Coefficient of friction calculation

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The coefficient of [static or dynamic] friction is not ordinarily something that one calculates from first principles. It does not arise from some simple underlying physics. It should be regarded instead as an engineering approximation that is observed to hold in a many circumstances. Instead of calculating it, you measure it.
 
The coefficient of friction is the ratio of the friction force to the normal force. To measure it you apply a normal force (eg a weight) and measure the friction force (eg max force that can be applied before the object moves). Work out the ratio.