Rolling Oscillator on a Slippable Platform

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118.Rolling Oscillator on a Slippable Platform

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A uniform solid disk of mass

m = 1\ \text{kg}

and radius

r

rests on top of a horizontal platform of mass

M = 4\ \text{kg}

. The disk is connected by a horizontal spring of constant

k = 600\ \text{N/m}

to a rigid post fixed to the platform; the spring lies in the plane of the disk. There is sufficient friction between the disk and platform for rolling without slipping, but the coefficient of static friction between the platform and the ground is only

\mu = 0.1

. The disk is placed at the spring's natural length and given a horizontal velocity

v

, rolling without slipping from the start. For what

v

will the platform never slip on the ground? Use

g = 10\ \text{m/s}^2

$A horizontal rectangular platform of mass $M$ resting on a hatched ground surface (indicating friction $\mu$). On top of the platform sits a uniform disk of mass $m$ and radius $r$, on the left side. A short vertical post is fixed to the platform near the right end, rising to about the height of the disk's center. A horizontal spring of constant $k$ connects the center (axle) of the disk to the top of the post. Label the disk $m$, the platform $M$, the spring $k$, and mark the ground contact with $\mu$. An arrow on the disk indicates initial velocity $v$ to the right.$

v \leq 1/6\ \text{m/s}

v \leq 1/5\ \text{m/s}

v \leq 1/4\ \text{m/s}

v \leq 1/3\ \text{m/s}

v \leq 1/2\ \text{m/s}

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