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

Weighing - 140Kg

This disc is fixed to a 75mm diameter shaft.

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

To calculate the radial force generated by gyroscopic precession, we need to use the formula that relates angular velocity, moment of inertia, and the rate of change of angular momentum. Gyroscopic forces occur due to the changing orientation of a spinning body, and they are related to the rotational speed and the geometry of the disc.

### Steps:
1. **Moment of Inertia of the Disc (I):**
   The moment of inertia of a solid disc rotating about its central axis is given by:

   \[
   I = \frac{1}{2} M R^2
   \]

   Where:
   - \( M \) is the mass of the disc (140 kg)
   - \( R \) is the radius of the disc, which is half the diameter (600 mm = 0.6 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 in radians per second can be found from the given RPM (850 RPM). We convert RPM to rad/s:

   \[
   \omega = \frac{850 \, \text{rpm} \times 2 \pi}{60} = 88.97 \, \text{rad/s}
   \]

3. **Precessional Angular Velocity (\( \Omega \)):**
   The precessional angular velocity depends on how fast the axis of rotation is changing orientation. If you have a specific precession rate, we can denote it as \( \Omega \). However, without more detail on how the disc is being tilted or the angular rate of precession, we cannot compute a numerical value for the gyroscopic torque or force yet.

4. **Gyroscopic Torque (T):**
   The torque due to gyroscopic precession is calculated as:

   \[
   T = I \omega \Omega
   \]

   Where \( \Omega \) is the precessional angular velocity. Once \( \Omega \) is known, you can compute the gyroscopic torque \( T \).

5. **Radial Force (F):**
   The radial force exerted by gyroscopic precession on the shaft can be related to the torque by:

   \[
   F = \frac{T}{r}
   \]

   Where \( r \) is the radius of the shaft (half of the 75 mm diameter = 37.5 mm = 0.0375 m).

   Therefore:

   \[
   F = \frac{I \omega \Omega}{r}
   \]

   After calculating the precessional velocity \( \Omega \), you can substitute the values of \( I \), \( \omega \), and \( r \) to get the force.

---

Since the precessional angular velocity \( \Omega \) is not given, this is as far as we can go without more information on how the axis of the disc is changing its orientation. If you know the rate of precession, we can continue the calculation.

Would you like help calculating further, or more details on this concept?

Here are 5 follow-up questions:
1. What is the precession rate of the disc in terms of angular velocity?
2. Are there any external factors contributing to the disc's precession, like vibrations?
3. How is the shaft mounted, and is it subject to external torques?
4. Are you analyzing this in the context of a specific mechanical system (e.g., a gyroscope)?
5. How does the material or flexibility of the shaft affect the precessional force?

**Tip:** For rotating systems, it's always important to consider the material properties of the shaft as it could bend under gyroscopic forces, affecting the overall dynamics.

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

Learn how to calculate the radial force generated by gyroscopic precession on a shaft for a spinning disc with a diameter of 1200mm and weight of 140kg at 850 RPM. This detailed step-by-step guide covers key concepts like moment of inertia, angular velocity, and gyroscopic torque, suitable for engineering and physics students.

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