Connected particles: pulleys and tow-bars

Connected-particle questions come with a short list of modelling words. Each one tells you something you can use:

• Light string or rod: its mass is ignored, so it doesn't need any force to accelerate it. The pull is the same all along it.
• Inextensible: it doesn't stretch. So the two objects always move together, with the same speed and the same size of acceleration.
• Smooth pulley: no friction at the pulley, so the tension (the pulling force in the string, $T$) is the same on both sides.
• Particle: treat each object as a dot, so you ignore its size and rotation.
• Driving force: the forward push from a car's engine. Resistance: the backward forces (air resistance, friction) that oppose the motion.

Throughout, we use Newton's second law, $F = ma$: the resultant (overall) force on an object equals its mass times its acceleration. Take $g = 10\text{ m s}^{-2}$, as Cambridge 9709 does unless a question says otherwise.

Method 1: one object at a time. Apply $F = ma$ to each object separately, using only the forces on that object. You get two equations in two unknowns (usually $a$ and $T$). Solve them together.

Method 2: the whole system. Treat the two objects as one big object of mass $m_1 + m_2$. The tension pulls one forward and the other back by the same amount, so inside the system it cancels. Only the outside (external) forces count:

$\text{resultant external force} = (m_1 + m_2)\,a.$

Method 2 gets $a$ in one line, but it can't give you $T$, because $T$ has cancelled. So a good routine is: whole system for $a$, then one object for $T$.

Choose a positive direction for each object: the way it moves. For the pulley, that's up for the lighter object and down for the heavier one.