Determine the Acceleration of point A

[Northbysouth]

Homework Statement

The uniform 22-kg bar is supported on the horizontal surface at A by a small roller of negligible mass. If the coefficient of kinetic friction between end B and the vertical surface is 0.19, calculate the initial acceleration of end A as the bar is released from rest in the position shown. The acceleration is positive if to the left, negative if to the right.

I have attached an image of the question

Homework Equations

The Attempt at a Solution

I summed the forces in the x and y directions

ƩF_x = ma_x = -N_B

ƩF_y = ma_y = N_A - mg - F

F = u_kN

ƩF_y = ma_y= N_A - mg + u_kN_B

So, I have 4 unknowns: a_x, N_B, N_A and a_y

and two equations. How do I find the other two equations needed to solve this problem?

I considered taking the moment about B, but this then adds in two more unknowns: α and a_G. I suspect that I may need to use relative accelerations equations, but is there another way?

 

[tiny-tim]

Hi Northbysouth!

So, I have 4 unknowns: a_x, N_B, N_A and a_y

No, you have 3 independent unknowns.

(and an equation relating a_x and a_y)

I considered taking the moment about B, but this then adds in two more unknowns: α and a_G.

No, they also depend on a_x

 

[Northbysouth]

I'm not sure I follow you when you say that I have an equation relating a_x and a_y. Are you talking about

a = (a²_x + a²_y)^1/2

The questions asks me to find the acceleration of point a which I know is only in the x direction, and point B has an acceleration which only acts in the y direction.

Could you expand a little on what you said, please?

Thank you

 

[tiny-tim]

Are you talking about

a = (a²_x + a²_y)^1/2

No, I'm saying a = (x² + y²)^1/2.

(and so y = √(a² - x²))

 

[Northbysouth]

I'm sorry but I'm still not following you.

When you use x and y, are you referring to the horizontal and vertical distances respectively?

 

[haruspex]

I think tiny-tim is using x for the horizontal position of the roller, y for the vertical position of end B, and a (confusingly) for the length of the bar. The point is that the bar is of constant length, which gives you an equation relating a_x to a_y.
You will need to take moments too, since the moment of inertia of the bar plays a role. As with a_x and a_y, there is a direct relationship between these and angular acceleration.