There are two theorems that are useful in particle kinematics that allow us to compute velocities and accelerations by transferring them to another point. They are discussed below.
This theorem is useful for determining the velocity (and/or acceleration) of one point on a rigid body by using (or transferring to) another point on the same rigid body.
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In the above figure, $P$ and $Q$ are two points that are rigidly fixed to the same rigid body, $B$. Further, $B$ is in freemotion relative to $A$. In other words, $B$ is moving freely in $A$; in other words, it is rotating and translating when viewed from $A$.
In such a scenario, the following two relationships can be used to compute the velocity and acceleration of $P$ in $A$.
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where,
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If B is purely in translation:
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If B is in simple rotation about a fixed axis. E.g., a simple pendulum, (see figure).
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Consider the case shown in the above figure containing two discs, $A$ and $B$. The two discs are rolling on each other and maintain a single point of contact $P$. Their motion is observed from a fixed frame $N$.
As the point $P$ is clearly shared between the 2 bodies, we introduce the following notation:
If it is said that A rolls without slip on B, then the following assumption can be made in tackling a problem:
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In the figure above, the disk $D$ rolls without slip on $A$. $P$ is a point on the disk’s circumference.
:::{admonition} Solution :class: tip, dropdown
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:::
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In the above figure, $P$ is a point that translates relative to body, $B$. Point $Q$ is rigidly attached to $B$. The one-point theorem allows computation of the velocity and acceleration of $P$ relative to the frame $A$, while accounting for the relative motion of $P$ as seen from $B$. The velocity formula is given by Equation $7.5$.
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${}^{B}\vec{v}^{P}$ is the velocity of $P$ in $B$. This term is also called relative velocity.
The highlighted term is referred to as coincident point velocity. This highlighted set of terms are identical in form to that seen in the two-point theorem.
The acceleration in this scenario is computed from Equation $7.6$ (also called five-term beast):
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The terms in the acceleration equation have special terminology; these are shown below. Also, note that the ‘transport acceleration’ term below is identical in form to the two-point theorem acceleration formula.
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