Measuring μs, the static friction coefficient

Put two objects together and increase the tilt until the top object slides. The critical angle &thetac at which slipping occurs determines the static friction coefficient μs.

To relate the critical angle to μs, use DID TASC with fs = fs max = μs N.

  1. Diagram and coordinate systems
  2. Isolate the systems
  3. Draw all forces acting
  4. Take components
  5. Apply F=ma and constraints
  6. Solve
  7. Check!

Steps 1-4: See diagram and transparency

Step 5: Apply Newton's 2nd law
ΣFx = m ax = 0 | ΣFy = m ay = 0
mg sin θc - fs = 0 | N - mg cos θc = 0
mg sin θc = μs N | N = mg cos θc

Step 6: Solve

Substitute for N= mg cos θc into the left equation gives:
mg sin θc = μs mg cos θc   or   tan θc= &mus

A similar method gives μk. Adjust the angle so that a pushed object continues to slide with constant speed. When this occurs, the tangent of the angle equals μk.