Rotating governor (rotational motion)

[theydidntname]

Homework Statement

A rotating governator is designed as shown. A 8.66 lb collar at the bottom can slide freely along the shaft (no friction). Initially the collar is 2.00ft from the top. As the governor rotates, the collar is pulled upwards, and when it reaches a distance of 1.732 from the top, it will activate a shut off switch. Two small balls of just the right mass will be used to shut power when the rotational speed is 120 rpm. Each mass, M, is held with two light rigid rods of length, 1.00 ft.

a. Draw FBD of the collar, and FBD of the mass M (no problems here)
b. Find the tension in the lower rod (no problems here)
c. Find the mass required (problem starts here)

Homework Equations

for x or y directoin:
$$\sum F_{x} = ma_{radial}$$
$$\sum F_{y} = 0$$

$$a_{radial} = Rw^2$$

The Attempt at a Solution

I used triangles to find the radius from the balls to the center. its 0.50004 ft. for the collar i used the sum of the forces in the y direction to find the tension in the lower rod.

T1 is the tension in the lower rod, T2 is the tension in the upper rod. $$m_{b}$$
is the mass of the ball which i need to find

$$T_{1} = 5^{lb}$$

Which is correct. I have the correct answers, I am trying to figure out how to get them however.

I have 2 unknowns now, $$T_{2} and m_{b}$$

from my fbd of the ball, i have 2 equations, one x and one y

x:
$$T_{1x} + T_{2x} = m_{b}a_{radial}$$ ($$a_{radial} = Rw^2$$)
$$T_{1} + T_{2} = 157.91367^{rad^2/sec^2}$$


and y:
$$T_{2y} - T_{1y} - W_{ball} = 0$$
$$T_{2}(0.8660018) - T_{1}(0.8660018) - m_{ball}g = 0$$
$$T_{2} = T_{1} + (m_{ball}g)/0.8660018$$
$$T_{2} = T_{1} + 36.951424^{ft/sec^2}m_{ball}$$

I guess this is more of a algebra problem then physics. i have t1, but when i try to solve for t2, then use the last equation to solve for m, i get the wrong units and can't solve it. in the y equation, the number is in ft/sec^2, but in my x equation it is rad^2/sec^2. my professor every once in awhile tells us that its okay to just erase the 'rad' in some situations because it doesn't matter. to be honest though i don't totally understand what he meant by that. in physics you can't just combine the two numbers right, because the units don't match up? I am really confused and I've been working on this problem for a long time, i appreciate any advice.


the mass is supposed to equal 0.08267 slugs

 

[LowlyPion]

Just a thought ... but maybe a drawing would prove useful?