## Torques in Equilibrium

#### This was merely an exercise to satisfy my curiosity. In any case, it's an interesting experiment, I thought. Just ... interesting.

During the review session we encountered some strangenss in the results of one of the practice-test questions. Namely, that of forces & torques in static equilibrium (Problem 2, diagram below). We found mathematically that given that the weights were attached equidistant from the center of rotation, w1 and w2 must be equal. The angle θ (Theta) was not a factor in determining the value of w2 in terms of w1. Intuitively, one would think that if both blocks weigh the same, and are attached equidistant from the axis of rotation, the rod would level out, making θ effectively 0 (zero). In addition, we calculated mathematically that any weights of unequal weights would result in an exactly vertical orientation of the beam (i.e., θ = 90°). Therefore, according to the results of the problem, the situation depicted by the diagram is impossible, if angle θ is not zero. But, we've probably all seen counter-weight decorations with this problem's parameters, and have probably experienced the slanting of a balance when the weight on one side is different than the other (eg - see-saw). Thus our confusion.

So I decided to perform an experiment to confirm ... one thing or the other.

Equipment:

Equipment:

- Digital scale (to measure weights)
- Cardboard rod (4g)
- Two felt-tip pens (17g)
- Bluetooth wireless headset (20g)
- String (mass ignored: combined weight is less than 1g)
- Metric ruler

Procedure: Attached string to center of rod, and 10cm from either side of center. Attached the two weights (pens & headset) to the two hanging strings.

By letting it hang, I observed that indeed(!) there was a noticable and measurable tilt in the rod (approximately 20 degrees above horizontal). So this definitely contradicted the calculated results. Just to make sure everything was in order, I took approximate measurements of the moment arms. The moment arm of the elevated side ("w2", the two pens) measured to be just over 10cm. The lower side ("w1", the headset), however, measured almost 9cm. Clearly, the two values of (l/2)*cos(θ) were not equal for some strange reason.

Then I noticed that the junctions of the two weights and the axis of rotation were not along the same one-dimensional line. Because of the location of the knot in the strings, the axis of rotation was at the middle of the rod,

**above the rod**, while the junctions of the weights were 10cm away from the center,**under the rod**. I thought that perhaps this was affecting the value of the moment arm, thus affecting the torques.
I re-aligned the knots, making sure again that the two outer strings were equidistant from the center. I released, and behold! The entire system immediately turned exactly vertical. So this confirms that, in a two-weight balanced system with equal moment arms, there can be non-vertical equilibrium only if the two weights are equal. And such equilibrium would level the balancing rod/beam/whatever to the horizontal.

In conclusion, we (meaning I) hope you enjoyed this presentation of a confusing concept made confusinger or less confusing. Really, I hope I didn't make things worse. If you have any complaints, please file the proper paperwork and notify the authorities. My boundless curiosity has taken me to 04:30. I'm getting silly and it's far too late to write any more.

###### Jul 22, 2004

For the more intensely curious such as myself, I investigated the physics behind this further. Check it out.