Nick Connor

Specific Internal Energy

The specific internal energy (u) of a substance is its internal energy per unit mass. It is an intensive property. It equals to the total internal energy (U) divided by the total mass (m).

u = U/m

where:

u = specific internal energy (J/kg)

U = internal energy (J)

m = mass (kg)

Internal Energy – Thermal Energy

Potential energy and kinetic energy, that was discussed in previous chapters, are macroscopic forms of energy. They are dependent on macroscopic variables such as the position and the velocity of objects.

In thermodynamics, internal energy (also called the thermal energy) is defined as the energy associated with microscopic forms of energy. It is an extensive quantity, it depends on the size of the system, or on the amount of substance it contains. The SI unit of internal energy is the joule (J). It is the energy contained within the system, excluding the kinetic energy of motion of the system as a whole and the potential energy of the system. Microscopic forms of energy include those due to the rotation, vibration, translation, and interactions among the molecules of a substance. None of these forms of energy can be measured or evaluated directly, but techniques have been developed to evaluate the change in the total sum of all these microscopic forms of energy.

In addition, energy is can be stored in the chemical bonds between the atoms that make up the molecules. This energy storage on the atomic level includes energy associated with electron orbital states, nuclear spin, and binding forces in the nucleus.

Internal energy is represented by the symbol U, and the change in internal energy in a process is U₂ – U₁.

Microscopic Energy

Internal energy involves energy on the microscopic scale. It may be divided into microscopic potential energy, U_pot, and microscopic kinetic energy, U_kin, components:

U = U_pot + U_kin

where the microscopic kinetic energy, U_kin, involves the motions of all the system’s particles with respect to the center-of-mass frame. For an ideal monatomic gas, this is just the translational kinetic energy of the linear motion of the atoms. Monoatomic particles do not rotate or vibrate. The behavior of the system is well described by kinetic theory of gases. Kinetic theory is based on the fact that during an elastic collision between a molecule with high kinetic energy and one with low kinetic energy, part of energy will transfer to the molecule of lower kinetic energy. However, for polyatomic gases there is rotational and vibrational kinetic energy as well.

The microscopic potential energy, U_pot, involves the chemical bonds between the atoms that make up the molecules, binding forces in the nucleus and also the physical force fields within the system (e.g. electric or magnetic fields).

In liquids and solids there is significant component of potential energy associated with the intermolecular attractive forces.

See also:

Internal Energy

Microscopic Energy

Internal energy involves energy on the microscopic scale. It may be divided into microscopic potential energy, U_pot, and microscopic kinetic energy, U_kin, components:

U = U_pot + U_kin

where the microscopic kinetic energy, U_kin, involves the motions of all the system’s particles with respect to the center-of-mass frame. For an ideal monatomic gas, this is just the translational kinetic energy of the linear motion of the atoms. Monoatomic particles do not rotate or vibrate. The behavior of the system is well described by kinetic theory of gases. Kinetic theory is based on the fact that during an elastic collision between a molecule with high kinetic energy and one with low kinetic energy, part of energy will transfer to the molecule of lower kinetic energy. However, for polyatomic gases there is rotational and vibrational kinetic energy as well.

The microscopic potential energy, U_pot, involves the chemical bonds between the atoms that make up the molecules, binding forces in the nucleus and also the physical force fields within the system (e.g. electric or magnetic fields).

In liquids and solids there is significant component of potential energy associated with the intermolecular attractive forces.

Internal Energy of an Ideal Gas

Monatomic Gas

For a monatomic ideal gas (such as helium, neon, or argon), the only contribution to the energy comes from translational kinetic energy. The average translational kinetic energy of a single atom depends only on the gas temperature and is given by equation

K_avg = 3/2 kT.

The internal energy of n moles of an ideal monatomic (one atom per molecule) gas is equal to the average kinetic energy per molecule times the total number of molecules, N:

E_int = 3/2 NkT = 3/2 nRT

where n is the number of moles. Each direction (x, y, and z) contributes (1/2)nRT to the internal energy. This is where the equipartition of energy idea comes in – any other contribution to the energy must also contribute (1/2)nRT. As can be seen, the internal energy of an ideal gas depends only on temperature and the number of moles of gas.

Diatomic Molecule

If the gas molecules contain more than one atom, there are three translation directions, and rotational kinetic energy also contributes, but only for rotations about two of the three perpendicular axes. The five contributions to the energy (five degrees of freedom) give:

Diatomic ideal gas:

E_int = (5/2)NkT = (5/2)nRT

This is only an approximation and applies at intermediate temperatures. At low temperatures only the translational kinetic energy contributes, and at higher temperatures two additional contributions (kinetic and potential energy) come from vibration.

The internal energy will be greater at a given temperature than for a monatomic gas, but it will still be a function only of temperature for an ideal gas.

The internal energy of real gases also depends mainly on temperature, but similarly as the Ideal Gas Law, the internal energy of real gases depends also somewhat on pressure and volume. All real gases approach the ideal state at low pressures (densities). At low pressures molecules are far enough apart that they do not interact with one another. The internal energy of liquids and solids is quite complicated, for it includes electrical potential energy associated with the forces (or “chemical” bonds) between atoms and molecules.

Joule’s Second Law

For any gas whose equation of state is given exactly by pV = nRT (or pv = RT), the specific internal energy depends on temperature only. This rule was originally found in 1843 by Joule experimentally for real gases and is known as Joule’s second law:

The internal energy of a fixed mass of an ideal gas depends only on its temperature (not pressure or volume).

The specific enthalpy of a gas described by pV = nRT also depends on temperature only. Note that  the enthalpy is the thermodynamic quantity equivalent to the total heat content of a system. It is equal to the internal energy of the system plus the product of pressure and volume. In intensive variables the Joule’s second law is therefore given by h = h(T) = u(T) + pv = u(T) + RT.

These three equations constitute the ideal gas model, summarized as follows:

pv = RT

u = u(T)

h = h(T) = u(T) + RT

See also:

Internal Energy

Microscopic Energy

Internal energy involves energy on the microscopic scale. It may be divided into microscopic potential energy, U_pot, and microscopic kinetic energy, U_kin, components:

U = U_pot + U_kin

where the microscopic kinetic energy, U_kin, involves the motions of all the system’s particles with respect to the center-of-mass frame. For an ideal monatomic gas, this is just the translational kinetic energy of the linear motion of the atoms. Monoatomic particles do not rotate or vibrate. The behavior of the system is well described by kinetic theory of gases. Kinetic theory is based on the fact that during an elastic collision between a molecule with high kinetic energy and one with low kinetic energy, part of energy will transfer to the molecule of lower kinetic energy. However, for polyatomic gases there is rotational and vibrational kinetic energy as well.

The microscopic potential energy, U_pot, involves the chemical bonds between the atoms that make up the molecules, binding forces in the nucleus and also the physical force fields within the system (e.g. electric or magnetic fields).

In liquids and solids there is significant component of potential energy associated with the intermolecular attractive forces.

Specific Internal Energy

The specific internal energy (u) of a substance is its internal energy per unit mass. It is an intensive property. It equals to the total internal energy (U) divided by the total mass (m).

u = U/m

where:

u = specific internal energy (J/kg)

U = internal energy (J)

m = mass (kg)

Internal Energy of an Ideal Gas

Monatomic Gas

For a monatomic ideal gas (such as helium, neon, or argon), the only contribution to the energy comes from translational kinetic energy. The average translational kinetic energy of a single atom depends only on the gas temperature and is given by equation

K_avg = 3/2 kT.

The internal energy of n moles of an ideal monatomic (one atom per molecule) gas is equal to the average kinetic energy per molecule times the total number of molecules, N:

E_int = 3/2 NkT = 3/2 nRT

where n is the number of moles. Each direction (x, y, and z) contributes (1/2)nRT to the internal energy. This is where the equipartition of energy idea comes in – any other contribution to the energy must also contribute (1/2)nRT. As can be seen, the internal energy of an ideal gas depends only on temperature and the number of moles of gas.

Diatomic Molecule

If the gas molecules contain more than one atom, there are three translation directions, and rotational kinetic energy also contributes, but only for rotations about two of the three perpendicular axes. The five contributions to the energy (five degrees of freedom) give:

Diatomic ideal gas:

E_int = (5/2)NkT = (5/2)nRT

This is only an approximation and applies at intermediate temperatures. At low temperatures only the translational kinetic energy contributes, and at higher temperatures two additional contributions (kinetic and potential energy) come from vibration.

The internal energy will be greater at a given temperature than for a monatomic gas, but it will still be a function only of temperature for an ideal gas.

The internal energy of real gases also depends mainly on temperature, but similarly as the Ideal Gas Law, the internal energy of real gases depends also somewhat on pressure and volume. All real gases approach the ideal state at low pressures (densities). At low pressures molecules are far enough apart that they do not interact with one another. The internal energy of liquids and solids is quite complicated, for it includes electrical potential energy associated with the forces (or “chemical” bonds) between atoms and molecules.

Internal Energy and the First Law of Thermodynamics

In thermodynamics the concept of energy is broadened to account for other observed changes, and the principle of conservation of energy is extended to include a wide variety of ways in which systems interact with their surroundings. The only ways the energy of a closed system can be changed are through transfer of energy by work or by heat. Further, based on the experiments of Joule and others, a fundamental aspect of the energy concept is that energy is conserved. This principle is known as the first law of thermodynamics. The first law of thermodynamics can be written in various forms:

In words:

Equation form:

∆E_int = Q – W

where E_int represents the internal energy of the material, which depends only on the material’s state (temperature, pressure, and volume). Q is the net heat added to the system and W is the net work done by the system. We must be careful and consistent in following the sign conventions for Q and W. Because W in the equation is the work done by the system, then if work is done on the system, W will be negative and E_int will increase.

Similarly, Q is positive for heat added to the system, so if heat leaves the system, Q is negative. This tells us the following: The internal energy of a system tends to increase if heat is absorbed by the system or if positive work is done on the system. Conversely, the internal energy tends to decrease if heat is lost by the system or if negative work is done on the system. It must be added Q and W are path dependent, while E_int is path independent.

Differential form:

dE_int = dQ  – dW

The internal energy E_int of a system tends to increase if energy is added as heat Q and tends to decrease if energy is lost as work W done by the system.

See also: Open System – Closed System – Isolated System

See also:

Internal Energy

Internal Energy – Thermal Energy

Potential energy and kinetic energy, that was discussed in previous chapters, are macroscopic forms of energy. They are dependent on macroscopic variables such as the position and the velocity of objects.

In thermodynamics, internal energy (also called the thermal energy) is defined as the energy associated with microscopic forms of energy. It is an extensive quantity, it depends on the size of the system, or on the amount of substance it contains. The SI unit of internal energy is the joule (J). It is the energy contained within the system, excluding the kinetic energy of motion of the system as a whole and the potential energy of the system. Microscopic forms of energy include those due to the rotation, vibration, translation, and interactions among the molecules of a substance. None of these forms of energy can be measured or evaluated directly, but techniques have been developed to evaluate the change in the total sum of all these microscopic forms of energy.

In addition, energy is can be stored in the chemical bonds between the atoms that make up the molecules. This energy storage on the atomic level includes energy associated with electron orbital states, nuclear spin, and binding forces in the nucleus.

Internal energy is represented by the symbol U, and the change in internal energy in a process is U₂ – U₁.

Microscopic Energy

Internal energy involves energy on the microscopic scale. It may be divided into microscopic potential energy, U_pot, and microscopic kinetic energy, U_kin, components:

U = U_pot + U_kin

where the microscopic kinetic energy, U_kin, involves the motions of all the system’s particles with respect to the center-of-mass frame. For an ideal monatomic gas, this is just the translational kinetic energy of the linear motion of the atoms. Monoatomic particles do not rotate or vibrate. The behavior of the system is well described by kinetic theory of gases. Kinetic theory is based on the fact that during an elastic collision between a molecule with high kinetic energy and one with low kinetic energy, part of energy will transfer to the molecule of lower kinetic energy. However, for polyatomic gases there is rotational and vibrational kinetic energy as well.

The microscopic potential energy, U_pot, involves the chemical bonds between the atoms that make up the molecules, binding forces in the nucleus and also the physical force fields within the system (e.g. electric or magnetic fields).

In liquids and solids there is significant component of potential energy associated with the intermolecular attractive forces.

Specific Internal Energy

The specific internal energy (u) of a substance is its internal energy per unit mass. It is an intensive property. It equals to the total internal energy (U) divided by the total mass (m).

u = U/m

where:

u = specific internal energy (J/kg)

U = internal energy (J)

m = mass (kg)

Internal Energy of an Ideal Gas

Monatomic Gas

For a monatomic ideal gas (such as helium, neon, or argon), the only contribution to the energy comes from translational kinetic energy. The average translational kinetic energy of a single atom depends only on the gas temperature and is given by equation

K_avg = 3/2 kT.

The internal energy of n moles of an ideal monatomic (one atom per molecule) gas is equal to the average kinetic energy per molecule times the total number of molecules, N:

E_int = 3/2 NkT = 3/2 nRT

where n is the number of moles. Each direction (x, y, and z) contributes (1/2)nRT to the internal energy. This is where the equipartition of energy idea comes in – any other contribution to the energy must also contribute (1/2)nRT. As can be seen, the internal energy of an ideal gas depends only on temperature and the number of moles of gas.

Diatomic Molecule

If the gas molecules contain more than one atom, there are three translation directions, and rotational kinetic energy also contributes, but only for rotations about two of the three perpendicular axes. The five contributions to the energy (five degrees of freedom) give:

Diatomic ideal gas:

E_int = (5/2)NkT = (5/2)nRT

This is only an approximation and applies at intermediate temperatures. At low temperatures only the translational kinetic energy contributes, and at higher temperatures two additional contributions (kinetic and potential energy) come from vibration.

The internal energy will be greater at a given temperature than for a monatomic gas, but it will still be a function only of temperature for an ideal gas.

The internal energy of real gases also depends mainly on temperature, but similarly as the Ideal Gas Law, the internal energy of real gases depends also somewhat on pressure and volume. All real gases approach the ideal state at low pressures (densities). At low pressures molecules are far enough apart that they do not interact with one another. The internal energy of liquids and solids is quite complicated, for it includes electrical potential energy associated with the forces (or “chemical” bonds) between atoms and molecules.

Joule’s Second Law

For any gas whose equation of state is given exactly by pV = nRT (or pv = RT), the specific internal energy depends on temperature only. This rule was originally found in 1843 by Joule experimentally for real gases and is known as Joule’s second law:

The internal energy of a fixed mass of an ideal gas depends only on its temperature (not pressure or volume).

The specific enthalpy of a gas described by pV = nRT also depends on temperature only. Note that  the enthalpy is the thermodynamic quantity equivalent to the total heat content of a system. It is equal to the internal energy of the system plus the product of pressure and volume. In intensive variables the Joule’s second law is therefore given by h = h(T) = u(T) + pv = u(T) + RT.

These three equations constitute the ideal gas model, summarized as follows:

pv = RT

u = u(T)

h = h(T) = u(T) + RT

Internal Energy and the First Law of Thermodynamics

In thermodynamics the concept of energy is broadened to account for other observed changes, and the principle of conservation of energy is extended to include a wide variety of ways in which systems interact with their surroundings. The only ways the energy of a closed system can be changed are through transfer of energy by work or by heat. Further, based on the experiments of Joule and others, a fundamental aspect of the energy concept is that energy is conserved. This principle is known as the first law of thermodynamics. The first law of thermodynamics can be written in various forms:

In words:

Equation form:

∆E_int = Q – W

where E_int represents the internal energy of the material, which depends only on the material’s state (temperature, pressure, and volume). Q is the net heat added to the system and W is the net work done by the system. We must be careful and consistent in following the sign conventions for Q and W. Because W in the equation is the work done by the system, then if work is done on the system, W will be negative and E_int will increase.

Similarly, Q is positive for heat added to the system, so if heat leaves the system, Q is negative. This tells us the following: The internal energy of a system tends to increase if heat is absorbed by the system or if positive work is done on the system. Conversely, the internal energy tends to decrease if heat is lost by the system or if negative work is done on the system. It must be added Q and W are path dependent, while E_int is path independent.

Differential form:

dE_int = dQ  – dW

The internal energy E_int of a system tends to increase if energy is added as heat Q and tends to decrease if energy is lost as work W done by the system.

See also: Open System – Closed System – Isolated System

Distinguishing Temperature, Heat, and Internal Energy

Using the kinetic theory, a clear distinction between these three properties can be made.

• Temperature is related to the kinetic energies of the molecules of a material. It is the average kinetic energy of individual molecules.
• Internal energy refers to the total energy of all the molecules within the object. It is an extensive property, therefore when two equal-mass hot ingots of steel may have the same temperature, but two of them have twice as much internal energy as one does.
• Finally, heat is the amount of energy flowing from one body to another spontaneously due to their temperature difference.

It must be added, when a temperature difference does exist heat flows spontaneously from the warmer system to the colder system. Thus, if a 5 kg cube of steel at 100°C is placed in contact with a 500 kg cube of steel at 20°C, heat flows from the cube at 300°C to the cube at 20°C even though the internal energy of the 20°C cube is much greater because there is so much more of it.

A particularly important concept is thermodynamic equilibrium. In general, when two objects are brought into thermal contact, heat will flow between them until they come into equilibrium with each other.

See also:

Energy

Gravitational Potential Energy

In classical mechanics, the gravitational potential energy (U) is energy an object possesses because of its position in a gravitational field. Gravitational potential (V; the gravitational energy per unit mass) at a location is equal to the work (energy transferred) per unit mass that would be needed to move the object from a fixed reference location to the location of the object. The most common use of gravitational potential energy is for an object near the surface of the Earth where the gravitational acceleration can be assumed to be constant at about 9.8 m/s².

U = mgh

Conservation of Mechanical Energy

First the principle of the Conservation of Mechanical Energy was stated:

The total mechanical energy (defined as the sum of its potential and kinetic energies) of a particle being acted on by only conservative forces is constant.

See also: Conservation of Mechanical Energy

An isolated system is one in which no external force causes energy changes. If only conservative forces act on an object and U is the potential energy function for the total conservative force, then

E_mech = U + K

The potential energy, U, depends on the position of an object subjected to a conservative force.

It is defined as the object’s ability to do work and is increased as the object is moved in the opposite direction of the direction of the force.

The potential energy associated with a system consisting of Earth and a nearby particle is gravitational potential energy.

The kinetic energy, K, depends on the speed of an object and is the ability of a moving object to do work on other objects when it collides with them.

 K = ½ mv²

The above mentioned definition (E_mech = U + K) assumes that the system is free of friction and other non-conservative forces. The difference between a conservative and a non-conservative force is that when a conservative force moves an object from one point to another, the work done by the conservative force is independent of the path.

In any real situation, frictional forces and other non-conservative forces are present, but in many cases their effects on the system are so small that the principle of conservation of mechanical energy can be used as a fair approximation. For example the frictional force is a non-conservative force, because it acts to reduce the mechanical energy in a system.

Note that non-conservative forces do not always reduce the mechanical energy. A non-conservative force changes the mechanical energy, there are forces that increase the total mechanical energy, like the force provided by a motor or engine, is also a non-conservative force.

Block sliding down a frictionless incline slope

The 1 kg block starts out a height H (let say 1 m) above the ground, with potential energy mgH and kinetic energy that is equal to 0. It slides to the ground (without friction) and arrives with no potential energy and kinetic energy K = ½ mv². Calculate the velocity of the block on the ground and its kinetic energy.

E_mech = U + K = const

=> ½ mv² = mgH

=> v = √2gH = 4.43 m/s

=> K₂ = ½ x 1 kg x (4.43 m/s)² = 19.62 kg.m².s^-2 = 19.62 J

Pendulum

Assume a pendulum (ball of mass m suspended on a string of length L that we have pulled up so that the ball is a height H < L above its lowest point on the arc of its stretched string motion. The pendulum is subjected to the conservative gravitational force where frictional forces like air drag and friction at the pivot are negligible.

We release it from rest. How fast is it going at the bottom?

The pendulum reaches greatest kinetic energy and least potential energy when in the vertical position, because it will have the greatest speed and be nearest the Earth at this point. On the other hand, it will have its least kinetic energy and greatest potential energy at the extreme positions of its swing, because it has zero speed and is farthest from Earth at these points.

If the amplitude is limited to small swings, the period T of a simple pendulum, the time taken for a complete cycle, is:

where L is the length of the pendulum and g is the local acceleration of gravity. For small swings the period of swing is approximately the same for different size swings. That is, the period is independent of amplitude.

See also:

Potential Energy

Examples of Potential Energy

Gravitational potential energy:

In classical mechanics, the gravitational potential energy (U) is energy an object possesses because of its position in a gravitational field. Gravitational potential (V; the gravitational energy per unit mass) at a location is equal to the work (energy transferred) per unit mass that would be needed to move the object from a fixed reference location to the location of the object. The most common use of gravitational potential energy is for an object near the surface of the Earth where the gravitational acceleration can be assumed to be constant at about 9.8 m/s².

U = mgh

Elastic potential energy:

Elastic potential energy is potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring. It is dependent upon the spring constant k as well as the distance stretched.

U = 1/2 kx²

Electric potential energy:

Electric potential energy is a potential energy that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. For example, if a positive charge Q is fixed at some point in space, any other positive charge which is brought close to it will experience a repulsive force and will therefore have potential energy.

U = kQq/r

Block sliding down a frictionless incline slope

The 1 kg block starts out a height H (let say 1 m) above the ground, with potential energy mgH and kinetic energy that is equal to 0. It slides to the ground (without friction) and arrives with no potential energy and kinetic energy K = ½ mv². Calculate the velocity of the block on the ground and its kinetic energy.

E_mech = U + K = const

=> ½ mv² = mgH

=> v = √2gH = 4.43 m/s

=> K₂ = ½ x 1 kg x (4.43 m/s)² = 19.62 kg.m².s^-2 = 19.62 J

Pendulum

Assume a pendulum (ball of mass m suspended on a string of length L that we have pulled up so that the ball is a height H < L above its lowest point on the arc of its stretched string motion. The pendulum is subjected to the conservative gravitational force where frictional forces like air drag and friction at the pivot are negligible.

We release it from rest. How fast is it going at the bottom?

The pendulum reaches greatest kinetic energy and least potential energy when in the vertical position, because it will have the greatest speed and be nearest the Earth at this point. On the other hand, it will have its least kinetic energy and greatest potential energy at the extreme positions of its swing, because it has zero speed and is farthest from Earth at these points.

If the amplitude is limited to small swings, the period T of a simple pendulum, the time taken for a complete cycle, is:

where L is the length of the pendulum and g is the local acceleration of gravity. For small swings the period of swing is approximately the same for different size swings. That is, the period is independent of amplitude.

See also:

Potential Energy

What is Potential Energy

Potential energy, U, is defined as the energy stored in an object subjected to a conservative force. Common types include the gravitational potential energy, the elastic potential energy of an extended spring, and the electric potential energy of an electric charge in an electric field and so on.

Let assume the mechanical energy (E_mech), which is the energy associated with the motion and position of an object usually in some force field (e.g. gravitational field). Mechanical energy (and also the thermal energy) can be separated into two categories, transient and stored. Transient energy is energy in motion, that is, energy being transferred from one place to another. Stored energy is the energy contained within a substance or object. Transient mechanical energy is commonly referred to as work. Stored mechanical energy exists in one of two forms: kinetic or potential:

• Potential energy. Potential energy, U, is defined as the energy stored in an object subjected to a conservative force. Common types include the gravitational potential energy of an object that depends on its mass and its distance from the center of mass of another object.
• Kinetic energy. The kinetic energy, K, is defined as the energy stored in an object because of its motion. It depends on the speed of an object and is the ability of a moving object to do work on other objects when it collides with them.

Examples of Potential Energy

Gravitational potential energy:

In classical mechanics, the gravitational potential energy (U) is energy an object possesses because of its position in a gravitational field. Gravitational potential (V; the gravitational energy per unit mass) at a location is equal to the work (energy transferred) per unit mass that would be needed to move the object from a fixed reference location to the location of the object. The most common use of gravitational potential energy is for an object near the surface of the Earth where the gravitational acceleration can be assumed to be constant at about 9.8 m/s².

U = mgh

Elastic potential energy:

Elastic potential energy is potential energy stored as a result of deformation of an elastic object, such as the stretching of a spring. It is dependent upon the spring constant k as well as the distance stretched.

U = 1/2 kx²

Electric potential energy:

Electric potential energy is a potential energy that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. For example, if a positive charge Q is fixed at some point in space, any other positive charge which is brought close to it will experience a repulsive force and will therefore have potential energy.

U = kQq/r

Conservation of Mechanical Energy

First the principle of the Conservation of Mechanical Energy was stated:

The total mechanical energy (defined as the sum of its potential and kinetic energies) of a particle being acted on by only conservative forces is constant.

See also: Conservation of Mechanical Energy

An isolated system is one in which no external force causes energy changes. If only conservative forces act on an object and U is the potential energy function for the total conservative force, then

E_mech = U + K

The potential energy, U, depends on the position of an object subjected to a conservative force.

It is defined as the object’s ability to do work and is increased as the object is moved in the opposite direction of the direction of the force.

The potential energy associated with a system consisting of Earth and a nearby particle is gravitational potential energy.

The kinetic energy, K, depends on the speed of an object and is the ability of a moving object to do work on other objects when it collides with them.

 K = ½ mv²

The above mentioned definition (E_mech = U + K) assumes that the system is free of friction and other non-conservative forces. The difference between a conservative and a non-conservative force is that when a conservative force moves an object from one point to another, the work done by the conservative force is independent of the path.

In any real situation, frictional forces and other non-conservative forces are present, but in many cases their effects on the system are so small that the principle of conservation of mechanical energy can be used as a fair approximation. For example the frictional force is a non-conservative force, because it acts to reduce the mechanical energy in a system.

Note that non-conservative forces do not always reduce the mechanical energy. A non-conservative force changes the mechanical energy, there are forces that increase the total mechanical energy, like the force provided by a motor or engine, is also a non-conservative force.

Block sliding down a frictionless incline slope

The 1 kg block starts out a height H (let say 1 m) above the ground, with potential energy mgH and kinetic energy that is equal to 0. It slides to the ground (without friction) and arrives with no potential energy and kinetic energy K = ½ mv². Calculate the velocity of the block on the ground and its kinetic energy.

E_mech = U + K = const

=> ½ mv² = mgH

=> v = √2gH = 4.43 m/s

=> K₂ = ½ x 1 kg x (4.43 m/s)² = 19.62 kg.m².s^-2 = 19.62 J

Pendulum

Assume a pendulum (ball of mass m suspended on a string of length L that we have pulled up so that the ball is a height H < L above its lowest point on the arc of its stretched string motion. The pendulum is subjected to the conservative gravitational force where frictional forces like air drag and friction at the pivot are negligible.

We release it from rest. How fast is it going at the bottom?

The pendulum reaches greatest kinetic energy and least potential energy when in the vertical position, because it will have the greatest speed and be nearest the Earth at this point. On the other hand, it will have its least kinetic energy and greatest potential energy at the extreme positions of its swing, because it has zero speed and is farthest from Earth at these points.

If the amplitude is limited to small swings, the period T of a simple pendulum, the time taken for a complete cycle, is:

where L is the length of the pendulum and g is the local acceleration of gravity. For small swings the period of swing is approximately the same for different size swings. That is, the period is independent of amplitude.

See also:

Energy

What is Mechanical Energy

In physics, mechanical energy (E_mech) is the energy associated with the motion and position of an object usually in some force field (e.g. gravitational field). Mechanical energy (and also the thermal energy) can be separated into two categories, transient and stored. Transient energy is energy in motion, that is, energy being transferred from one place to another. Stored energy is the energy contained within a substance or object. Transient mechanical energy is commonly referred to as work. Stored mechanical energy exists in one of two forms: kinetic or potential:

• Potential energy. Potential energy, U, is defined as the energy stored in an object subjected to a conservative force. Common types include the gravitational potential energy of an object that depends on its mass and its distance from the center of mass of another object.
• Kinetic energy. The kinetic energy, K, is defined as the energy stored in an object because of its motion. It depends on the speed of an object and is the ability of a moving object to do work on other objects when it collides with them.

Example: Block sliding down a frictionless incline slope

The 1 kg block starts out a height H (let say 1 m) above the ground, with potential energy mgH and kinetic energy that is equal to 0. It slides to the ground (without friction) and arrives with no potential energy and kinetic energy K = ½ mv². Calculate the velocity of the block on the ground and its kinetic energy.

E_mech = U + K = const

=> ½ mv² = mgH

=> v = √2gH = 4.43 m/s

=> K₂ = ½ x 1 kg x (4.43 m/s)² = 19.62 kg.m².s^-2 = 19.62 J

Example: Pendulum

Assume a pendulum (ball of mass m suspended on a string of length L that we have pulled up so that the ball is a height H < L above its lowest point on the arc of its stretched string motion. The pendulum is subjected to the conservative gravitational force where frictional forces like air drag and friction at the pivot are negligible.

We release it from rest. How fast is it going at the bottom?

The pendulum reaches greatest kinetic energy and least potential energy when in the vertical position, because it will have the greatest speed and be nearest the Earth at this point. On the other hand, it will have its least kinetic energy and greatest potential energy at the extreme positions of its swing, because it has zero speed and is farthest from Earth at these points.

If the amplitude is limited to small swings, the period T of a simple pendulum, the time taken for a complete cycle, is:

where L is the length of the pendulum and g is the local acceleration of gravity. For small swings the period of swing is approximately the same for different size swings. That is, the period is independent of amplitude.

See also:

Mechanical Energy

Mechanical Energy Formula

In physics, mechanical energy (E_mech) is the energy associated with the motion and position of an object usually in some force field (e.g. gravitational field). Mechanical energy (and also the thermal energy) can be separated into two categories, transient and stored. Transient energy is energy in motion, that is, energy being transferred from one place to another. Stored energy is the energy contained within a substance or object. Transient mechanical energy is commonly referred to as work. Stored mechanical energy exists in one of two forms: kinetic or potential:

• Potential energy. Potential energy, U, is defined as the energy stored in an object subjected to a conservative force. Common types include the gravitational potential energy of an object that depends on its mass and its distance from the center of mass of another object.
• Kinetic energy. The kinetic energy, K, is defined as the energy stored in an object because of its motion. It depends on the speed of an object and is the ability of a moving object to do work on other objects when it collides with them.

See also: Conservation of Mechanical Energy

An isolated system is one in which no external force causes energy changes. If only conservative forces act on an object and U is the potential energy function for the total conservative force, then

E_mech = U + K

The potential energy, U, depends on the position of an object subjected to a conservative force.

It is defined as the object’s ability to do work and is increased as the object is moved in the opposite direction of the direction of the force.

The potential energy associated with a system consisting of Earth and a nearby particle is gravitational potential energy.

The kinetic energy, K, depends on the speed of an object and is the ability of a moving object to do work on other objects when it collides with them.

 K = ½ mv²

The above mentioned definition (E_mech = U + K) assumes that the system is free of friction and other non-conservative forces. The difference between a conservative and a non-conservative force is that when a conservative force moves an object from one point to another, the work done by the conservative force is independent of the path.

Block sliding down a frictionless incline slope

The 1 kg block starts out a height H (let say 1 m) above the ground, with potential energy mgH and kinetic energy that is equal to 0. It slides to the ground (without friction) and arrives with no potential energy and kinetic energy K = ½ mv². Calculate the velocity of the block on the ground and its kinetic energy.

E_mech = U + K = const

=> ½ mv² = mgH

=> v = √2gH = 4.43 m/s

=> K₂ = ½ x 1 kg x (4.43 m/s)² = 19.62 kg.m².s^-2 = 19.62 J

See also:

Mechanical Energy

Kinetic Energy Formula

The kinetic energy, K, is defined as the energy stored in an object because of its motion. An object in motion has the ability to do work and thus can be said to have energy. It is called kinetic energy, from the Greek word kinetikos, meaning “motion.”

The kinetic energy depends on the speed of an object and is the ability of a moving object to do work on other objects when it collides with them. On the other hand, the kinetic energy of an object represents the amount of energy required to increase the velocity of the object from rest (v = 0) to its final velocity. The kinetic energy also depends linearly on the mass, which is a numerical measure of object’s inertia and the measure of an object’s resistance to acceleration when a force is applied.

Kinetic Energy Formula:

K = ½ mv²

It is the translational kinetic energy of the object. It must be added, it is called the “translational” kinetic energy to distinguish it from rotational kinetic energy.

Relativistic vs. Non-relativistic Kinetic Energy Formula

The previous relationship between work and kinetic energy are based on Newton’s laws of motion. When we generalize these laws according to the principle of relativity, we need a corresponding generalization of the equation for kinetic energy. If an object’s speed is close to the speed of light, it is necessary to use relativistic mechanics to calculate its kinetic energy.

In classical mechanics, kinetic energy and momentum are expressed as:

Derivation of its relativistic relationships is based on the relativistic energy-momentum relation:

It can be derived, the relativistic kinetic energy and the relativistic momentum are:

The first term (ɣmc²) of the relativistic kinetic energy increases with the speed v of the particle. The second term (mc²) is constant; it is called the rest energy (rest mass) of the particle, and represents a form of energy that a particle has even when at zero velocity. As velocity of an object approaches the speed of light, the kinetic energy approaches infinity. It is caused by the Lorentz factor, which approaches infinity for v → c. Therefore the speed of light cannot be reached by any massive particles.

The first term (ɣmc²) is known as the total energy E of the particle, because it equals the rest energy plus the kinetic energy:

E = K + mc²

For a particle at rest, i.e. K is zero, so the total energy is its rest energy:

E = mc²

This is one of the striking results of Einstein’s theory of relativity is that mass and energy are equivalent and convertible one into the other. Equivalence of the mass and energy is described by Einstein’s famous formula E = mc². This result have been experimentally confirmed countless times in nuclear and elementary particle physics. For example, see Positron-electron Pair Production or Conservation of Energy in Nuclear Reactions.

See also: Relativistic Mass

See also:

Kinetic Energy