Connected particles

In a nutshell

In some problems, particles interact with other particles, causing their forces to interact. Particles connected together can be considered as separate particles or as one single particle if their motion is along a straight line. According to Newton's third law, any particle will exert an equal and opposite force to any force that acts on it.

Equations

DESCRIPTION	EQUATION
Newton's second law	$F=ma$

Variables

Quantity name	symbol	unit name	unit symbol
$Force$	$F$	$Newton$	$N$
$Mass$	$m$	$Kilograms$	$kg$
$Acceleration$	$a$	$Metres\space per\space second\space squared$	$ms^{-2}$
$Tension$	$T$	$Newton$	$N$
$Normal \space Force$	$N$	$Newton$	$N$

Newton's third law

Newton's third and final law of motion states that:

"For every action there is an equal and opposite reaction"

Consider a stationary object of mass $m$ resting on a table.

Maths; Forces and motion; KS5 Year 12; Connected particles

The object experiences a weight force due to gravity, $mg$ , which if unstopped would pull the object towards the Earth's core. It is halted by the table, which exerts an opposite reaction force, the normal force $N$ . As this object is in equilibrium, these forces cancel out. Therefore:

$\boxed{N=mg}$

Connected particles

In some cases, particles will be operated on by multiple forces, and interact with multiple particles. In cases where connected particles are moving along one dimension, consider all connected objects as one object, and assume the forces affecting the multiple object affect this one object. You need to be able to think as connected particles as one single object or as many different parts depending on what you are trying to find.

Example 1

Two particles of mass $4\space kg$ and $6\space kg$ are joined by a light, inextensible string in the manner shown below. The $6\space kg$ block is pulled to the right with a force of $20\ N$ .

$(i)$ What is the acceleration $a$ felt by the particles?

$(ii)$ What is the tension $T$ in the string connecting the particles?

Maths; Forces and motion; KS5 Year 12; Connected particles

To find the acceleration, consider the two objects as one. As such, the masses add together and the tension forces $T$ cancel each other out.

$\begin{aligned}&F=ma\\&F=20\\&m=4+6=10\\2&0=10a\\&\underline{a=2\ ms^{-2}}\\\end{aligned}$ 

To find the tension $T$ , consider one particle by itself. Using the $6\space kg$ , find the tension:

$\begin{aligned}F&=ma\\F&=20-T\\m&=6\\a&=2\\20-T&=6\times 2\\20-T&=12\\-T&=12-20\\T&=8\ N\\\end{aligned}$

Verify the answer by using the $4\space kg$ particle, as the tension force should be equal.

$\begin{aligned}&F=ma\\&F=T\\&m=4\\&a=2\\&T=4\times2\\&\underline{T=8\ N}\end{aligned}$ 