A uniform cylinder of radius $R$ is spinned about its axis to the angular velocity $ω_{0}$ and then placed into a corner. The coefficient of friction between the corner walls and the cylinder is $μ_{2}$ . How many turns will the cylinder accomplish before it stops?

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Solution

From the condition of translational equilibrium for the cylinder.

$m g = N_{1} + K N_{2}; N_{2} = K N_{1} N_{1} = \frac{m g}{1 + K^{2}}; N_{2} = K \frac{m g}{1 + K^{2}}$

For pure rotation the cylinder about its rotation axis,

$N_{2} = I β_{2} o r, - K N_{1} R - K N_{2} R = \frac{m R^{2}}{2} β_{2} o r, - \frac{K m g R (1 + K)}{1 + K^{2}} = \frac{m R^{2}}{2} β_{2} o r, β_{2} = - \frac{2 K (1 + K) g}{(1 + K^{2}) R}$

Now from the kinematical equation

$w^{2} = w_{0}^{2} + 2 β_{2} Δ ϕ Δ ϕ = \frac{w_{0}^{2} (1 + K^{2}) R}{4 K (1 + K) g}$

because $w = 0$

Hence the sought number of turns,

$n = \frac{Δ ϕ}{2 π} = \frac{w_{0}^{2} (1 + K^{2}) R}{8 π K (1 + K) g}$

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Q. A uniform cylinder of radius

R

is spinned about its axis to the angular velocity

ω_{0}

and then placed into a corner. The coefficient of friction between the corner walls and the cylinder is

μ_{k}

. How many turns will the cylinder accomplish before it stops?

A uniform cylinder of radius R is spinned about its axis to the angular velocity $ω_{0}$ and then placed into a corner, see the figure. The coefficient of friction between the corner walls and the cylinder is $μ_{k}$ . How many turns will the cylinder accomplish before it stops?