Kinematics of a point on a vinyl record
Let’s consider a vinyl record rotating with a constant angular velocity \boldsymbol{\omega} = \omega \mathbf{k}_2. A point P moves radially outward with a constant velocity \mathbf{u} = u \mathbf{i}_2 relative to the record. My objective is to determine the velocity and acceleration of point P with respect to the ground (reference frame 1).
The velocity of point P with respect to reference frame 1, \mathbf{v}_{P/1}, is given by the following equation:
\mathbf{v}_{P/1} = \mathbf{v}_{O_2/1} + \mathbf{v}_{P/2} + \boldsymbol{\omega}_{2/1} \times \mathbf{r}_{P/O_2}
- \mathbf{v}_{O_2/1} is the velocity of the origin of the rotating frame (frame 2) relative to the ground (frame 1). Since I assume the center of the record is not moving, then \mathbf{v}_{O_2/1} = \mathbf{0},
- \mathbf{v}_{P/2} is the velocity of point P relative to frame 2, which is given by \mathbf{u} = u \mathbf{i}_2,
- \boldsymbol{\omega}_{2/1} is the angular velocity of frame 2 with respect to frame 1, \omega \mathbf{k}_2,
- \mathbf{r}_{P/O_2} is the position vector of point P with respect to the origin of frame 2, which is r \mathbf{i}_2.
Substituting these into the velocity equation, I obtain:
\mathbf{v}_{P/1} = u \mathbf{i}_2 + \omega \mathbf{k}_2 \times r \mathbf{i}_2 = u \mathbf{i}_2 + \omega r \mathbf{j}_2
The acceleration of point P with respect to frame 1, \mathbf{a}_{P/1}, is given by:
\mathbf{a}_{P/1} = \mathbf{a}_{O_2/1} + \mathbf{a}_{P/2} + \boldsymbol{\alpha}_{2/1} \times \mathbf{r}_{P/O_2} + 2 \boldsymbol{\omega}_{2/1} \times \mathbf{v}_{P/2} + \boldsymbol{\omega}_{2/1} \times (\boldsymbol{\omega}_{2/1} \times \mathbf{r}_{P/O_2})
- \mathbf{a}_{O_2/1} is the acceleration of the origin of frame 2 relative to frame 1, which is zero,
- \mathbf{a}_{P/2} is the acceleration of P with respect to frame 2, which is zero since the radial velocity u is constant,
- \boldsymbol{\alpha}_{2/1} is the angular acceleration of frame 2 relative to frame 1, which is zero, since \omega is constant.
The acceleration simplifies to:
\mathbf{a}_{P/1} = \omega u \mathbf{j}_2 - \omega^2 r \mathbf{i}_2
The acceleration \mathbf{a}_{P/1} has two terms:
- The term -\omega^2 r \mathbf{i}_2 is the centripetal acceleration, which points towards the center of rotation.
- The term 2 \omega u \mathbf{j}_2 is the Coriolis acceleration, which is perpendicular to the velocity \mathbf u and the rotation axis.
These terms account for the motion of the point P in both the rotating frame and relative to the ground.
For more insights into this topic, you can find the details here.