Deflection and oscillations in a homemade microgram balance.

[SporkInTucson]

Hi,

As part of a larger hobby project, I'm building a microgram balance based on the one described at Sci-Toys.com, but I believe my questions are appropriate for the "general physics" forum.

Background: I purchased a 36" threaded steel rod, connected it to a razor, and then put it on a couple glasses for support (I haven't yet started the upturned razors support). I'm amazed at how long it takes the rod's oscillations to damp down to equilibrium. The longer the rod oscillates, the less convenient the scale will be to use, so I've been wondering what I can do about it. The first thing that came to mind was using a threaded titanium rod instead of a threaded steel rod -- they cost 10 times more but they're available online.

Q1: Am I correct in assuming that if two scales both start at the same non-equilibrium initial condition, the one with a titanium rod will damp out to equilibrium before the one with a steel rod, all other aspects of the scales, samples, and reference weights being the same? I think this would be so because the steel version would start with more gravitational potential energy that needs to be damped out.

Q2: Will the endpoints of a titanium rod sag about the same amount as the endpoints of a steel rod, even though the stiffness of steel is about twice that of titanium? I'm guessing this is true because titanium has about half the density of steel, so the decrease in weight per unit length cancels out the decrease in elasticity in the equation for deflection of a cantilevered beam under a uniform load.

Let me know if my intuition and limited physics knowledge are steering me wrong.

 

[Bobbywhy]

Many balance beam instruments use “Eddy-Current Damping” to reduce these oscillations. A conducting plate (aluminum, for example) is attached to the balance arm and placed between the poles of a magnet. As the arm oscillates the plate moves in and out of the magnetic field between the poles of the magnet. This oscillation produces a changing magnetic flux; the magnetic flux induces an emf (electromotive force); this induced emf cause eddy currents to flow on the surface of the plate. The eddy currents generate a magnetic force; the magnetic force acting on the eddy currents must oppose the flux change, according to Lenz's Law, so it must oppose the motion of the balance arm and plate through the magnet; this, in turn, decreases the oscillations of the arm and it comes to rest sooner than without this mechanism.

Here are many images of a wide variety of eddy-current dampers:
eddy current damping - Google Search

Here’s a photo of a magnetic damper for a balance beam in figure 20.24:
http://physics.bu.edu/~duffy/EssentialPhysics/chapter20/section20dash5.pdf

 

[sophiecentaur]

Any good balance needs to be in an enclosure, to avoid draughts. Air damping can be achieved using wide vanes on either end of the balance beam. The enclosure is even more important here though.

 

[SporkInTucson]

Thanks Bobby, Centaur. I wasn't considering a damping mechanism but Eddy Current Damping sounds relatively simple so maybe it would be fun to try.

Any comments on reducing the mass of the beam? If using a material that weighs half as much means that there's half as much energy to damp out on the way to equilibrium, I'll take that.

 

[pmacias]

Hi there,

Thank you for sharing your project and questions with us. As a fellow scientist, I am always excited to see people exploring and experimenting with scientific concepts on their own.

To answer your first question, yes, you are correct in assuming that a titanium rod will damp out to equilibrium faster than a steel rod, all other factors being the same. This is because the titanium rod has a lower density and thus a lower mass, which means it will have a lower moment of inertia and less potential energy to dissipate through oscillations. Additionally, titanium has a higher strength-to-weight ratio than steel, meaning it can withstand greater stress and strain without deforming. This can also contribute to faster damping of oscillations.

As for your second question, the endpoints of a titanium rod will not sag as much as the endpoints of a steel rod, even though titanium is less stiff than steel. This is because, as you mentioned, the decrease in weight per unit length cancels out the decrease in elasticity in the equation for deflection of a cantilevered beam under a uniform load. However, it is important to note that the stiffness of a material is not the only factor that affects deflection. Other factors such as the shape and cross-sectional area of the rod also play a role.

Overall, your intuition and understanding of physics seem to be on the right track. However, I would also recommend doing some research and calculations to get a more accurate understanding of the behavior of different materials in this type of setup. Good luck with your project!