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## Finding the instantaneous axis of rotation (dynamics - circular and linear motion)

1. The problem statement, all variables and given/known data

Shaft's AC movement is regulated by runways A and B. The angular velocity of the shaft is 3 rad/sec, counterclockwise. When angle psi = 40 degrees. Calculate:

A) The velocities of pistons A and B (Va, Vb)
B) The velocity of point C (Vc) - magnitude and direction towards the horizontal axis.

3. The attempt at a solution

My problem is finding P - the instantaneous axis of rotation. I picked this:

Picking it, I got the correct Va, the correct Vb, and the correct Vc. (I'm still working on the angle of Vc to the horizontal axis - so far it's not correct but maybe I'm doing something obviously wrong.)

And the solution manual picked that:

Are they both correct?

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 There is a very straightforward way to relate Vc to Va and Vb. The book solutions are correct but they're complicated
 Recognitions: Homework Help The instantaneous axis is the center of movement. This means that all points of the body make a circular motion around this axis. In your case you know the direction of the speed at 2 points. Are they making a circular motion around your P?

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## Finding the instantaneous axis of rotation (dynamics - circular and linear motion)

Well I just wanna know how come P is in the opposite direction to my P? It's pretty easy to solve this the way the manual did actually. You just find the distance from point B to P and the distance from point A to P and the distance from point C to P and do angular velocity times this distance and you get the answer. Finding the angle of C is a bit more tricky, but I first of all wanna know if I got the correct P because it gives me the same result as they....

Edit: Nevermind, I get it. It depends on the direction of the rotation! :)

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 Quote by Femme_physics Edit: Nevermind, I get it. It depends on the direction of the rotation! :)
Hmm, I would have worded it differently.

I'd have said that any speed vector must be perpendicular....

 Recognitions: Gold Member Fair enough, but if my P is where the solution manual says, then I get opposite results for Va and Vb to what the manual says I should get!

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 Quote by Femme_physics Fair enough, but if my P is where the solution manual says, then I get opposite results for Va and Vb to what the manual says I should get!
Perhaps if you switched the letters A and B around?

You might say of course that you're not allowed to do that.
But then, if I look at the original problem, it seems to me that they did exactly that!

 Recognitions: Gold Member *an embarrassing look* Oh. *slaps forehead* Ouch. I really gotta stop doing that... Thanks ILS :) Problema el solva (I have no idea if what I wrote is lingually correct)......

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 Quote by Femme_physics *slaps forehead* Ouch. I really gotta stop doing that...
What? The switching around of letters and similar stuff?
Or the slapping?

You can slap me if you want!

 Quote by Femme_physics Thanks ILS :) Problema el solva (I have no idea if what I wrote is lingually correct)......
If it is or not, it sounds nice!

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Actually, still one problem. Finding the angle of Vc to the horizontal axis. I get 61.4 but the manual gets 59.21... I feel it's too much to chalk it up to "rounding errors".

 What? The switching around of letters and similar stuff? Or the slapping?
Both!

 You can slap me if you want!
Slap you? But why?!? I can't. I'm too gentle and soft. And you're too harmless. Well, maybe a little. *soft slap*... *little less of a soft slap*...

*slaps!*
*SLAPS!!*
*THUMPS!!!!*
*MONSTER THUMP!*

Woah, sorry, lost control

Told you I'm a fast learner and adapter. ^^

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 Quote by Femme_physics Actually, still one problem. Finding the angle of Vc to the horizontal axis. I get 61.4 but the manual gets 59.21... I feel it's too much to chalk it up to "rounding errors".
Yes dear madam, it is too much for rounding errors. ;)

How did you get the angle of 28.5 degrees?

 Quote by Femme_physics Both!
Aha, now I understand the head band.
It is both to protect your forehead from the slap, and to hide the bruises under it!

 Quote by Femme_physics Slap you? But why?!? I can't. I'm too gentle and soft. And you're too harmless. Well, maybe a little. *soft slap*... *little less of a soft slap*... *slaps!* *SLAPS!!* *THUMPS!!!!* *MONSTER THUMP!* Woah, sorry, lost control Told you I'm a fast learner and adapter. ^^
aw, aw, aw, aw, Aw, Aw, AW, AWW, AWWWW, AIEEEH!

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 Yes dear madam, it is too much for rounding errors. ;) How did you get the angle of 28.5 degrees?
Law of sines.

BP/sin(beta) = CP/sin(130)

When I know that CP = 641.45
And BP = 306.4

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 Quote by Femme_physics Law of sines. BP/sin(beta) = CP/sin(130) When I know that CP = 641.45 And BP = 306.4
All right, so how did you get BP and CP?

EDIT: Did you perhaps take BP from the drawing where you switched the letters A and B around?

EDIT2: You wrote in your problem that the angle is "psi", but actually that is "phi".
$\phi$ or $\varphi$ is phi
and $\psi$ is psi

 Recognitions: Gold Member I most certainly did not switch A and B ar...no wait...wait...I most certainly did! Your corrections are all correct-- as always. All solved ;) thanks!