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7: General Rotational Motion

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    • 7.1: Linear and Angular Velocity
      We related the linear and angular velocities of a rotating object in two dimensions in Section 5.1. There, we also already stated the relation between the linear velocity vector and rotation vector in three dimensions. It is not hard to see that this expression indeed simplifies to the scalar relationship for rotations in a plane, with the right sign for the linear velocity.
    • 7.2: Rotating Reference Frames
      In this section, we’ll consider a rotating reference frame, where instead of co-moving with a linear velocity, we co-rotate with a constant angular velocity. Rotating reference frames are not inertial frames, as to keep something rotating (and thus change the direction of the linear velocity) requires the application of a net force. Instead, as we’ll see, in a rotating frame of reference we’ll get all sorts of fictitious forces - forces that have no real physical source, like gravity or electros
    • 7.3: Rotations About an Arbitrary Axis
      In Chapter 5, we studied the rotation of rigid bodies about an axis of symmetry. For these cases, we have \(\boldsymbol{L} = I \boldsymbol{\omega}\), where I is the moment of inertia with respect to the rotation axis. In this section, we’ll derive the more general form, in which the number I is replaced by a 2-tensor, i.e., a map from a vector space (here \(\mathbb{R}^{3}\)) into itself, represented by a 3×3 matrix.
    • 7.E: General Rotational Motion (Exercises)

    Thumbnail: A gyroscope is a device used for measuring or maintaining orientation and angular velocity. It is a spinning wheel or disc in which the axis of rotation (spin axis) is free to assume any orientation by itself. When rotating, the orientation of this axis is unaffected by tilting or rotation of the mounting, according to the conservation of angular momentum. (Public Domain; LucasVB).

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