Problem Statement
A particle of mass m moves in a horizontal circle of radius R on a rough horizontal surface (coefficient of kinetic friction μk), with initial speed v₀. Find: (a) how long it moves; (b) total path length.
Given
m, R, v₀, μk. Circular motion with friction.
Concepts & Formulas
Friction decelerates: F = μk·mg. Deceleration a = μk·g. Speed: v = v₀ − μk·g·t. Path length until stop: s = v₀t − ½μk·g·t².
Step-by-Step Solution
Step 1: t_stop = v₀/(μk·g).
Step 2: s = v₀·t_stop − ½μk·g·t_stop² = v₀²/(μk·g) − v₀²/(2μk·g) = v₀²/(2μk·g).
Step 3: Number of revolutions: N = s/(2πR) = v₀²/(4πRμk·g).
Worked Calculation
t = v₀/(μk·g). s = v₀²/(2μk·g).
Boxed Answer
t = v₀/(μk·g); s = v₀²/(2μk·g)
Physical Interpretation
The result is identical to linear deceleration — the path shape (circle) is irrelevant; only friction magnitude and initial speed matter for total distance.
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