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Question Number 52043 by ajfour last updated on 02/Jan/19

Commented by ajfour last updated on 02/Jan/19

Except ground all suraces are  frictionless. An wedge has to  support two cyliners one top of  another against wall. Find  minimum μ on ground for  needed equilibrium.

$${Except}\:{ground}\:{all}\:{suraces}\:{are} \\ $$$${frictionless}.\:{An}\:{wedge}\:{has}\:{to} \\ $$$${support}\:{two}\:{cyliners}\:{one}\:{top}\:{of} \\ $$$${another}\:{against}\:{wall}.\:{Find} \\ $$$${minimum}\:\mu\:{on}\:{ground}\:{for} \\ $$$${needed}\:{equilibrium}. \\ $$

Answered by mr W last updated on 02/Jan/19

N=normal force between ball and wedge  N cos α=(m+m_0 )g  ⇒N=(((m+m_0 )g)/(cos α))  N sin α≤μ(M+m+m_0 )g  ⇒μ≥(((m+m_0 )tan α)/(M+m+m_0 ))

$${N}={normal}\:{force}\:{between}\:{ball}\:{and}\:{wedge} \\ $$$${N}\:\mathrm{cos}\:\alpha=\left({m}+{m}_{\mathrm{0}} \right){g} \\ $$$$\Rightarrow{N}=\frac{\left({m}+{m}_{\mathrm{0}} \right){g}}{\mathrm{cos}\:\alpha} \\ $$$${N}\:\mathrm{sin}\:\alpha\leqslant\mu\left({M}+{m}+{m}_{\mathrm{0}} \right){g} \\ $$$$\Rightarrow\mu\geqslant\frac{\left({m}+{m}_{\mathrm{0}} \right)\mathrm{tan}\:\alpha}{{M}+{m}+{m}_{\mathrm{0}} } \\ $$

Commented by ajfour last updated on 02/Jan/19

Very nice Sir, i mistook it might  be little interesting.

$${Very}\:{nice}\:{Sir},\:{i}\:{mistook}\:{it}\:{might} \\ $$$${be}\:{little}\:{interesting}. \\ $$

Commented by mr W last updated on 02/Jan/19

it makes no difference if it is one ball  or more balls, i think.

$${it}\:{makes}\:{no}\:{difference}\:{if}\:{it}\:{is}\:{one}\:{ball} \\ $$$${or}\:{more}\:{balls},\:{i}\:{think}. \\ $$

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