Math Formula for Equilibrium

$Math Formula for Equilibrium - Formula Quest Mania$

Math Formula for Equilibrium

Introduction to Equilibrium

In mathematics and physics, equilibrium refers to a state in which opposing forces or influences are balanced. It plays a crucial role in mechanics, economics, and chemistry.

Types of Equilibrium

1. Static Equilibrium

Static equilibrium occurs when an object is at rest and the sum of all forces and torques acting on it is zero.

The mathematical condition for static equilibrium is given by:

Force equilibrium: $ \sum F = 0 $

Torque equilibrium: $ \sum \tau = 0 $

2. Dynamic Equilibrium

Dynamic equilibrium occurs when an object moves at a constant velocity with no net force acting upon it.

Newton’s First Law states:

\[ \sum F = 0 \Rightarrow \text{Constant velocity} \]

Mathematical Formulation

Force Equilibrium

For a system in equilibrium, the sum of all forces in each direction must be zero:

\[ \sum F_x = 0, \quad \sum F_y = 0, \quad \sum F_z = 0 \]

Torque Equilibrium

For rotational equilibrium, the sum of all torques about any axis must be zero:

\[ \sum \tau = r F \sin\theta = 0 \]

Example Problem

Example 1: Balancing a Beam

A uniform beam of length 10 m and mass 20 kg is supported at both ends. A person of mass 50 kg stands 4 m from one end. Find the reaction forces at the supports.

Solution:

Using torque equilibrium about one end:

\[ R_2 \times 10 - (20 \times 9.8 \times 5) - (50 \times 9.8 \times 4) = 0 \]

Solving for $ R_2 $, then using $ R_1 + R_2 = 686N $, we find the reaction forces.

Further Applications of Equilibrium

Mechanical Engineering

In mechanical engineering, equilibrium analysis helps in designing stable structures, bridges, and vehicles. Engineers ensure that forces and torques are balanced to prevent structural failure.

Economic Equilibrium

In economics, equilibrium refers to a state where supply equals demand. The mathematical representation is:

\[ Q_d = Q_s \]

where $ Q_d $ is the quantity demanded and $ Q_s $ is the quantity supplied.

Chemical Equilibrium

In chemistry, equilibrium occurs when the forward and reverse reaction rates are equal. The equilibrium constant $ K $ is given by:

\[ K = \frac{[C]^c [D]^d}{[A]^a [B]^b} \]

where $ A $ and $ B $ are reactants, and $ C $ and $ D $ are products.

Example Problem 2: Economic Equilibrium

Suppose a market has a demand function $ Q_d = 100 - 2P $ and a supply function $ Q_s = 20 + 3P $. Find the equilibrium price and quantity.

Solution:

Setting $ Q_d = Q_s $:

\[ 100 - 2P = 20 + 3P \]

Solving for $ P $:

\[ 5P = 80 \Rightarrow P = 16 \]

Substituting $ P = 16 $ into either equation to find $ Q $:

\[ Q = 100 - 2(16) = 68 \]

Thus, the equilibrium price is $ P = 16 $ and equilibrium quantity is $ Q = 68 $.

Conclusion

Equilibrium is a fundamental concept in various disciplines, from physics to economics and chemistry. Understanding equilibrium principles allows for better problem-solving in real-world applications. Whether balancing forces in a mechanical system, analyzing market dynamics, or determining reaction rates, equilibrium equations play a vital role.