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?

 

[nlightner]

I can provide some guidance on how to approach this problem. First, it's important to understand the physical principles involved. In this case, the key concepts are rotational motion and forces (specifically centripetal force).

Next, it's important to carefully draw a free body diagram for each component (the collar and the ball) and clearly label all the forces acting on them. This will help you set up the equations of motion for each component.

In the x direction, you should have two equations (one for the collar and one for the ball) that involve the tension forces and the centripetal acceleration (which is given by a = Rw^2). You can then solve for the unknown tension in the upper rod (T2) by setting the two equations equal to each other and solving for T2.

In the y direction, you should have two equations as well (one for the collar and one for the ball). The collar equation should involve the weight of the collar (8.66 lbs) and the tension in the lower rod (T1). The ball equation should involve the weight of the ball (m*g, where m is the unknown mass and g is the acceleration due to gravity) and the tension in the upper rod (T2).

At this point, you should have two equations with two unknowns (T2 and m). You can then use algebra to solve for both variables. It's important to pay attention to units and make sure they are consistent throughout your calculations. If they are not, you may need to convert units to make them match.

I hope this helps guide you towards a solution. Remember, as a scientist, it's important to carefully analyze the problem and use fundamental principles to solve it. Good luck!