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You will be making a mess as you refinish the coffee table. Work outside or somewhere like a garage if you can, or put down a drop sheet, tarp, or plastic sheet to catch the dust from sanding and protect the floor from spills. If you don't have a good work area or some kind of drop sheet to use, you could put down 2-3 overlapping layers of newspaper or even create a drop sheet by taping some plastic garbage bags together. Mix 2-3 drops of gentle soap, such as a mild liquid dish detergent, into a cup or bowl of water. Soak a clean cloth in the soap solution and wring out excess water until it's not dripping. Rub down the entire coffee table with the damp cloth to clean off dust and grime. It's important to clean off the surface very well so there are no grains of dirt that you might end up grinding into the wood as you sand. Attach a sheet of coarse-grit sandpaper, like 40- or 60-grit sandpaper, to a sanding block or electric sander. Start sanding, going with the grain of the wood, until you remove all the finish and expose all the bare wood. Wear a face mask and goggles whenever you're sanding to keep wood dust out of your mouth, nose, and eyes. If there are any hard-to-reach areas, such as engraved legs or sides of the coffee table, fold up a piece of sandpaper into a small square and sand those areas by hand. If your sandpaper stops removing the finish at any point, stop sanding and feel the sandpaper with your fingers. If it is smooth, it has worn out and you need to replace it with a fresh sheet. Use a tack cloth or other clean, soft cloth to wipe away all the dust from sanding. This will allow you to better see the exposed wood and continue with sanding and preparing the surface for a new finish. A tack cloth is a cloth specifically made for cleaning up wood dust. You can purchase one at a home improvement center, a woodworking supply shop, or online. If you don't have one, something like a microfiber cloth or an old cut-up T-shirt will work. You could also use a vacuum with a hose attachment to suck up the dust. If you notice any spots where there is still paint, varnish, or stain showing after you wipe away the dust, go back over them with the coarse-grit sandpaper until the wood is bare. Put a sheet of fine-grit sandpaper, like 120- or 240-grit sandpaper, on your sanding block or electric sander. Sand down the entire coffee table, going with the grain, until it is completely smooth. Stop sanding periodically and run a hand over the whole coffee table to feel for rough spots. Keep sanding and feeling the table until it feels like it has a uniform smoothness. Wipe off the loose dust with your tack cloth or other soft cloth. Dampen a rag with mineral spirits and wipe down the entire coffee table to clean it and prepare it for finishing. Wear a face mask when you apply mineral spirits so you don't inhale the fumes. If you are working inside, open up any doors and windows you can to ventilate the space. Mineral spirits are also known as mineral turps or mineral turpentine. The liquid is very similar to paint thinner, which you can use as a substitute if that's all you have.
Pick an easy-to-clean area to work in or put down a drop sheet. Clean the surface of the table with a mild soap and water solution. Remove the finish by sanding the coffee table with a coarse-grit sandpaper. Wipe down the entire coffee table with a clean cloth to remove the wood dust. Smooth the bare wood surface by sanding it again with fine-grit sandpaper. Clean off the coffee table with mineral spirits to get rid of fine wood dust.