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 2.50 m in diameter (fig. e9.18). 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. (a) at how many rpm must the disk turn to raise the elevator at 25.0 cm>s? (b) to start the elevator moving, it must be accelerated at 1 8 g. what must be the angular acceleration of the disk, in rad>s 2 ? (c) through what angle (in radians and degrees) has the disk turned when it has raised the elevator 3.25 m between floors?

Answer 1

To raise the elevator at a speed of 25.0 cm/s, the disk must turn at a speed of 3.18 RPM. The angular acceleration of the disk is 0.71 rad/s^2. The angle through which the disk has turned when it has raised the elevator 3.25 m between floors is 1.30 radians.

Step-by-step explanation:

(a) To determine the RPM at which the disk must turn to raise the elevator at 25.0 cm/s, we can use the formula:

V = πDn,

Where V is the linear velocity of the elevator, D is the diameter of the rotating disk, and n is the RPM of the disk. Rearranging the formula, we get:

n = V / (πD),

Substituting the given values into the formula, we have:

n = (25.0 cm/s) / (π * 2.50 m) * (100 cm/m)

= 3.18 RPM.

(b) To find the angular acceleration of the disk, we can use the formula:

α = a / r,

Where α is the angular acceleration, a is the linear acceleration of the elevator, and r is the radius of the rotating disk. Substituting the given values into the formula, we have:

α = (1.8 g) / (2.50 m) * (9.81 m/s^2/g)

= 0.71 rad/s^2.

(c) To find the angle through which the disk has turned when it has raised the elevator 3.25 m between floors, we can use the formula:

θ = s / r,

Where θ is the angle in radians, s is the height raised by the elevator, and r is the radius of the rotating disk. Substituting the given values into the formula, we have:

θ = (3.25 m) / (2.50 m)

= 1.30 radians.