Rotational and Translational Equilibrium Help Needed

[hjcrbwg]

Homework Statement

I am making a hanging mobile which needs to be done mathematically by calculating torque. The problem is, I can't seem to figure out how to solve for the distance.
You see, all of the problems we did in class talked about finding mass, but to do this project, I already know the mass, I just need to know the distance between the objects that are hanging.

Note: 0.0284 kg is the weight of the rod that is holding up the objects and the string holding the rod up is in the center of the rod, which is why the force up is the same distance away from the pivot point as the rod.

Homework Equations

Here is what my teacher gave me to work on this:

1. Force_up = Force_down
2. τ_clockwise = τ_{counter clockwise}
3. Use an object as the pivot point if there is more than one object to solve for.

The Attempt at a Solution

Here is what I filled in those steps:

1. Force_up = (2.3814 kg + .0452 kg + .0878 kg + 0.0284 kg)9.8 m/s² = 3.96312 N.
2. 2.3814 kg(9.8 m/s²)(0 m.) + 0.0284 kg(9.8 m/s²)(.125 m.) + 0.0878 kg(9.8 m/s²)(x m.) + 0.0452 kg(9.8 m/s²)(y m.) = 3.96312 N.(.125 m.)
which is, if my math is correct:
0.03479 Nm. + 0.86044 N (x m.) + 0.44296 N (y m.) = 0.49539 Nm.

So my problem lies here: what do I do with the two different variables? I don't have a second equation nor do I know of a second equation that I can use to calculate the distance needed between them. What can I do to make this work?

EDIT: Here is what the structure looks like: a string is holding up a rod from the middle of the rod so that the rod will be level when held from the string. There are three objects hanging from that rod, and the pivot point is the heaviest of the hanging objects.

 

[haruspex]

Please describe the structure. It's hard to figure out from the numbers.

 

[hjcrbwg]

Please describe the structure. It's hard to figure out from the numbers.

Ok, just updated my question. Let me know if you need any more information

 

[haruspex]

Are you putting one object at each end, and just want to know where to place the third? If so, don't put the heaviest at one end if it weighs more than the other two put together.
The point where the string is attached is necessarily the 'pivot point', but I guess you mean the point you would take moments about. I don't see that it matters which point you take moments about. The point of string attachment looks the simplest, since then you don't care about the weight of the rod (which you don't, since it is attached in its middle, so it balances itself).

 

[Nickie]

First of all, it's great that you are using the equations your teacher provided and trying to apply them to your project. It shows that you are understanding the concepts of rotational and translational equilibrium. However, there are a few things that need to be clarified in order to solve your problem.

1. The equations you listed are correct, but they are not the only equations that can be used to solve for the distance. In fact, the second equation you listed is not entirely correct. The correct equation should be: τclockwise = τcounter clockwise + τpivot. This accounts for the torque exerted by the pivot point, which is the string in this case.

2. In order to solve for the distance, you will need to use a combination of these equations and also include the information about the weights and distances of the hanging objects. It's important to note that the equation for torque is: τ = rFsinθ, where r is the distance from the pivot point to the point where the force is applied, F is the force, and θ is the angle between the force and the lever arm (the distance r). This will be helpful in solving for the distance.

3. It's also important to clarify what the variables x and y represent in your equations. From your description, it seems like x and y are the distances from the pivot point to the two hanging objects. In this case, you will need to set up two equations, one for each hanging object, and solve them simultaneously in order to find the values of x and y.

4. Finally, make sure you are using the correct units in your calculations. The units for torque should be Nm, not N/m.

Overall, solving for the distance in this problem will require a combination of the equations given by your teacher, as well as the equation for torque and the information about the weights and distances of the hanging objects. Don't worry if you don't have a second equation, you can still solve for the distance using the information provided. Keep in mind that solving for the distance may require some trial and error, so don't get discouraged if your initial calculations don't give you the correct answer. Keep working through the problem and check your calculations to make sure they are correct. Good luck!