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Further kinematics

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Connected particles

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Functions of time: Displacement, velocity, acceleration

Using differentiation to find velocity and acceleration

Maxima and minima problems

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Constant acceleration

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Constant acceleration formulae: SUVAT

Vertical motion under gravity

Modelling in mechanics

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Proof II

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Proof I

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Explainer Video

Tutor: Daniel

Summary

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=maF=maF=ma​​

​

Variables

Quantity name

symbol

unit name

unit symbol

​ForceForceForce​​
​FFF​​
​NewtonNewtonNewton​​
​NNN​​
​MassMassMass​​
​mmm​​
​KilogramsKilogramsKilograms​​
​kgkgkg​​
​AccelerationAccelerationAcceleration​​
​aaa​​
​Metres per second squaredMetres\space per\space second\space squaredMetres per second squared​​
​ms−2ms^{-2}ms−2​​
​TensionTensionTension​​
​TTT​​
​NewtonNewtonNewton​​
​NNN​​
​Normal ForceNormal \space ForceNormal Force​​
​NNN​​
​NewtonNewtonNewton​​
​NNN​​

​


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 mmm​ resting on a table. 


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

The object experiences a weight force due to gravity, mgmgmg, 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 NNN. As this object is in equilibrium, these forces cancel out. Therefore:

N=mg\boxed{N=mg}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 kg4\space kg4 kg and 6 kg6\space kg6 kg are joined by a light, inextensible string in the manner shown below. The 6 kg6\space kg6 kg​ block is pulled to the right with a force of 20 N20\ N20 N. 

(i)(i)(i)​ What is the acceleration aaa felt by the particles?

(ii)(ii)(ii)​ What is the tension TTT in the string connecting the particles?



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

F=maF=20m=4+6=1020=10aa=2 ms−2‾\begin{aligned}&F=ma\\&F=20\\&m=4+6=10\\2&0=10a\\&\underline{a=2\ ms^{-2}}\\\end{aligned}2​F=maF=20m=4+6=100=10aa=2 ms−2​​​ 


To find the tension TTT, consider one particle by itself. Using the 6 kg6\space kg6 kg, find the tension:

F=maF=20−Tm=6a=220−T=6×220−T=12−T=12−20T=8 N\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}FFma20−T20−T−TT​=ma=20−T=6=2=6×2=12=12−20=8 N​


Verify the answer by using the 4 kg4\space kg4 kg particle, as the tension force should be equal.

F=maF=Tm=4a=2T=4×2T=8 N‾\begin{aligned}&F=ma\\&F=T\\&m=4\\&a=2\\&T=4\times2\\&\underline{T=8\ N}\end{aligned}​F=maF=Tm=4a=2T=4×2T=8 N​​​​



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FAQs - Frequently Asked Questions

How do you solve problems with connected particles?

Connected particles, if their motion occurs in a straight line, can be dealt with as one single whole. This simplifies the problem and allows you to be flexible in what you are trying to solve.

What forces act on stationary object on a table?

A weight acting vertically downwards and an equal opposite normal reaction force acting vertically upwards.

What is Newton's third law?

Newton's third law states that "For every action there is an equal and opposite reaction".

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