A demonstration gyroscope consists of a uniform disk with a 39-cm radius, mounted at the midpoint of a light 56-cm axle. The axle is supported at one end while in a horizontal position. How fast is the gyroscope precessing, in units of rad/s, if the disk is spinning around the axle at 34 rev/s?
The equation you need can be found at
http://physics-animations.com/Physics/English/gyro_txt.htm
The mass will cancel out when you insert (1/2)M R^2 for the moment of inertia.
r in the reference above is the distance from the support point to the rotor. wp is the precession rate that you want
what radius should we use again?
To find the precession rate of the gyroscope, we need to relate the rotational speed of the disk to the precession speed.
First, let's find the angular speed of the disk in radians per second. We are given that the disk is spinning at 34 revolutions per second. Since 1 revolution is equal to 2π radians, the angular speed of the disk is:
Angular speed of the disk = 34 rev/s * 2π rad/rev
Next, let's calculate the moment of inertia of the disk. The moment of inertia of a uniform disk rotating about its axis is given by the formula:
Moment of inertia (I) = (1/2) * mass * radius^2
Since the disk is uniform, we can also write the moment of inertia as:
Moment of inertia (I) = (1/2) * mass * radius^2 = (1/2) * density * volume * radius^2
But since the density, volume, and mass are not provided, we can simplify the calculation by assuming the mass is 1 (unit mass). So the moment of inertia simplifies to:
Moment of inertia (I) = (1/2) * radius^2
Now, let's find the torque exerted on the gyroscope due to its weight. The torque is given by the formula:
Torque (τ) = moment of inertia (I) * angular acceleration (α)
In this case, since the gyroscope is precessing at a constant speed, the angular acceleration is zero (α = 0). Hence, the torque exerted by the weight is also zero (τ = 0).
Now, let's relate the angular speed of the disk (ω) to the precession rate (ω_p) using the formula for torque:
τ = I * α
Since τ = 0, we have:
0 = I * α
So, the angular acceleration (α) is zero.
Finally, the precession rate (ω_p) is given by the formula:
ω_p = α / Ω
where Ω is the angular speed of the disk. Since α = 0, the precession rate (ω_p) is also zero.
Therefore, the gyroscope is not precessing, and the precession rate is zero rad/s.