# Proof of equilibrium equations?

• bhsmith
In summary, the proof of equilibrium equations involves understanding that a system in equilibrium has an acceleration of zero, which implies that the mass must also be zero. The moment can be determined using Newton's first and third laws, where the resultant force on the body is zero and reaction forces are considered.
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.

## 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!

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

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?

bhsmith said:
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 1st 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 3rd Law - You'd pretty much use this to get the sum of forces, for example reaction forces.

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

## What is the definition of equilibrium?

Equilibrium is a state of balance or stability in a system, where the forces acting on the system are equal and opposite, resulting in no net change in the system.

## How are equilibrium equations used in science?

Equilibrium equations are used to describe and predict the behavior of systems in balance, such as in physics, chemistry, and engineering. They help us understand the forces and interactions at play in a system and how they affect its overall state.

## What are the different types of equilibrium?

There are three types of equilibrium: stable, unstable, and neutral. In stable equilibrium, a system returns to its original state after being disturbed. In unstable equilibrium, a system moves away from its original state after being disturbed. In neutral equilibrium, a system remains in its new state after being disturbed.

## How do we know if a system is in equilibrium?

A system is in equilibrium when all the forces acting on it are balanced and there is no net movement or change in the system. This can be determined by analyzing the forces acting on the system and ensuring that they are equal and opposite.

## What are the fundamental equilibrium equations?

The fundamental equilibrium equations are the equations of static equilibrium, which state that the sum of all forces acting on a system must be equal to zero, and the sum of all torques acting on a system must also be equal to zero. These equations are used to analyze and predict the behavior of objects in balance.

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