stable equilibrium

Examples of stable equilibrium in the following topics:

• Thermal Instability

  • If the power absorbed and generated within the gas equals the power emitted by the gas, the temperature of the gas will remain constant and equilibrium is achieved.
  • The question remains whether this equilibrium is stable.
  • Heuristically we can see that if the cooling rate increases faster with temperature than the heating rate, then a slight increase in temperature will result in the gas cooling faster and the temperature returning to its equilibrium value.

• Stability, Balance, and Center of Mass

  • An object in static equilibrium remains in the same state forever, but not all forms of equilibrium are the same.
  • When the first derivative is zero, we can take the second derivative to find whether the equilibrium is stable or unstable.
  • \frac{d^2 U(x)} {dx^2} \right|_{x_0} > 0$, then the system is stable; conversely, if the potential is concave-down, then the equilibrium is unstable.
  • If the second derivative is zero or does not exist, then the equilibrium is neutral—neither stable nor unstable.
  • This is an example of unstable equilibrium.

• Period of a Mass on a Spring

  • Once released, the restoring force causes the ruler to move back toward its stable equilibrium position, where the net force on it is zero.
  • The maximum displacement from equilibrium is known as the amplitude X .
  • The object's maximum speed occurs as it passes through equilibrium.
  • (d) The equilibrium point is reach again, this time with momentum to the right.
  • It stops the ruler and moves it back toward equilibrium again.

• Problems

  • This is the thickness of a layer in thermal equilibrium.
  • If the surface luminosity increases faster with temperature than the helium burning rate, then the layer is stable.

• First Condition

  • The first condition of equilibrium is that the net force in all directions must be zero.
  • For an object to be in equilibrium, it must be experiencing no acceleration.
  • The condition $F_\text{net} = 0$ must be true for both static equilibrium, where the object's velocity is zero, and dynamic equilibrium, where the object is moving at a constant velocity.
  • Below, the motionless person is in static equilibrium.
  • This car is in dynamic equilibrium because it is moving at constant velocity.

• Radioactive Decay Series: Introduction

  • Most radioactive elements do not decay directly to a stable state; rather, they undergo a series of decays until eventually a stable isotope is reached.
  • The daughter isotope may be stable, or it may itself decay to form a daughter isotope of its own.
  • When equilibrium is achieved, a granddaughter isotope is present in proportion to its half-life.

• The Zeroth Law of Thermodynamics

  • The Zeroth Law of Thermodynamics states that systems in thermal equilibrium are at the same temperature.
  • If A and C are in thermal equilibrium, and A and B are in thermal equilibrium, then B and C are in thermal equilibrium.
  • Temperature is the quantity that is always the same for all systems in thermal equilibrium with one another.
  • The double arrow represents thermal equilibrium between systems.
  • If systems A and C are in equilibrium, and systems A and B are in equilibrium, then systems B and C are in equilibrium.

• Translational Equilibrium

  • This implies two types of possible equilibrium.
  • Static or dynamic, these kinds of equilibrium can be categorized as translational equilibrium.
  • Examples of translational equilibrium are all around us.
  • These six forces are in equilibrium.
  • Assess the role each type of equilibrium plays in mechanical devices

• A Review of the Zeroth Law

  • Zeroth law justifies the use of thermodynamic temperature, defined as the shared temperature of three designated systems at equilibrium.
  • The Zeroth Law of Thermodynamics states: If two systems, A and B, are in thermal equilibrium with each other, and B is in thermal equilibrium with a third system, C, then A is also in thermal equilibrium with C.
  • Two systems are in thermal equilibrium if they could transfer heat between each other, but don't.
  • Indeed, experiments have shown that if two systems, A and B, are in thermal equilibrium with each other, and B is in thermal equilibrium with a third system C, then A is also in thermal equilibrium with C.
  • The objects are then in thermal equilibrium, and no further changes will occur.

• Second Condition

  • The second condition of static equilibrium says that the net torque acting on the object must be zero.
  • A child's seesaw, shown in , is an example of static equilibrium.
  • An object in static equilibrium is one that has no acceleration in any direction.
  • If a given object is in static equilibrium, both the net force and the net torque on the object must be zero.
  • The system is in static equilibrium, showing no acceleration in any direction.