direction is: mg − N = m v^2 / r.
Solving for N gives: N = mg − (m v^2) / r.
Now substitute the given values: m = 48.42 kg, g ≈ 9.8 m/s^2, v = 9.14 m/s, r = 80 m.
Calculate mg = 48.42 × 9.8 ≈ 474.52 N. Next, compute m v^2 / r: v^2 = 9.14^2 ≈ 83.54, so m v^2 / r ≈ (48.42 × 83.54) / 80 ≈ 50.56 N.
Subtract to find N: N ≈ 474.52 − 50.56 ≈ 423.96 N, which can be rounded to 424 N.
In summary, at the top, the normal reaction is slightly less than the weight because part of the weight provides the required centripetal force, with the remaining portion supported by the floor as the cabin floor.

A 48.42 kg passenger is in a cabin of a Ferris wheel rotating in a vertical circular path with a uniform speed of 9.14 m/s. If the Ferris wheel has a radius of 80 m, find the normal reaction (in Newtons) exerted on the passenger by the cabin floor at the top of the circular path.Numerical

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