Inertia and Torque given Angular Velocity

In summary, the conversation involves a system consisting of a spinning rotor with a moment of inertia IR and outer radius R, a small constant friction torque τf, a string attached to the rotor and a weight hanging from a massless frictionless pulley. The system is released from rest and the rotor's angular velocity increases linearly at a rate α1 until the string detaches, after which the rotor's angular acceleration becomes α2. The goal is to find expressions for I0 and τf in terms of α1, α2, m, R, and g. There is some confusion about the equations involving inertia and torque.
  • #1
Lucidity
1
0

Homework Statement



A rotor of moment of inertia IR and outer radius R spins about a vertical axis. The rotor mount is not friction free. Rather, there is a small constant friction torque of magnitude τf . A string of negligible mass is wound around the outside of the rotor. The string is attached to the rotor in a way that allows it to come free when the end is reached. The other end of the string is attached to a weight of mass m hanging from a massless frictionless pulley. When the system is released from rest the angular velocity of the rotor increases linearly at a rate α1 under the influence of the tension in the string and the torque due to friction. After the string detaches from the rotor, the rotor’s angular acceleration becomes α2 (a negative quantity) due to the friction torque.

Find expressions for I0 and τf in terms of some or all of the quantities α1, α2, m, R, and g the acceleration of gravity.

Homework Equations


I=mR^2
τ=Iα

The Attempt at a Solution


For inertia, would it simply be mR^2? I remember something about a parallel line to the axis of rotation, involving a cross product and all of that... mR^2 seems too easy to be correct...
but if inertia = mR^2, would τ simply be mR^2(α1-α2)?
The τ equation doesn't sound right at all, and I'm somewhat confused on what to do here.

Thanks!
 
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  • #2
what is I0 ?
 
  • #3
OP, do you go to MIT by chance?
 

1. What is inertia and how does it relate to angular velocity?

Inertia is the resistance of an object to change its state of motion. In the case of angular velocity, inertia refers to the tendency of an object to maintain its rotational motion. The greater the inertia of an object, the more difficult it is to change its angular velocity.

2. How is torque related to angular velocity?

Torque is the measure of the force that causes an object to rotate. It is directly proportional to the angular velocity of an object, meaning that the greater the torque applied, the greater the resulting angular velocity will be.

3. How does angular velocity affect the stability of an object?

Angular velocity plays a key role in determining the stability of an object. The higher the angular velocity, the more stable the object will be. This is because objects with higher angular velocities have more momentum and are less likely to be affected by external forces.

4. Can angular velocity be changed without applying an external force?

No, angular velocity can only be changed by applying a torque or an external force. This is due to the law of inertia, which states that an object will maintain its state of motion unless acted upon by an external force.

5. How does the distribution of mass affect inertia and torque given angular velocity?

The distribution of mass affects both the inertia and torque of an object given a certain angular velocity. Objects with a greater mass are more difficult to change their state of motion, resulting in a higher inertia. Additionally, the distribution of mass affects the magnitude and direction of the torque, which in turn affects the resulting angular velocity.

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