Q3. Consider two disks in essentially the same plane, and rotating about the commoncenter O with different but constant angular velocities. Let the angular velocities by wand w'. The radii of the disks are r and r'. Two points P and P' are located at the outeredges of the disks. Assume that point O is fixed. Find the velocity and acceleration of P'relative to P at the time instant when the two points have the least separation, in thefollowing two cases: (a) the observer is fixed to the ground, or nonrotating; (b) theobserver is rotating with the unprimed system, i.e., the disk with radius r and angularvelocity w. Express the results in terms of the unit vectors e; and en which rotate withthe unprimed system. (Problem 2-15 in the text).

Answered: Q3. Consider two disks in essentially…

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Q3. Consider two disks in essentially the same plane, and rotating about the common
center O with different but constant angular velocities. Let the angular velocities by w
and w'. The radii of the disks are r and r'. Two points P and P' are located at the outer
edges of the disks. Assume that point O is fixed. Find the velocity and acceleration of P'
relative to P at the time instant when the two points have the least separation, in the
following two cases: (a) the observer is fixed to the ground, or nonrotating; (b) the
observer is rotating with the unprimed system, i.e., the disk with radius r and angular
velocity w. Express the results in terms of the unit vectors e; and en which rotate with
the unprimed system. (Problem 2-15 in the text).

Expert Solution

Step 1 Concept

As given in the problem is based on the concept of rotational motion.

The point P is located on the disk of radius r and it is rotating with constant angular velocity $ω$ .

$T h e v e l o c i t y o f p o i n t P i s v_{P} = r ω \vec{e_{t}} T h e a c c e l e r a t i o n o f p o i n t P i s; a_{P} = - ω^{2} r \vec{e_{n}}$

Now the point P' is located on the disk of radius r' and it is rotating with angular velocity $ω'$ .

As mentioned in the problem P and P' are separated by least separation.

Therfore consider the angular position of P separated by P' at this instant is $ϕ .$

Then at point P, the normal unit vector is given by;

$\begin{array}{rcl} \vec{e_{n}} & = & c o s (ϕ) \vec{i} + s i n (ϕ) \vec{j} \\ S i m i l a r l l y t h e \tan g e n t i a l u n i t v e n c t o r i s g i v e n b y; \\ \vec{e_{t}} & = & - s i n (ϕ) \vec{i} + c o s (ϕ) \vec{j} \\ Then the position vector OP at instant t is given by; \\ \vec{r'} & = & \vec{OP} \\ = & OP \vec{e_{n}} \\ Therefore; \\ \overset{\to'}{r} & = & r' (\cos (ϕ) \vec{i} + \sin (ϕ) \vec{j}) \\ As velocity is rate of change of distance; \\ Therefore differentiating r with respect to time we will get velocity . \\ \vec{v'} & = & \frac{d \vec{r'}}{dt} \\ = & \frac{d}{dt} r' (\cos (ϕ) \vec{i} + \sin (ϕ) \vec{j}) \\ = & r' [\vec{i} (- \sin (ϕ) \frac{dϕ}{dt}) + \vec{j} (\cos (ϕ) \frac{dϕ}{dt})] \\ But as we know; \\ angular velocity is given by \\ ω' & = & rate of change of angular distance \\ ω' & = & \frac{dϕ}{dt} \\ ∴ \vec{v'_{P'}} & = & r' ω' [\vec{i} (- \sin (ϕ)) + \vec{j} (\cos (ϕ))] \\ But \\ \vec{e_{t}} & = & - \sin (ϕ) \vec{i} + \cos (ϕ) \vec{j} \\ ∴ \vec{v'_{P'}} & = & r' ω' \vec{e_{t}} \\ This is the velocity of point P' . . . . . . . . . . . . . . . . . (1) \\ Now the acceleration at point P can be calculated . \\ As acceleration is given by rate of change of velocity, \\ \vec{a'_{P'}} & = & \frac{d \vec{v'_{P'}}}{dt} \\ Therefore acceleration at point P at instant t is; \\ ∴ \vec{a'_{P'}} & = & \frac{d \vec{v'_{P'}}}{dt} \\ As we have, \vec{v'_{P'}} & = & r' ω' [\vec{i} (- \sin (ϕ)) + \vec{j} (\cos (ϕ))] \\ ∴ \vec{a'_{P'}} & = & \frac{d}{dt} r' ω' [\vec{i} (- \sin (ϕ)) + \vec{j} (\cos (ϕ))] \\ Now here as we know \\ ω' & = & \frac{dϕ}{dt} it is also changing with time . \\ ∴ \vec{a'_{P'}} & = & r' \frac{d}{dt} ω' [\vec{i} (- \sin (ϕ)) + \vec{j} (\cos (ϕ))] \\ = & r' [\frac{dω'}{dt} [\vec{i} (- \sin (ϕ)) + \vec{j} (\cos (ϕ))] + ω' \frac{d}{dt} [\vec{i} (- \sin (ϕ)) + \vec{j} (\cos (ϕ))]] \\ As \vec{e_{t}} & = & - \sin (ϕ) \vec{i} + \cos (ϕ) \vec{j} \\ = & r' [\frac{dω}{dt} \vec{e_{t}} + ω' [\vec{i} (- \cos (ϕ) \frac{dϕ}{dt}) + \vec{j} (- \sin (ϕ) \frac{dϕ}{dt})]] \\ = & r' \frac{dω}{dt} \vec{e_{t}} - r' ω' \frac{dϕ}{dt} [\vec{i} (\cos (ϕ)) + \vec{j} (\sin (ϕ))] \\ = & r' \frac{dω}{dt} \vec{e_{t}} - r' ω'^{2} [\vec{i} (\cos (ϕ)) + \vec{j} (\sin (ϕ))] \\ But \vec{e_{n}} & = & \cos (ϕ) \vec{i} + \sin (ϕ) \vec{j} \\ ∴ \vec{a'_{P'}} & = & r' \frac{dω'}{dt} \vec{e_{t}} - r' ω'^{2} \vec{e_{n}} \\ But; \frac{dω'}{dt} & = & 0 \\ A s ω' i s c o n s \tan t \\ ∴ \vec{a} & = & - r' ω'^{2} \vec{e_{n}} \\ ∴ \vec{a} & = & - r' ω'^{2} \vec{e_{n}} \\ This is an acceleration of point P' \dots \dots \dots \dots (2) \end{array}$

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