Two bodies of masses m_1 = 1 kg and m_2 = 5 kg are connected…

Two bodies of masses $m_1 = 1\ \mathrm{kg}$ and $m_2 = 5\ \mathrm{kg}$ are connected by a light inextensible rope as shown in the diagram; one end of the rope is fixed. The body $m_1$ slides down a frictionless incline of angle $\alpha = 30^\circ$. Find the acceleration $a_2$ of the body $m_2$. Neglect friction and the mass of the pulley.

Explanation of Method

Two bodies on a rope over an incline.

Step 1: The diagram is missing

The statement says one end of the rope is fixed, which usually means a movable pulley, where the two accelerations differ by a factor of 2. Without the diagram, the arrangement cannot be reconstructed with certainty. The original solution treated it as a simple rope over a fixed pulley at the top of the incline, with $m_2$ hanging; that case is worked out here.

Step 2: Simple pulley arrangement

Take $m_2$ moving down as positive: $m_2 g - T = m_2 a$.

\[ T - m_1 g\sin\alpha = m_1 a. \]

\[ a = \frac{m_2 - m_1\sin\alpha}{m_1 + m_2}\,g = \frac{5 - 0.5}{6} \cdot 9.81. \]

\[\begin{gathered} a_2 \approx 7.36\ \mathrm{m/s^2} \\ (m_2 \text{ down, } m_1 \text{ up the incline}) \end{gathered}\]

Rope tension: $T = m_2(g - a) \approx 12.3\ \mathrm{N}$.

$7.36\ \mathrm{m/s^2}$ $5\ \mathrm{kg}$ $1\ \mathrm{kg}$ $30^\circ$ $\text{in this arrangement } m_1 \text{ is pulled UP the incline}$

The arrangement used in the original solution (a simple rope over a pulley). The 5 kg weight outweighs the 4.9 N slope force on the 1 kg body, so the hanging body falls and drags $m_1$ up the incline. The actual figure, with one rope end fixed, is not available.

Result

\[\boxed{\,\begin{gathered} a_2 \approx 7.36\ \mathrm{m/s^2} \\ (m_2 \text{ down, } m_1 \text{ up the incline}) \end{gathered}\,}\]

Remark

About the original solution. Its numbers are right for this arrangement, but they contradict the statement: with a hanging 5 kg body, $m_1$ is pulled up the incline, not down. So the real figure must be different, most likely with $m_2$ on a horizontal surface or on a movable pulley. The answer should be redone once the figure is available.