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Finding Friction's Magnitude

A solid spher...

Question

A solid sphere of mass 0.50 $k g$ is kept on a horizontal surface. The coefficient of static friction between the surfaces in contact is 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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Solution

Given: $μ = \frac{2}{7}$ and $m = 0.50 k g$
According to the question,
$Σ T = 0$
$\Rightarrow$ $F \times R = l_{a} \times F \times R$ ( $a_{c} = R α$ )
$\Rightarrow$ $F = \frac{2}{5} m R α + μ m g$ ........... $(i)$

Now, $F = m a_{c} - μ m g$
$\Rightarrow$ $a_{c} = \frac{F + μ m g}{m}$ .............. $(i i)$

From $(i)$ and $(i i)$ , we get

$\Rightarrow$ $F = \frac{2}{5} m (\frac{F + μ m g}{m}) + μ m g$

$\Rightarrow$ $F = \frac{2}{5} (F + μ m g) + μ m g$

Putting all the values, we have
$\Rightarrow$ $F = \frac{5 \times 2}{3} = \frac{10}{3} = 3.3 N$

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