Question

A ball of mass m1 and a block of mass m2 are attached by a lightweight cord that passes
over a frictionless pulley of negligible mass, as in figure below. The block lies on a frictionless incline of
angle θ. Find the magnitude of the acceleration of the two objects and the tension in the cord.

Answer 1

Write all forces by their axes along which the acceleration is directed according to Newton's second law:

$$\text{Block: }m_2a=m_2g\text{ sin}\theta-T,\\ \text{Ball: }m_1a=T-m_1g.$$

Divide one by another:

$$\frac{m_2}{m_1}=\frac{m_2g\text{ sin}\theta-T}{T-m_1g},\\ \space\\ T=g\cdot\frac{m_1m_2(1+\text{sin}\theta)}{m_1+m_2}.$$

Acceleration, on the other hand, will be

$$a=\frac{T}{m_1}-g,\\ \space\\ a=g\bigg[\frac{m_2(1+\text{sin}\theta)}{m_1+m_2}-1\bigg],\\ \space\\ a=g\cdot\frac{m_2\text{ sin}\theta-m_1}{m_1+m_2}.$$