A block of mass m rests at radius r on a horizontal rotating platform and is connected to the rotation axis by a radial spring of force constant k and natural length l₀, where r>l₀. The coefficient of static friction is μ. The maximum angular speed before the block starts slipping outward is:
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Question

A block of mass m rests at radius r on a horizontal rotating platform and is connected to the rotation axis by a radial spring of force constant k and natural length l₀, where r>l₀. The coefficient of static friction is μ. The maximum angular speed before the block starts slipping outward is:

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G1 Physics · Accepted Answer

Required inward centripetal force is mω²r. The stretched spring already supplies inward force k(r-l₀). At the limiting condition for outward slipping, friction acts inward at its maximum μmg. Thus mω²r = k(r-l₀)+μmg. Therefore ω_max = √{[k(r-l₀)+μmg]/(mr)}.