Question

A particle P of mass 2m lies on a rough horizontal table. P is connected by a light inextensible string passing over a smooth pulley fixed at the edge of the table to another particle Q of mass 5m, hanging freely. The system is released from rest with the string taught and the hanging part vertical. If the acceleration of the system is of magnitude 4g/7, find
(a) the tension in the string
(b) the coefficient of friction between the first particle and the table.

Answer 1

Let's solve the problem.

we have $a = \large\frac{4g}{7}$

first of all we mark the directions of x and y forces which impact to P and Q particles

 for P particle: $N = 2mg \space\space\space\space 2ma = T - F_{ff} = T - \mu N = T - \mu 2mg$

for Q particle: $5ma = 5mg - T$

$\begin{cases} 2ma = T - \mu 2mg \\ 5ma = 5mg - T \end{cases}$    $+ \to 7ma = 5mg-2\mu mg / : m$

                                                                  $7a = 5g - 2\mu g$

                                                                $\mu = \large\frac{5g-7a}{2g}$ $= 2.5-3.5\large\frac{a}{g}$ = $\large\frac{5}{2}-\frac{7}{2}*\frac{4}{7}$ $= 0.5$

(b) the coefficient of friction between the first particle and the table is 0.5

(a)  the tension in the string is $T = 5m(a+g) = 5m(g + \large\frac{4g}{7}) = \frac{55mg}{7}$