A solid sphere of mass 0.50kg is kept on a horizontal surface. The coefficient of static friction between the surfaces in contact is $\frac{2}{7}.$ What maximum force can be applied at the highest point in the horizontal direction so that the sphere does not slip on the surface?
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If we take moment about the centre, then
$\text{F}\times\text{R}=\ell\alpha\times\text{f}\times\text{R}$
$\Rightarrow\text{F}=\frac{2}{5}\text{mR}\alpha+\mu\text{mg}\ \dots(1)$
Again, $\text{F}=\text{ma}_{\text{c}}-\mu\text{mg}\ \dots(2)$
$\Rightarrow\text{a}_\text{c}=\frac{\text{F}+\mu\text{mg}}{\text{m}}$
Putting the value $a_c$ in eq(1) we get
$\Rightarrow\frac{2}{5}\times\text{m}\times\Big(\frac{\text{F}+\mu\text{mg}}{\text{m}}\Big)+\mu\text{mg}$
$\Rightarrow\frac{2}{5}(\text{F}+\mu\text{mg})+\mu\text{mg}$
$\Rightarrow\text{F}=\frac{2}{5}\text{F}+\frac{2}{5}\times0.5\times10+\frac{2}{7}\times0.5\times10$
$\Rightarrow\frac{\text{3F}}{5}=\frac{4}{7}+\frac{10}{7}=2$
$\Rightarrow\text{F}=\frac{5\times2}{3}=\frac{10}{3}=3.33\text{N}$
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