Definition of mechanical equilibrium?

Question

The common definition (e.g.: Wikipedia) is that: mech eq is achieved when net force and net torque on EACH PARTICLE of the system is zero. My question is: why is a ball spinning at constant angular velocity and zero translational motion at mech eq?

At first it makes sense, since translation velocity and angular velocity are both constant (hence net force and torque are zero). But the definition says that EVERY PARTICLE of the system must have zero translational and angular acceleration. If we consider the particles off the centre of the ball, they are actually accelerating centripetally as the ball spins, even though their speed is constant; these particles are therefore not in equilibium. So why do we still say that the ball itself is at equilibrium?

Is the issue here just a case of simplifying the ball into a single indivisible body?

Edit: sorry lithiumdeuteride, what's the difference between angular acceleration and rotational acceleratio? I always thought they are the same thing.

lithiumdeuteride: in any case, I'm not saying that there is rotational acceleration. I'm saying that there is centripetal acceleration (at the off-centre particles), which means that net force is non-zero, yet Wiki (and all other sources) say that net force must be zero.

Steph&glad: the definition you wrote seems quite different from others, and a bit counterintuitive. E.g.: a charged particle in a magnetic field can move with no pattern at all, like a bee zipping randomly around in the air, yet there is never ever any work done on the particle. Based on your definition, this particle woud be in equilibrium (since work is zero at any point in time), even though it's moving so unpredictably and randomly all over the place.

supastremph:
That sounds pretty nice actually for this example. But by that definition, an asteroid flying towards Earth would be at equilibrium (if we consider the asteroid and Earth together to be the system), right?

Answer 1

You are correct. When you combine all the particles into one system, you can say that the system as a whole is at equilibrium even if individual particles are accelerating.

Answer 2

Mechanical Equilibrium

Answer 3

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Depends on your definition of Mechanical Equilibrium. In general ,yes. BUT Some people define equilibrium to include constant acceleration, though this case is usually called Dynamic Equilibrium The earth in orbit around the sun is in (Dynamic) Equilibrium. I define Equilibrium as: Sum Forces _ x = 0 Sum Forces _ y = 0 Sum Forces_z = 0 Sum Moments _ x = 0 Sum Moments_y = 0 Sum Moments_z = 0 That covers both Static and Dynamic Equilibrium

Answer 4

How's about:

. . . when the net EXTERNAL force and net EXTERNAL torque on each particle is zero. With a rigid body or collection of particles, there are always internal forces, but they do not change the SYSTEM'S properties on the whole. Only an external force can change the motion of the center of mass of the ensemble, and only an external torque can change the angular momentum of the ensemble. This definition then takes equilibrium as "no changes to center of mass motion or net angular momentum".

Yes, that's correct about the asteroid. Usually in mechanics, you're more interested in rigid bodies, but you can think of the asteroid, or the solar system as a "dynamic equilibrium", that is, that the motion is cyclic, and therefore although the PARTS of the system change with time, the system does not--it will always revert back to the original state at some point . . . like a mass on a spring. (I'm assuming no second law energy dissipation here, e.g. if the asteroid crashes into the earth or that the mass on the spring eventually stops.)

Answer 5

Mechanical equilibrium means that the TOTAL work in the system is zero.

This means that the sum of the all the individual displacements from the force upon each particle in the system add up to zero.

Total Work = ∑  Fd 's.

Since a spinning ball is symmetric, summing over time, for every particle that is changing direction there is another particle that is displaced in a directly "opposite" position in the sphere. Or it could be described that each particle will move in its opposite direction as the sphere spins. To see the problem, placing the sphere in a coordinate axis will show a negatively directed point to correspond to every positively directed point at every given moment in time, thus the sum is zero.

Answer 6

Perhaps you should read the next line in the Wikipedia article:

"A particle in mechanical equilibrium is neither undergoing linear nor rotational acceleration; however it could be translating or rotating at a constant velocity."

It does not say the angular acceleration must be zero, only that the rotational acceleration must be zero. Rotational acceleration is what changes the speed of rotation. Hence the spinning ball is in mechanical equilibrium.

Answer 7

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RE:
Definition of mechanical equilibrium?
The common definition (e.g.: Wikipedia) is that: mech eq is achieved when net force and net torque on EACH PARTICLE of the system is zero. My question is: why is a ball spinning at constant angular velocity and zero translational motion at mech eq?

At first it makes sense, since translation...

Answer 8

in that case, pls read again and make neccessary corrections if the wikipedia is editable.