In the system shown in diagram E, the masses are m_1 = 10 kg…
In the system shown in diagram E, the masses are $m_1 = 10\ \mathrm{kg}$ and $m_2 = 5\ \mathrm{kg}$; the coefficient of friction between $m_1$ and its surface is $\mu = 0.2$, and the incline angle is $\alpha = 30^\circ$. Find (a) the acceleration of the system and (b) the rope tension.
Explanation of Method
A body on a table pulled by a body on an incline.
Step 1: Assumed arrangement
Diagram E is not available. As in the original solution, $m_1$ lies on a rough horizontal table and is pulled over a pulley by $m_2$, which slides down a smooth incline of angle $\alpha$.
\[ m_2 a = m_2 g\sin\alpha - T, \]
\[ m_1 a = T - \mu m_1 g. \]
Step 2: (a) Acceleration
\[ a = \frac{m_2 g\sin\alpha - \mu m_1 g}{m_1 + m_2} = \frac{24.53 - 19.62}{15}. \]
\[a \approx 0.327\ \mathrm{m/s^2}\]
Step 3: (b) Tension
\[ T = m_1 a + \mu m_1 g = 3.27 + 19.62. \]
\[T \approx 22.9\ \mathrm{N}\]
Result
\[\boxed{\,a \approx 0.327\ \mathrm{m/s^2}\,}\]
\[\boxed{\,T \approx 22.9\ \mathrm{N}\,}\]
Remark
Check. For this arrangement the original results are correct. If instead $m_1$ were on the incline and $m_2$ hanging, then $m_1 g\sin\alpha = m_2 g = 49.05\ \mathrm{N}$ exactly: the pull and the slope force would balance, friction would hold everything at rest ($a = 0$, $T = 49.05\ \mathrm{N}$). The answer depends on the figure.