ohgodamnitw0

2022-09-06

Do Newton's laws of motion apply on rigid bodies? If they apply on rigid bodies, would we consider forces acting in any direction or on any part of the body, and consider only the centre of mass when we talk about its momentum or the body being at rest or in uniform motion?
Because I used to regard Newton's laws as only applying to point masses, but I'm not sure if that's the case. For example, I read that the following statement can be justified using Newton's first law:
Consider a lever on a fulcrum with weights ${W}_{1}$ and ${W}_{2}$ on either side of the fulcrum, where the lever is in balance; the force exerted by the tip of the fulcrum on the lever is ${W}_{1}+{W}_{2}$

Conor Daniel

Absolutely Newton's laws apply for rigid bodies. There are extensions to F=ma attributed to Euler that describe the rotational equations of motion.
To be fully desciptive use point C to designate the center of mass and write
Momentum of body from the motion of the center of mass
$\begin{array}{}\text{(1)}& \stackrel{\to }{p}=m\phantom{\rule{thinmathspace}{0ex}}{\stackrel{\to }{v}}_{\mathrm{C}}\end{array}$
Newton's 2nd law as the time derivative of the above
$\begin{array}{}\text{(2)}& \stackrel{\to }{F}=m\phantom{\rule{thinmathspace}{0ex}}{\stackrel{\to }{a}}_{\mathrm{C}}\end{array}$
where $\stackrel{\to }{F}$ is the net force acting the body (applied anywhere, including external and reaction forces). Also ${a}_{\mathrm{C}}$ is the acceleration of the center of mass only.
Angular momentum about the center of mass is
$\begin{array}{}\text{(3)}& {L}_{\mathrm{C}}={\mathrm{I}}_{\mathrm{C}}\stackrel{\to }{\omega }\end{array}$
Where $\stackrel{\to }{\omega }$ is the rotationa velocity of the body (shared among all point on the body) and ${\mathrm{I}}_{C}$ is the mass moment of inertia (tensor) summed at the center of mass.
The motion about the center of mass is described by Euler law of rotation which is the time derivative of the above
$\begin{array}{}\text{(4)}& {M}_{\mathrm{C}}={\mathrm{I}}_{\mathrm{C}}\stackrel{\to }{\alpha }+\stackrel{\to }{\omega }×{\mathrm{I}}_{\mathrm{C}}\stackrel{\to }{\omega }\end{array}$
Where $\stackrel{\to }{\alpha }$ is the rotational acceleration of the body and ${M}_{\mathrm{C}}$ the net torque about the center of mass.
All of the above can be easily derived when considering a rigid body as a collection of finite particles each moving with a translation of the center of mass and a rotation about the center of mass. Every "Introduction to Dynamics" book out there should have this somewhere in the first chapters.

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