Proof of equilibrium equations?

[bhsmith]

proof of equilibrium equations??

Homework Statement

I have a statics test tomorrow, and I have to give a proof type answer of how the equilibrium equations (sum of forces and sum of moment are zero) are derived.

Homework Equations

The Attempt at a Solution

I'm not sure how to do this, I understand that

because the system is in equilibrium then the acceleration is zero and Force=mass(aceleration) which would imply that mass is zero. But I don't think I'm going about that the right way.

As for the moment, I'm not sure what to do.

The way he did it in class was using Newtons first and third laws to prove it, but I didn't quite understand how he went about it.

Thanks for any help!

 

[rock.freak667]

I would have thought it would have been a simple case of Newton's 2^nd Law where ∑F_n = ma_n (acceleration in the 'n' direction) and in equilibrium, a_n = 0 m/s².

 

[bhsmith]

Ok, Maybe I will look into that more. I wasn't sure if i was going down the right path with that thought.
But do you have any thoughts on the moment?

 

[rock.freak667]

Ok, Maybe I will look into that more. I wasn't sure if i was going down the right path with that thought.
But do you have any thoughts on the moment?

Well for Newton's 1^st Law to apply, you'd need to know the resultant force on the body is zero for it to be at rest or in motion (in equilibrium, so you know a = 0)

For Newton's 3^rd Law - You'd pretty much use this to get the sum of forces, for example reaction forces.

 

[DeeNos]

I can provide a response to this content by explaining the mathematical derivation of the equilibrium equations.

First, it is important to understand that equilibrium means that the object or system is in a state of balance, where all forces acting on it are equal and opposite, and there is no net torque (or moment) acting on it.

The sum of forces in the x-direction (Fx) and the sum of forces in the y-direction (Fy) can be written as:

ΣFx = m * ax = 0
ΣFy = m * ay = 0

where m is the mass of the object and ax and ay are the accelerations in the x and y directions, respectively. Since the object is in equilibrium, its acceleration is zero, therefore the sum of forces in both directions must also be zero.

Similarly, the sum of moments (or torques) around a point can be written as:

ΣM = I * α = 0

where I is the moment of inertia and α is the angular acceleration. Again, since the object is in equilibrium, its angular acceleration is zero, therefore the sum of moments must also be zero.

To prove these equations, we can use Newton's second law (F=ma) and the definition of torque (M=Fr), where F is the force, m is the mass, a is the acceleration, r is the distance from the point of rotation, and α is the angular acceleration.

For the sum of forces in the x-direction, we can write:

ΣFx = F1x + F2x + ... + Fnx = ma = 0

where F1x, F2x, etc. are the forces acting in the x-direction. Since the object is in equilibrium, the sum of these forces must be zero.

Similarly, for the sum of forces in the y-direction, we can write:

ΣFy = F1y + F2y + ... + Fny = ma = 0

And for the sum of moments, we can write:

ΣM = r1F1sinθ1 + r2F2sinθ2 + ... + rnfnsinθn = Iα = 0

where rn is the distance from the point of rotation to each force, and θn is the angle between the force and the line connecting it to the point of rotation.

By setting these equations equal