A block of mass m lies at distance L on a horizontal ruler rotating about one end. The friction coefficient is µ. Find (a) the maximum constant angular speed without slip; (b) if angular speed increases from zero with constant angular acceleration α, the angular speed at which slipping begins.
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Question

A block of mass m lies at distance L on a horizontal ruler rotating about one end. The friction coefficient is µ. Find (a) the maximum constant angular speed without slip; (b) if angular speed increases from zero with constant angular acceleration α, the angular speed at which slipping begins.

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G1 Physics · Accepted Answer

For constant ω, friction supplies mω²L, so ω(max)=√(µg/L). With angular acceleration, friction supplies perpendicular radial and tangential components mω²L and mαL; thus (µg/L)²=ω⁴+α² and ω=[(µg/L)²-α²]^(1/4).