# 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 W1 and W2 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 W1+W2.

• You can think of any rigid body as a sum of many point like particles and work with that approximations with N-element-sums or integrals in some cases, this is a subject for more advanced classical mechanics, you can check Taylor's or Goldstein's books if interested. Commented Apr 18, 2022 at 12:26
• How is this not a duplicate? A candidate (though probably a duplicate itself): Why are Newton's laws valid only for rigid bodies? Commented Apr 19, 2022 at 2:32
• physics.stackexchange.com/questions/578535/… Commented Apr 19, 2022 at 15:06
• Commented Jul 19, 2023 at 14:38

Absolutely Newton's laws apply for rigid bodies. There are extensions to $$F=m a$$ 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

1. Momentum of body from the motion of the center of mass

$$\vec{p} = m\, \vec{v}_{\rm C} \tag{1}$$

2. Newton's 2nd law as the time derivative of the above

$$\vec{F} = m\, \vec{a}_{\rm C} \tag{2}$$

where $$\vec{F}$$ is the net force acting the body (applied anywhere, including external and reaction forces). Also $$a_{\rm C}$$ is the acceleration of the center of mass only.

3. Angular momentum about the center of mass is

$$L_{\rm C} = \mathrm{I}_{\rm C} \vec{\omega} \tag{3}$$

Where $$\vec{\omega}$$ 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.

4. The motion about the center of mass is described by Euler law of rotation which is the time derivative of the above

$$M_{\rm C} = \mathrm{I}_{\rm C} \vec{\alpha} + \vec{\omega} \times \mathrm{I}_{\rm C} \vec{\omega} \tag{4}$$

Where $$\vec{\alpha}$$ is the rotational acceleration of the body and $$M_{\rm C}$$ 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.

If a single force is applied to a rigid body which is free to move, the center of mass will move in the direction of the force with an acceleration which is proportional to the force. If the line of action of the force does not pass through the center of mass, then it will also produce a torque which will cause an angular acceleration about the center of mass.

Newton's laws have been applied in detail to a system of particles. A rigid body is a special case of a system of particles for which the distance among the particles comprising the rigid body are fixed. Advanced treatments of rigid bodies use the Euler angles and Euler equations, and Lagrangian/Hamiltonian approaches, but all of these are built upon Newton's laws. See any intermediate/advanced physics mechanics text for details, such as Mechanics by Symon or Classical Mechanics by Goldstein.