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For a project I have to cut an acrylic glass rod (diameter = 5mm) into a few small chunks. Then, I have to sand the cut surfaces and polish them. Since the surface to be polished is very small, I don't want to buy a special polishing agent. So is there a household substance that I can use as a substitute?

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If you don't have a suitable abrasive substance nor wish to purchase some, you really have only toothpaste as a possible solution. Some products are not abrasive and will accomplish nothing. Some may be excessively abrasive and result in a poorer surface than desired. Obviously, testing is indicated if you try toothpaste.

On the other hand, flame polishing may provide what you require. A butane torch burns quite cleanly and can be controlled to the necessary extent. A gas range also burns cleanly, but control of such a large surface of burning propellant is going to be less easily managed.

Heat guns are another clean route, but if you have none, it's a purchase you wish to avoid.

Other heat sources are going to deposit soot or other impurities.

If you're able to locate a suitable heat source, experimentation is again indicated. It's easy to overheat and distort the plastic, especially 5 mm material. Aim for minimal application, as you can more easily repeat the heating than you can remove too much! Laser cut acrylic has nearly glass smooth edges, due to careful application of heat.

10 mm laser cut acrylic cube

This image is of a 10 mm laser cut acrylic cube. Due to the high magnification, one can see the striations caused by the stepper motors. If a smooth surface was required, I'd sand it with the highest grit sandpaper I can find (1200) followed by micromesh (undesired purchase for you) with a water bath. The result would be glass smooth at the expense of some elbow grease. Using heat would be much faster and less laborious.

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    It's not quite "just toothpaste" in that a baking soda slurry can also be used for polishing and is made from kitchen ingredients. Mechanical polishing ought to maintain a flatter surface than flame polishing, which can end up a little wavy - of course that might not matter, and practice would minimise that
    – Chris H
    Dec 24, 2022 at 10:17
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    @ChrisH And for a small item, I suppose some sort of sled to stop it from wobbling could be useful, if flat surfaces were desired. Maybe squeeze it between two bits of wood. Dec 24, 2022 at 20:46
  • It's not out of the question that a flat surface is required. 5 mm diameter is pretty small.
    – fred_dot_u
    Dec 24, 2022 at 21:35
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    Drill a 5.1 mm hole in a block of wood, stick the rod through and it will hold and guide for a level surface.
    – Willeke
    Dec 25, 2022 at 8:00
  • @Willeke Yes, that's a good hint — otherwise the edges get rounded and especially on a small diameter that can be quite noticeable!
    – Madamadam
    Dec 25, 2022 at 22:30

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