Question

Two blocks with masses m1 and m2 are connected by a massless string over a frictionless pulley. Block 1 sits on a frictionless horizontal surface and block 2 sits on a plane inclined at an angle θ above the horizontal. The coefficient of friction between block 2 and the incline is µk. The pulley, which is a uniform disk, has a mass mp and a radius R. When you release the blocks, both blocks slide without the string slipping on the pulley. In terms of the given parameters and physical constants, what is the magnitude of the acceleration of the blocks?

Answer 1

From the given,

$m_{1} a_{1} = T_{1}$  

Force acting on the mass $m_{1}$,  

$m_{2} a_{2} = m_{2}g\ [sin \ sin \theta- \mu_k cos \ cos\theta-T_2]$  

$T_1$ and $T_2$ are tension on the string along the masses,

$(T_1-T_2)R = I*\frac {a} {R}$  

$T_1-T_2=\frac {ma} {2}$  

$m_1a -m_2 \ \{g [sin\ sin\theta- \mu_k cos\ cos\theta]-a\} = \frac {mp}{a} {2}$  

Here, $a_1$ and $a_2$ are equal as a.

$a = \frac {m_2g [ sin\ sin\theta- \mu_k\ cos\ cos\theta]} {m_1+m_2-m_p}$