Determine the absolute Velocity and Acceleration of Particle A

[Northbysouth]

Homework Statement

The disk rolls without slipping on the horizontal surface, and at the instant represented, the center O has the velocity v_O = 2.2 m/s and acceleration a_O = 5.5 m/s² with directions shown in the figure. For this instant, the particle A has the indicated speed u = 3.2 m/s and time rate of change of speed = 6.6 m/s², both relative to the disk with directions shown in the figure. Determine the absolute velocity v_A and acceleration a_A of particle A.

I have attached an image of the question

Homework Equations

V_A = V_O + wXr + V_rel

The Attempt at a Solution

I'm honestly confused as to how to start this question. I'm not given the angular velocity so initially I tried to solve for it.

v = wXr

I know that the velocity at point O is -2.2i but how should I determine r? Or should I use the velocity at A instead?

Any help would be appreciated.

 

[haruspex]

Forget the particle for the moment. You have a disc of known radius rolling at known linear speed...

 

[Northbysouth]

So, you're saying that the angular velocity, w, is;

v = wXr

w = v/r

w = 2.2/0.39

w = 5.641 CCW

Right?

 

[Northbysouth]

After this part I'm fairly certain that I need to use:

V_A = V_O + wXr + v_rel

Which I think should give me:

V_A = -2.2i + (5.641k)X(0.3j) + 1i

v_A = -2.8923 i

But it says this is not correct. I'm thinking that my mistake is with the v_rel, how should I interpret this term, because I thought it was 1 because part A is traveling faster in the x direction by 1i. Am I mistaken?

 

[haruspex]

I'm thinking that my mistake is with the v_rel, how should I interpret this term, because I thought it was 1 because part A is traveling faster in the x direction by 1i. Am I mistaken?

v_rel is given as 3.2 m/s to the right.

 

[Northbysouth]

I've managed to get V_A = -0.6923i + 0j

But I'm stuck on a_A

I think I need to use the equation:

a_A = a_O + αXr + wXwXr + 2wXv_rel + a_rel

I found α with

α = a/r

α = 5.5/0.3

α = -14.1k

a_A = 5.5i + (-14.1k)X(0.3j) + (5.641k)X[(5.641k)X(0.3j)] + 2(5.641k)X(3.2i) - 6.6i

a_A = 3.13i + 23.69j

I know the j part is incorrect but I don't know about the i part. I don't understand where I'm going wrong. Am I misinterpreting the a_rel?

 

[Northbysouth]

My i component is correct at 3.13i and I finally managed to get the j component but it doesn't make sense.

the values for my j component are:

<5.641k>X[<5.641k>X<0.3k>] + 2<5.641k>X<3.2i> - 3.2²/0.3

= -7.57719 j which is correct

I can see where the first two terms come from in the equation I gave above, but where does the 3.2²/0.3 come from?

 

[haruspex]

I can see where the first two terms come from in the equation I gave above, but where does the 3.2²/0.3 come from?

That's the centripetal acceleration from the fact that it is moving at 3.2m/s relative to the disc but constrained to move in an arc radius .3m relative to the disc centre.