By wrapping something around the bottle, you’ve essentially increased it’s diameter. By increasing it’s diameter you can apply more torque (rotational force) on the bottle top. Torque is measured by a force applied at a radius from the center point (T = F x r). This is where you get foot-pounds of torque in a car. Feet of the radius, pounds of the force applied at that radius. In smaller applications like a bottle top you’d measure ounce-inches of torque but the same thing applies. Think about using a wrench on a bolt. The longer the wrench, the more force you apply to the bolt head. By wrapping the towel around the bottle, the “wrench” is just a little bit longer and it helps get that little bit more you need.
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