Question

In a charming 19th-century hotel, an old-style elevator is connected to a counterweight by a cable that passes over a rotating disk 2.9m

2.9m in diameter. The elevator is raised and lowered by turning the disk, and the cable does not slip on the rim of the disk but turns with it. At how many rpm

rpm(revolutions per minute) must the disk turn to raise the elevator at 14.7

14.7 cm/s?

Answer 1

The length of the disk's circumference is:

$$l = 2\pi R$$

where $R = 2.9m/2 = 1.45m$. In order to make one revolution, it must pass the rope of length $l$ through itself. Thus, if the elevator was raised by $L = 14.7cm=0.147m$ per second the number of turn per second required for this is:

$$N = \dfrac{L}{l} = \dfrac{L}{2\pi R}\\ N = \dfrac{0.147m}{2\pi \cdot 1.45m} \approx 0.01614$$

Then the number of revolutions per minute is:

$$N\cdot 60 \approx 0.97$$

Answer. 0.97 revolutions per minute.