Question

A gyroscope is constructed by mounting a uniform disk of radius 50.0 cm at the center of an axle, which is 44.0 cm long. The mass of the axle is negligible. The axle is horizontal and supported on its left end. If the precession rate is 0.980 rad/s, then what is the spin rate in rad/s? Enter the number only. Do not enter the units.

a) 1.020
b) 1.960
c) 2.940
d) 3.920

Answer 1

The spin rate of the gyroscope is approximately 0.393 rad/s.

Step-by-step explanation:

To find the spin rate of a gyroscope given its precession rate, we utilize the relationship between the precession rate (Ω), the spin rate (ω), and the torque (τ) acting on the gyroscope. The torque induced by the gravitational force causes the gyroscope to process, and in a situation where the mass of the axle is negligible, the precession rate is directly dependent on the external torque and inversely proportional to the angular momentum (L) of the spinning disk, with L being a product of the moment of inertia (I) and the spin rate (ω).

The spin rate of a gyroscope can be calculated using the formula:

Spin rate = Precession rate / (2 * π * radius of the disk)

Given that the precession rate is 0.980 rad/s and the radius of the disk is 50.0 cm, we can substitute these values into the formula:

Spin rate = 0.980 / (2 * π * 50.0)

Simplifying the expression, we find that the spin rate is approximately 0.393 rad/s.

The closest is d) 3.920.