A 49.51 kg passenger is in a cabin of a Ferris wheel rotating in a vertical circular path with a constant speed of 14.65 m/s. At the bottom of the circular path, the passenger stands on an electronic balance which is calibrated to a gravitational field strength of 9.81 Nkg-1 . If the Ferris wheel has a radius of 78 m, calculate the reading of the balance in kg.

Question

BlackTom AI · Accepted Answer

Centripetal acceleration at the bottom is v^2/r, so the normal force (what the balance reads as a weight) must supply both the real weight and the upward net force to provide centripetal acceleration: N = m(g + v^2/r). The electronic balance reports mass by dividing the measured normal force by the calibration g (9.81 N kg^-1), so displayed mass = N / g.

1) Relevant concepts:
- Centripetal acceleration a_c = v^2 / r (directed toward centre).
- At the bottom of a vertical circle the normal force N = mg + m a_c.
- A balance calibrated in N kg^-1 displays mass = N / g.

2) Step-by-step calculation:
- Given: m = 49.51 kg, v = 14.65 m/s, r = 78 m, g = 9.81 N kg^-1.
- Compute v^2: 14.65^2 = 214.6225 m^2/s^2.
- Compute centripetal acceleration: a_c = v^2 / r = 214.6225 / 78 = 2.7515718 m/s^2 (≈ 2.7516 m/s^2).
- Sum accelerations for normal force: g + a_c = 9.81 + 2.7515718 = 12.5615718 m/s^2.
- Normal force: N = m (g + a_c) = 49.51 × 12.5615718 = 621.927 N (approximately).
- Displayed mass on balance = N / g = 621.927 / 9.81 = 63.3797 kg ≈ 63.4 kg.

3) Why this is correct:
At the bottom, the cabin occupant must be accelerated upward toward the circle centre; therefore the contact force (what the scale measures) exceeds mg by the amount m v^2/r. Converting that force into the balance’s mass units (dividing by 9.81 N/kg) yields the larger apparent mass.

4) Other options:
This is a numerical question with no alternative choices listed. Any lower value that ignored the centripetal term (i.e., simply 49.51 kg) would be incorrect; any value that added the wrong sign or used r incorrectly would also be wrong.

Conclusion: The balance reads approximately 63.4 kg, which matches the provided answer.

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