Question

Figure (10-E4) shows two blocks of masses m and M connected by a string passing over a pulley. The horizontal table over which the mass m slides is smooth. The pulley has a radius r and moment of inertia I about its axis and it can freely rotate about this axis. Find the acceleration of the mass M assuming that the string does not slip on the pulley.

Answer 1

According to the equation

$Mg-T_1=Ma$ ...(1)

$T_2=ma$ ...(2)

$\Rightarrow (T_1-T_2)=\frac{Ia}{r^2}$ [Because a = ra]....(3)

If we add the equation (1) and (2), we will get

$Mg+T_2-T_1=Ma+ma$ ...(4)

$\Rightarrow Mg-I\frac{a}{r^2}=Ma+ma$

$\Rightarrow (M+m+\frac{I}{r^2})a=Mg$

$\Rightarrow a=\frac{Mg}{M+m+\frac{I}{r^2}}$