Rotating Spring Arm

  A 5 kg collar slides along a rod that is rotating. The collar is also attached to the center with a spring. There is friction between the collar and the bar, μ = 0.3. At a particular time, dθ/dt = 1.2 rad/s, dr/dt = 0.9 m/s, d2r/dt2 = 1.3 m/s2, and r = 0.7 m. The angular velocity is constant. What is the force magnitude acting on the collar due to the spring? Neglect gravity.

Collar Free-body Diagram
with Accelerations


The acceleration in the radial and theta directions need to be determined before summing forces. The given information is,

Substituting into acceleration equations,

Summing forces and accelerations give,
     ΣF = maθ
     N = 5 (2.16)
     N = 10.8 N

     ΣF = mar
     -Fs - Ff = mar
     -Fs - μN = 5 (0.292)
     -Fs - 0.3 (10.8) = 1.46

Fs = -4.70 N or 4.70 N

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