A gyroscope consists of a rotating uniform disk with a 60-cm radius, suitably mounted at one end of a 15-cm-long axle (of negligible mass) so that the gyroscope can spin and precess freely. Its spin rate is 1540-rev/min. What is the rate of precession if the axle is supported at the other end and is horizontal?
To determine the rate of precession of the gyroscope, we can use the conservation of angular momentum.
The angular momentum of a spinning gyroscope is given by the equation:
L = Iω
Where L is the angular momentum, I is the moment of inertia, and ω is the angular velocity.
In this case, the gyroscope is rotating at a spin rate of 1540 revolutions per minute. To convert this to radians per second, we can use the following conversion factors:
1 revolution = 2π radians
1 minute = 60 seconds
So, the angular velocity is:
ω = (1540 rev/min) * (2π rad/rev) * (1 min/60 s)
= (1540 * 2π) / 60 rad/s
≈ 160.57 rad/s
The moment of inertia of a uniform disk rotating around its axis is given by the equation:
I = (1/2) * m * r^2
Where m is the mass and r is the radius.
In this case, the radius of the disk is 60 cm, or 0.6 meters. However, we need to convert it to kilograms, so we multiply by the density of the material of the disk. Assuming it is made of a material with a density of ρ = 1 g/cm^3 = 1000 kg/m^3 (e.g., aluminum), the mass can be calculated as:
m = ρ * V
= ρ * (π * r^2 * h)
= 1000 kg/m^3 * (π * (0.6 m)^2 * 0.1 m)
≈ 113.1 kg
Substituting these values into the equation for the moment of inertia, we get:
I = (1/2) * (113.1 kg) * (0.6 m)^2
≈ 20.37 kg*m^2
Finally, the rate of precession can be calculated using the equation:
ω_p = (M * g) / (L * ω)
Where ω_p is the rate of precession, M is the mass of the gyroscope, g is the acceleration due to gravity, L is the length of the axle, and ω is the angular velocity.
In this case, the axle is horizontal and supported at one end, so L is given as 15 cm, or 0.15 meters.
Substituting the values into the equation, we get:
ω_p = (113.1 kg * 9.8 m/s^2) / (20.37 kg*m^2 * 160.57 rad/s)
≈ 0.031 rad/s
Therefore, the rate of precession of the gyroscope is approximately 0.031 radians per second.