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Uniform circular motion with friction

  1. Jul 2, 2015 #1
    I recently took an exam in my engineering mechanics class and passed the exam except for one question. I have been trying to solve this question after the fact and determine the solution, however neither I nor several engineering tutors have had any success. The professor refuses to give assistance or clarification. Any assistance would be appreciated.

    1. The problem statement, all variables and given/known data

    A bead of mass 'm' slides on a rigid circular wire. The coefficient of kinetic friction between the bead and the wire is ## \mu ##. Assume Gravity is negligible.

    a. Is speed increasing or decreasing? [correct answer: decreasing]
    b. Find an expression for the rate of change of speed. [correct answer: ##\frac {dv} {dt} = \frac {\mu v^2} {r}##]
    c. If at t=0 the speed of the bead is ##v_o##, find the speed v(t) as a function of time t. [correct answer: ##v(t) = \frac {r v_o} {r + v_o \mu t}##]


    2. Relevant equations

    ##s = r \theta ##
    ##v = r \dot \theta ##
    ##\frac {dv} {dt} = r \ddot \theta ##
    ##\vec V = r \dot \theta \hat t ##
    ##\vec a = \frac {v^2} {r} \hat n + \frac {dv} {dt} \hat t ##


    3. The attempt at a solution

    a. The speed of the bead is decreasing due to the presence of friction [correct]

    b. Rate of change of speed is equal to ##\frac {dv} {dt}##

    Using normal-tangential coordinates, I determined that friction is opposing the direction of motion, in this case the velocity in the tangential direction, thus:
    ##\sum F_x = -F_f sin \theta ##
    ##\sum F_y = F_f cos \theta ##


    I am not even sure where to begin with this problem, although I am actively working on it now. The exam is over and I have lost all chance at points but I would still like to understand the concepts behind a problem like this. Any assistance would be appreciated; if there is anything I need to add or change in this problem statement please let me know. Thank you in advance!
     

    Attached Files:

  2. jcsd
  3. Jul 2, 2015 #2
    Have you drawn a free body diagram on the bead? What is the radial acceleration? What is the radial force balance in terms of N, the normal force between the bead and the wire? What is the force balance in the tangential direction, in terms of the frictional force F and the tangential acceleration? What is the kinematic equation for the tangential acceleration? What is the relationship between the frictional force F and the normal force N in terms of the coefficient of kinetic friction?

    Chet
     
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