How Much Clamping Force to Prevent a Ball from Slipping in Robotic Grippers?

[Cataracts]

Homework Statement

The static friction coefficient between the grippers and the ball is 0.5, and the grippers
hold the ball such that they touch the middle of the ball on opposite sides.

Find the clamping force required to keep the ball from slipping out, if the ball weighs 10 lb.

u_s= 0.5
Mass of ball = 10 lb
gravity = 9.81 m/s^2

Angular velocity (w_d) = 5 rad/s = Angular velocity of robot arm as they are attached.

Homework Equations

r_c=4cos45 + 2sin45 = 4.243 ft
Motion of point C
Vc = (w_r)(r_c)
a_r=$\alpha$r_c
a_n= w_r^2 (r_c)
a_c= sqrt(a_r^2 + a_n^2)

The Attempt at a Solution

Using the above equations and the givens I found the r_c which I used along with w_r to solve for Vc.

Vc = ( 5.0 rad/s) ( 4.243 ft) = 21.2 ft/s

Then solved for both components of acceleration where :

a_t= $\alpha$ r_c = 0
a_n= w_r^2 x r_c = 25 x 4.243ft = 106.07 ft/s^2

Then solved for magnitude which just equalled the normal acceleration

a_c= 106.07 rad/s^2

Currently stuck on what to do for the second part of the question. I understand that I need to do a force FBD as I am given u_s and the mass of the ball. Noting that the pincers touch the middle of the ball on both sides, wouldn't that mean the contact forces oppose each other ( equal and opposite direction ) ?

 

[aralbrec]

a_c= 106.07 rad/s^2

Currently stuck on what to do for the second part of the question. I understand that I need to do a force FBD as I am given u_s and the mass of the ball. Noting that the pincers touch the middle of the ball on both sides, wouldn't that mean the contact forces oppose each other ( equal and opposite direction ) ?

Yes.

The sum of the forces acting on that ball must produce that acceleration, otherwise the ball isn't going to follow the circular path as you've assumed. The grip will be equal, opposite and perpendicular to that acceleration so that alone cannot produce that acceleration. But then there's the friction...

 

[Cataracts]

I realized that movement was only constrained in the j direction, therefore there was a k and i direction force of friction. I then gathered components in those directions and solved the system giving me the answer ! Thanks for the guidance !