By Dr. Shahid Ali Yousafzai

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In rotational and translational dynamics, force is called the linear force while torque is called the rotational force or the rotational analogue of force. First understand what is force? We define force as $$F=\frac { \Delta p }{ \Delta t } =\frac { dp }{ dt }$$ where we measure force in N. $$p$$ is the linear momentum $$p=mv$$ $$m$$ is the mass, which is a material property. Both $$m$$ and $$v$$ are depending on time, so we will use the product rule, we can write this as $$F=\frac { d\left( mv \right) }{ dt } =m\frac { dv }{ dt } =v\frac { dm }{ dt }$$ In a situation where mass in independent of time, then $$F=m\frac { dv }{ dt } =ma$$ Which is the Newton's 2 law of motion and acceleration is $$a=\frac { F }{ m }$$ For a given force, if mass is greater, then it will acquire less acceleration. Mass is the resistance to linear motion. And now let's discuss what is torque? This is actually angular force. $$\tau =r\times F=\frac { \Delta L }{ \Delta t }$$ As linear force is the rate of change of linear momentum, then, torque is the rate of change of angular momentum. $$\vec { L } =\vec { r } \times \vec { p }$$ The maximum angular momentum will be $$L=rp$$ Thus, $$\tau =\frac { dL }{ dt } =\frac { d\left( rp \right) }{ dt }$$ $$\tau =\frac { d\left( rmv \right) }{ dt } =\frac { d\left( rmr\omega \right) }{ dt } \\ =\frac { d\left( m{ r }^{ 2 }\omega \right) }{ dt }$$ We consider $$m$$ and $$r$$ independent of time. $$=\frac { d\left( m{ r }^{ 2 }\omega \right) }{ dt } =m{ r }^{ 2 }\frac { d\left( \omega \right) }{ dt }$$ $$m{ r }^{ 2 }\equiv I$$ Where $$I$$ is the moment of inertia and if we are having the mass not uniformly distributed then we will consider the center of mass $$I=\sum _{ i=1 }^{ n }{ { m }_{ i }{ r }_{ i }^{ 2 } } \\ \tau =I\alpha \\ \alpha =\frac { \tau }{ I }$$ Torque is measured in Nm. The role that is played by mass $$m$$ in linear dynamics is played by $$I$$ in rotational dynamics. Moment of inertia is the resistance to angular motion.

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