# Solve Gyroscope Problem: Find Force & Angular Speed

• akozaro8
In summary, the rotor of a toy gyroscope has mass .130 kg, has a moment of inertia of 1.40 * 10^-4 kg·m^2, and is supported on a pivot with its center of mass 4.00 cm from the pivot. The gyroscope is precessing in a horizontal plane at the rate of one revolution in 2.50 s. The upward force exerted by the pivot is 1.91 N and the angular speed of the rotor is 2074.55 rev/min.
akozaro8

## Homework Statement

The rotor (flywheel) of a toy gyroscope has mass 0.130 kg. Its moment of inertia about its axis is 1.40 * 10^-4 kg·m^2. The mass of the frame is 0.0650 kg. The gyroscope is supported on a single pivot (Diagram: http://www.webassign.net/yf10/10-43.gif) with its center of mass a horizontal distance of 4.00 cm from the pivot. The gyroscope is precessing in a horizontal plane at the rate of one revolution in 2.50 s.

(a) Find the upward force exerted by the pivot.
(b) Find the angular speed with which the rotor is spinning about its axis, expressed in rev/min.

I know that I=(1/2)*M*R^2 so I found the radius of the rotor to be .0464 meters. I don't know where to go from there. Can anyone please help me with this?

What level of physics do you know? In particular, do you know what I mean by "Lagrangian"?

"Lagrangian" no I don't know. I'm taking AP Physics C in high school right now. If you don't know what it is, it's basic physics with a bit of calculus.

Ok then.

Since the center of mass of the gyroscope is not centered over the support, gravity will exert a torque on the gyroscope about the support. Can you determine what this torque is? What is the relevant equation relating a force and a torque?

Next set of questions relate to angular momentum, if you know what that is. Can determine the angular momentum of the gyroscope?

Well I know that torque = (lever arm) * (force)
So I set up my equation (for part a) and have torque = .04 * (.065+.13) * 9.8
Once I found my torque, though, I'm not sure how that helps me find the upward force exerted by the pivot.

Sorry, I thought that part was obvious. Gravity is pulling the gyroscope downwards and the pivot is pushing the top upwards. Since the gyroscope isn't moving at all vertically, it isn't accelerating vertically. What does this tell you about the upward force?

The torque will tell you about the precession. Getting back to the torque and angular momentum, (1) what is the angular momentum, and (2) what is the relevant equation relating angular momentum and torque?

I have torque=I*alpha and L=I*omega
Therefore I can say that torque/alpha = L/omega
Solving for omega, i get omega=(L*alpha)/torque
and since L=r*p and p=m*v, I get L=r*m*v
Also, I know that torque=r*F
Plugging the L and torque into the omega equation, I get omega = r^2*m*v*a/(r*F) which simplifies into omega=r*m*v*a/F

Now I'm confused as to which rotation this equation gets its variables from. The force I'm guessing is the upward force that the pivot exerts on the frame (I correctly found it to be 1.91 N). Omega is the angular velocity of the rotor around the frame. The radius is the given radius of orbit of the center of mass around the pivot. And i have no clue which velocity or acceleration to use. This whole paragraph is the part I am lost at.

I was hoping you might give me the rotational equivalent of Newton's second law,

$$\vec \tau = \frac{d\vec L} {dt}$$

Notice that the torque is at right angles to the angular momentum vector. When the time derivative of some vector is normal to the vector, the vector's length doesn't change but its direction does. It rotates or precesses. The precession is related to the torque by

$$\vec \tau = \vec \omega_p\times \vec L$$

You computed the torque, and you already know the precession rate. Since the vectors are at nice ninety degree angles, you can divide the torque by the precession rate to determine the angular momentum. From that point you should be able to compute the angular velocity.

If you continue in physics you will cover this subject at least two more times. When you learn about vector operations you will find out how to deal with cases where the gyroscope is at some angle other than horizontal. When you learn about the Lagrangian or Hamiltonian, you will learn even more.

Thanks a lot for the help, I really appreciate it. I correctly got (angular velocity of rotor around its axis) = (gravitational force of entire system)*(radius around pivot)/((given moment of interia)*(angular velocity around pivot)). My final answer turned out to be 2074.55 rev/min.
Thank you again for all the help.

## 1. What is a gyroscope and how does it work?

A gyroscope is a device that consists of a spinning wheel or disc that is mounted on an axis. It works by utilizing the principle of conservation of angular momentum, which means that an object will maintain its rotational motion unless acted upon by an external force. This allows a gyroscope to resist changes in its orientation and maintain a stable axis of rotation.

## 2. How do you find the force acting on a gyroscope?

The force acting on a gyroscope can be found using the equation F = ma, where F is the force, m is the mass of the gyroscope, and a is the acceleration of the gyroscope. The acceleration can be calculated using the equation a = ω^2r, where ω is the angular speed and r is the radius of the gyroscope. By plugging in the values for mass, angular speed, and radius, the force acting on the gyroscope can be determined.

## 3. What is the importance of solving gyroscope problems?

Solving gyroscope problems is important for understanding the behavior and stability of objects in motion. Gyroscopes are commonly used in navigation systems, aerospace technology, and engineering applications. By solving gyroscope problems, scientists and engineers can design and improve upon systems that rely on gyroscopes for accurate motion control.

## 4. What factors can affect the angular speed of a gyroscope?

The angular speed of a gyroscope can be affected by several factors, including the mass of the spinning wheel, the radius of the wheel, the force applied to the gyroscope, and external forces such as friction or air resistance. Additionally, the initial conditions and orientation of the gyroscope can also impact its angular speed.

## 5. How can you troubleshoot and solve a gyroscope problem?

To troubleshoot and solve a gyroscope problem, it is important to first understand the principles and equations that govern its motion. Then, the problem can be broken down into smaller parts and solved using mathematical equations and principles of physics. It may also be helpful to use visual aids, such as diagrams or models, to better understand the problem and potential solutions.

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