Finding velocity and acceleration vectors from rotational motion

[CGI]

Homework Statement

Homework Equations

Rotational Kinematic Equations
Kinematic Equations

The Attempt at a Solution

I honestly have no clue how to get a vector out of this.
I thought about an equation:

Θ = Θ(initial) + ω(initial)*t + .5αt^2

and how maybe that v = wr could play into this, but there is so much I don't know where to start.

Any help would be really appreciated!

 

[TSny]

http://ocw.mit.edu/courses/aeronaut...fall-2009/lecture-notes/MIT16_07F09_Lec05.pdf

 

[CGI]

Oh okay, so I should be thinking of something along those lines.

So if my theta = 0 and my R = .8ft, would my position vector just be r = .8êr?
what would my r dot and theta dot be? Sorry, this is still relatively new to me
and I've been watching videos on it as well. I just want to make sure I understand
this.

 

[haruspex]

Oh okay, so I should be thinking of something along those lines.

So if my theta = 0 and my R = .8ft, would my position vector just be r = .8êr?
what would my r dot and theta dot be? Sorry, this is still relatively new to me
and I've been watching videos on it as well. I just want to make sure I understand
this.

Assuming the answers are wanted in terms of the x, y coordinates:
Let the diagram position represent time 0. At time t, what will r and theta be? So what will x and y be?

 

[CGI]

Would the x just be rcosΘ and y be rsinΘ?

 

[haruspex]

Would the x just be rcosΘ and y be rsinΘ?

Yes, but you are trying to find velocities and accelerations, so you need to express them as functions of time.

 

[CGI]

Oh okay. Right. Could that just be rcostheta(t) and rsintheta(t)?

 

[haruspex]

Oh okay. Right. Could that just be rcostheta(t) and rsintheta(t)?

According to the problem statement, both r and theta vary with time. Write each as a function of t.

 

[CGI]

Hmmm...okay. Could I say that r = r_initial + vt and that theta = theta_inital + wt where w = 45 rev/min?

 

[haruspex]

Hmmm...okay. Could I say that r = r_initial + vt and that theta = theta_inital + wt where w = 45 rev/min?

Yes. Now express x and y that way and differentiate as necessary.

 

[CGI]

When you say express x and y in that way, do you mean that I can say

x = (r_initial + vt)cos(Θ_initial + ωt)

And the same for y, only with a "sin?"

 

[haruspex]

When you say express x and y in that way, do you mean that I can say

x = (r_initial + vt)cos(Θ_initial + ωt)

And the same for y, only with a "sin?"

Yes. Now differentiate.

 

[CGI]

Okay I was just double checking. So when I take the derivative with respect to t I get,

x = vcos(Θ_inital + ωt) - (r_initial + vt)sin(Θ_initial + ωt)*ω

y = vsin(Θ_inital + ωt) + (r_inital + vt)cos(Θ_initial + ωt)*ω

Does this look about right?

 

[haruspex]

Okay I was just double checking. So when I take the derivative with respect to t I get,

x = vcos(Θ_inital + ωt) - (r_initial + vt)sin(Θ_initial + ωt)*ω

y = vsin(Θ_inital + ωt) + (r_inital + vt)cos(Θ_initial + ωt)*ω

Does this look about right?

Yes.

 

[CGI]

Okay great. So would these two x and y be the vector for velocity?

 

[haruspex]

Okay great. So would these two x and y be the vector for velocity?

Small correction: in your post #13 I presume you meant $\dot x=$ etc., not x=. Similarly y.
If so, yes they would be the x and y components of velocity.