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Frictional force between pen and paper

A pen of mass ‘m’ is lying on a piece of paper of mass M placed on a rough table. If the cofficient of friction between the pen and paper and, the paper and the table are \(\mu_1\) and \(\mu_2\) respectively, what is the minimum horizontal force with which the paper has to be pulled for the pen to start slipping?

Solution:

For pen to start slipping, maximum horizontal acceleration for pen and paper to start slipping is \(f=\mu_1\)g
Therefore, \(a=\mu_1 g\) is the common acceleration for pen and paper.
If \(f_1\) and \(f_2\) be the frictional forces for pen and paper respectively, the net force for the system
$$ F=f_1+f_2+Ma
F=\mu_1mg+\mu_2mg+Ma$$
Now, \(a=\mu_1 g\)
Hence, $$ F=(m+M)(\mu_1+\mu_2)g$$

June 30, 2017

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