Collisions and Inertia

Imagine two balls. One elastic and one inelastic. The balls move towards a mass and the elastic one pushes it over. The inelastic ball is unable to push over the mass of wood. This is because it is an inelastic collision and the change in mechanical energy is not conserved. With the elastic collision, energy is conserved and returned to the bouncy ball.

Torque is Force * Radius. The larger the radius the more force needed to move an object over a distance. m and R are constants. RF (Force times radius) is torque. It is a force that goes in a circle. Torque is a measure of rotational inertia. The Movement of inertia is I=mR2.

  1. Radial Force equals mass times radius times acceleration, using angular acceleration Radial Force equals mass times radius squared times angular acceleration (which is equal to acceleration tangential over Radius)
  2. Magnitude of Torque equals magnitude of radius times magnitude of Force
  3. Torque equals mass times radius squared
  4. Inertia equals torque ie Mass times radius squared.
  5. Angular acceleration equals Torque over mass times radius squared.

    Now if we look at this in terms of Kinetic Energy, and imagine we have a bunch of masses rotating around a center of mass. Changing the center point will change the inertia of the system.

    1. Kinetic energy equals one half inertia times angular velocity squared.
    2. Inertia total equals sum of masses times Radius squared

    Moment of Inertia of a thin stick of length L and a total mass M, rotated about one end.

    So if we take a little mass from the thin stick, and call this little mass dm, which is x distance from the origin and we need to express dm as the running variable, x. We use a ratio of total length to total mass to do this, and then substitute it back into the derivative of dI. So The total inertia when the thin stick is rotated about one end is the sum of all those little dI’s, which is the integral of dI.

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