Connected Particles

Connected particles can be analysed using Newton’s laws of motion. It’s crucial to consider each particle separately, applying Newton’s second law to each.
In such scenarios, two or more particles are connected through some binding forces. These connections create internal forces which might be dependent on a tension (usually in a string or a cable) or a normal contact force (as in a block lying on a surface).
When particles are connected by a light inextensible string, the tension throughout the string is constant.
This is because a light string is one which is assumed to have no mass. As such, it does not affect the overall system dynamics, but transmits the force without any loss or delay – hence the force at both ends of the string will be the same.
An inextensible string means that it does not stretch or deform under tension. This assumption allows us to disregard factors of elasticity or damping.
In a problem involving connected particles, the system of particles can be at rest or in motion.
For each individual particle, analyse the forces acting on it, separate them into those acting parallel and perpendicular to the direction of motion, and apply F=ma to create equations.
When considering particles connected in a vertical plane, remember that the weight of each particle acts downwards
The weight (W) of a particle is given by the product of its mass (m) and the acceleration due to gravity (g), therefore W=mg.
Normal reaction is the contact force exerted by a surface on a particle lying on it. It acts perpendicular to the surface. It is also equal and opposite to the component of weight acting perpendicular to the surface.
Friction can also exist between a particle and the surface it’s on. It always acts to oppose motion, and for a rough surface, its maximum value is proportional to the normal reaction (F ≤ μR, where μ is the coefficient of friction and R is the normal reaction).
To solve problems involving connected particles, it’s useful to create a free body diagram. Label the forces on each particle and then apply Newton’s laws to create and solve simultaneous equations.
It’s essential when setting up the equations for connected particles that forces in one direction are treated as positive and those in the opposite direction as negative. It helps to choose a common direction for all particles, typically the direction of overall motion.
When particles are connected in an inclined plane, remember to resolve forces along the plane and perpendicular to the plane to accurately describe the overall behaviour of the system.
While analysing connected particles it’s preferable to assume the motion or direction of motion. If the assumption is wrong, it will be clear from the results (for example, you might end up with a negative acceleration). This does not invalidate the analysis but instead indicates that the assumed direction of motion was incorrect.

Keep revising these core principles and apply them to solve problems systematically to gain a deep understanding of the dynamics of connected particles.