kinematics

Rare Kinematic Formula

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Here we present some useful kinematic fact which is uncommon for textbooks in mechanics.
Consider a convex rigid body (RB) rolling without slipping on a fixed plane. (The plane can actually be replaced with a some other  fixed surface.)
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In the picture RB is a filled with dots oval .  At a given moment of time RB contacts with the plane by a point ##P##. The point ##P## belongs to RB.  So  in different moments of time we denote by  ##P##  the different points of  RB. Let ##A## stand for the point of contact, the point ##A## draws a curve on the plane as RB rolls.
Theorem. ##\boldsymbol a_P=-\boldsymbol\omega\times\boldsymbol v_A.##
Here ##\boldsymbol a_P## is the acceleration of the point ##P##; ##\boldsymbol v_A## is the velocity of the point ##A##; ##\boldsymbol\omega## is the angular velocity of RB.
Proof. Introduce a coordinate frame (say  ##Oxyz##) which is connected with RB. With respect to this frame we will consider relative and transport velocities and accelerations. By well-known formula we have
$$\boldsymbol v_A=\boldsymbol v_A^e+\boldsymbol v_A^r,$$ where by superscripts ##e,r## we denote transport velocity and relative velocity respectively. Since RB does not slip it follows that ##\boldsymbol v_A^e =0## and thus
$$\boldsymbol v_A=\boldsymbol v_A^r.\qquad (**)$$
Differentiate the last equality in time
$$\boldsymbol a_A=\frac{\delta}{\delta t}\boldsymbol v_A^r+\boldsymbol \omega\times \boldsymbol v_A^r=\boldsymbol a_A^r+\boldsymbol \omega\times \boldsymbol v_A^r,\qquad (*)$$
here ##\frac{\delta}{\delta t}## stands for derivative relatively the frame ##Oxyz##. On the other hand there is  standard formula
$$\boldsymbol a_A=\boldsymbol a_A^r+\boldsymbol a_A^e+2\boldsymbol \omega\times \boldsymbol v_A^r.$$
Observe that ##\boldsymbol a_A^e=\boldsymbol a_P## and by formula (**) we obtain
$$\boldsymbol a_A=\boldsymbol a_A^r+\boldsymbol a_P+2\boldsymbol \omega\times \boldsymbol v_A.$$
Combining the last formula with (*) we get the assertion of the theorem.
As an example of  application of this theorem we just quote  the following classical problem.
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Consider a fixed cone with angle ##\alpha##. A disk rolls around the cone without slipping such that   the center of the disk coincides with cone’s apex all the time.  The disk’s rim touches the cone in a point ##A##. The point ##A## belongs to the disk. The radius of the disk equals ##r##.
Let ##a_A## be a given acceleration of the point ##A## at a moment ##t=t_0##.  Find an angular velocity of the disk at the moment ##t_0##.
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