Put two objects together and increase the tilt until the top object slides. The critical angle &theta_{c} at which slipping occurs determines the static friction coefficient μ_{s}.
To relate the critical angle to μ_{s}, use DID TASC with f_{s} = f_{s max} = μ_{s} N.
Steps 1-4: See diagram and transparency
Step 5: Apply Newton's 2nd law
ΣF_{x} = m a_{x} = 0 | | | ΣF_{y} = m a_{y} = 0 |
mg sin θ_{c} - f_{s} = 0 | | | N - mg cos θ_{c} = 0 |
mg sin θ_{c} = μ_{s} N | | | N = mg cos θ_{c} |
Step 6: Solve
Substitute for N= mg cos θ_{c} into the left equation gives:
mg sin θ_{c} = μ_{s} mg
cos θ_{c} or tan
θ_{c}= &mu_{s}
A similar method gives μ_{k}. Adjust the angle so that a pushed object continues to slide with constant speed. When this occurs, the tangent of the angle equals μ_{k}.