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 find the rate of precession, we need to use the concept of angular momentum.
First, let's find the angular momentum of the rotating disk. The formula for angular momentum (L) is given by:
L = Iω
Where:
L is the angular momentum
I is the moment of inertia
ω is the angular velocity
The moment of inertia of a uniform disk rotating about an axis perpendicular to its plane is given by:
I = (1/2) * m * r^2
Where:
m is the mass of the disk
r is the radius of the disk
Given that the radius of the disk (r) is 60 cm, we need to convert it to meters by dividing by 100:
r = 60 cm / 100 = 0.6 m
The problem does not provide the mass of the disk, but it is not required to find the rate of precession. We can simplify the problem by assuming that the mass cancels out in the final calculation.
Next, we need to find the angular velocity (ω) of the spinning disk. The problem states that the spin rate is 1540 rev/min. To convert this to radians/second, we can use the conversion factor 1 rev = 2π radians:
ω = 1540 rev/min * (2π radians/rev) * (1 min/60 s)
= 1540 * 2π / 60 radians/s
≈ 161.74 radians/s
Now, we can calculate the angular momentum (L) of the disk:
L = (1/2) * m * r^2 * ω
L = (1/2) * 0.6^2 * 161.74
L ≈ 29.12 kg·m²/s
Since the gyroscope is mounted so that it can spin and precess freely, the angular momentum of the spinning disk is conserved. Therefore, the rate of precession (Ω) can be determined by dividing the angular momentum by the moment of inertia of the gyroscope system:
Ω = L / I_gyro
The moment of inertia of the gyroscope system (I_gyro) is the sum of the moment of inertia of the disk (I_disk) and the moment of inertia of the axle (I_axle). The moment of inertia of a thin rod rotating about one end is given by:
I_axle = (1/3) * m * L_axle^2
Where:
m is the mass of the axle
L_axle is the length of the axle
Given that the length of the axle (L_axle) is 15 cm, we convert it to meters:
L_axle = 15 cm / 100 = 0.15 m
Assuming negligible mass for the axle, we simplify the equation as:
I_axle = 0
Therefore, the moment of inertia of the gyroscope system (I_gyro) is simply equal to the moment of inertia of the disk (I_disk):
I_gyro = I_disk = (1/2) * m * r^2
Now we can plug in the values to find the rate of precession:
Ω = L / I_gyro
Ω = 29.12 / [(1/2) * m * r^2]
Since the problem does not provide the mass of the disk, we can cancel it out in the final calculation:
Ω = 29.12 / [(1/2) * r^2]
Substituting the value for r:
Ω = 29.12 / [(1/2) * 0.6^2]
Ω ≈ 120.3 rad/s
Therefore, the rate of precession of the gyroscope is approximately 120.3 radians per second.