I have a disc 1200mm diameter, 16mm thick, spinning at 850rpm.

Weighing 140Kgs

This disc is fixed to a 75mm long shaft.

What radial force is generated by gyroscopic precession on the shaft?

To calculate the radial force generated by **gyroscopic precession** on the shaft, we need to consider the properties of the disc and the rotational motion involved.

### Gyroscopic Precession
When a spinning disc is subjected to an angular change (for example, a change in direction of the axis of rotation), it experiences **gyroscopic precession**, which generates a **gyroscopic couple**. The force due to precession depends on several factors, including the angular velocity of the disc, its moment of inertia, and the rate of change of the axis direction.

The gyroscopic couple \( C \) is given by the formula:

\[
C = I \cdot \omega \cdot \Omega
\]

Where:
- \( I \) is the **moment of inertia** of the disc.
- \( \omega \) is the **angular velocity** of the disc (spin speed in radians per second).
- \( \Omega \) is the **angular velocity of precession** (rate of change of the direction of the spin axis).

### 1. **Moment of Inertia** of the disc \( I \):
For a solid disc, the moment of inertia about its axis of rotation is:

\[
I = \frac{1}{2} m r^2
\]

Where:
- \( m \) is the mass of the disc (140 kg).
- \( r \) is the radius of the disc, \( r = \frac{1200 \, \text{mm}}{2} = 0.6 \, \text{m} \).

\[
I = \frac{1}{2} \times 140 \, \text{kg} \times (0.6 \, \text{m})^2 = 25.2 \, \text{kg} \cdot \text{m}^2
\]

### 2. **Angular Velocity \( \omega \)**:
The angular velocity \( \omega \) in radians per second is calculated from the rotational speed in RPM (850 rpm):

\[
\omega = 2\pi \times \frac{850}{60} \, \text{rad/s} = 88.96 \, \text{rad/s}
\]

### 3. **Gyroscopic Couple \( C \)**:
To compute the gyroscopic couple, we also need the **angular velocity of precession** \( \Omega \). Since this depends on the rate at which the axis of rotation is changing, and this value was not provided, I will assume you need the relationship up to this point.

\[
C = 25.2 \times 88.96 \times \Omega = 2240.6 \times \Omega \, \text{Nm}
\]

The gyroscopic couple depends directly on the precession rate \( \Omega \). If you provide this rate or if you want to explore further aspects of this force, we can adjust the result.

Would you like to discuss the next steps or need more clarification on the precession rate?

### Relative Questions:
1. How does the gyroscopic precession affect stability in rotating systems?
2. Can we calculate the gyroscopic couple for different materials with varying densities?
3. How would the radial force change with an increase in disc diameter?
4. What are practical ways to reduce gyroscopic precession in mechanical systems?
5. How does the shaft length (75 mm in this case) influence the overall precession force?

### Tip:
Always ensure that your angular velocity of precession is measured or estimated accurately when calculating gyroscopic effects, as small errors in this value can lead to significant differences in force estimates.

I have a disc 1200mm diameter, 16mm thick, spinning at 850rpm. Weighing 140Kgs. This disc is fixed to a 75mm long shaft. What radial force is generated by gyroscopic precession on the shaft?

Explore how to calculate the radial force generated by gyroscopic precession on a 1200mm diameter, 140kg disc spinning at 850 RPM. Learn about gyroscopic precession, moment of inertia, and angular velocity in this detailed guide suitable for undergraduate students.

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