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Relativistic Mass and Energy

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Relativistic Mass and Energy

Mass and energy are related quantities that we can respectively convert to the other by using the speed of light. In physics, mass is considered to be a form of energy because the mass of a particle can be converted into other forms of energy such as thermal energy, kinetic energy, etc. In the same way, kinetic energy or radiant energy can form particles with mass.

Mass-energy conversion

There are many examples of mass-energy conversions that occur in daily life. One of them is the reason the universe exists, the Big Bang. According to the Big Bang theory, the universe was formed after a gigantic amount of energy was released and converted into mass.

Relativistic Mass and Energy. Formation of the universe. StudySmarterFigure 1. How the universe was formed is a great example of energy-mass conversion. Source: Hubble ESA, Flickr (CC BY 2.0).

Other great examples of energy-mass conversion include nuclear power plants, nuclear weapons, and the sun. However, there is a difference in the way they convert mass to energy. For instance, while the sun converts mass to energy via nuclear fusion, nuclear power plants do it via nuclear fission.

Relativistic Mass and Energy. Energy to mass conversion. StudySmarterFigure 2. The sun and nuclear weapons both convert mass into energy. Sources: NASA Goddard Space Flight Center and Steve Knight, Flickr (CC BY 2.0).

How can mass be converted into energy?

The first postulate of Einsteins theory of special relativity states that the laws of physics are the same for all inertial frames of reference. When properly studied, this theory yields an expression for relativistic energy whose dependence on the inertial frame of reference is captured by a relativistic factor.

In order to understand relativistic energy, we have to take into consideration total energy and rest energy.

Total energy

Total energy E is the sum of all energies that an object with mass carries. Mathematically, it can be defined as:

Here, m is the mass, c is the speed of light, and y can be calculated as follows:

Here again, v is the velocity of the moving object [m/s], while c is the speed of light in a vacuum [m/s].

Its clear that E is related to relativistic momentum, which is the momentum of an object travelling at a relativistic speed. However, if the velocity is zero, y will not be zero. It rather tells us that the object has some rest energy.

Rest energy

Rest energy can be defined as the energy of an object whose velocity with respect to the frame of reference is zero. It can be described mathematically as:

E0 is the rest energy in joules.

This might remind you of Einsteins most famous equation, which showed that when an object is at rest, its energy is proportional to its mass. Thus, when energy is stored in an object, its rest mass increases. This also shows that destroying mass can release energy. See the following example for further illustration.

Calculate the rest energy of a proton in joules.

The mass of a single proton is approximately 1.67 ⋅ 10 ^ -27 kg, and we know that the speed of light c is 3.00⋅10^8m/s. Lets put these values into the following equation:

This gives us:

Lets now convert the unit to joules to see how large the energy is. We know that:

The result, therefore, is:

This may seem small, but we need to keep in mind that this is only for a single proton. If you wanted to calculate the rest energy of 1 gram mass, you might be surprised that the result would be 9 ⋅ 10 ^ 13 J. We dont notice this because usually, the energy is not available.

How can energy be converted into mass?

Just as mass can be converted into energy, energy can also be converted into mass. In order to understand how this works, we need to look into the relationship between stored energy and potential energy.

Stored energy and potential energy

What happens to the energy that is stored in a compressed spring? It becomes part not only of the total energy but also of the mass of the spring. So, how come we dont notice these changes in mass? Lets look at the example below.

Lets say a big battery is able to move 700 ampere-hours (A ⋅ h) of charge at 15 volts. Calculate the difference in mass when the battery goes from fully discharged to fully charged.

First, using the equation below, lets find out how much energy can be stored in the battery in the form of electrical potential (PE) energy:

Here, q is the charge in ampere-hours, while V is the electrical potential difference in Volts.

The change of energy is in the form of electrical potential energy, and we want to determine the difference in mass. So, lets put together the previous equation and the one for finding the rest energy:

The charge q, we said, is 700 A⋅h, while V is 15 V. This gives us:

Next, we write amperes as coulombs per second and convert hours into seconds.

Knowing that 1kg ⋅ (m ^ 2 / s ^ 2) = 1 Joule, we can change the result to kilograms, which gives us the difference in mass as follows:

As you can see, there is only a very small amount of change in the mass, which explains why we dont notice this phenomenon in our daily lives.

Mass and Energy - Key takeaways

  • Mass and energy are the same quantities, which we can convert into each other by using the speed of light.
  • Total energy E is the sum of all energies that an object with mass carries. Mathematically, it can be defined as E = ymc ^ 2.
  • Rest energy can be defined as the energy of an object whose velocity is zero. It can be stated mathematically as E0 = mc ^ 2.
  • Just as mass can be converted into energy, energy can also be converted into mass. However, in everyday contexts, the change in mass is so small that we dont notice it.
  • The change in mass when there is excess energy stored can be determined with the equation E = PE = q⋅V = (Δm) ⋅c ^ 2.
  • The forming of the universe (Big Bang), nuclear power plants, and the sun are all examples of mass-energy conversion.

Frequently Asked Questions about Relativistic Mass and Energy

Mass can be converted into energy and vice versa. The relation between mass and energy based on Einstein’s famous equation is E = mc ^ 2.


Yes, it does. When mass converts into energy or vice versa, the total energy is conserved. 

The concept that explains mass being a form of energy is rest energy. When an object has the velocity of zero, it has a rest energy that is directly proportional to its mass.

Final Relativistic Mass and Energy Quiz

Question

 What is the rest energy of a 2.5-gram mass in Joules?

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Answer

 2.25 * 10 ^ 14 [J]

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Question

Which of the following is the reason we don’t notice a change in mass when stored energy increases?

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Answer

We don’t notice the change in mass because it is usually too little for everyday energy levels.

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Question

 Which of the following explains the relationship between mass and energy?


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Answer

Mass can be converted into energy and vice versa.

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Question

For relativistic energy, what is the given name to the energy an object has when its velocity is zero?


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Answer

Rest energy.

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Question

Which of the following is not an example of mass-energy conversion?

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Answer

A ball with high mass rolling down a hill.

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Question

What is the symbol for mass?


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Answer

m or M.

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Question

What is the name for the sum of all energies?

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Answer

Total energy.

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Question

Can mass-energy conversions be seen in our daily lives?

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Answer

Yes, it can.

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Question

How does the sun convert mass into energy?


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Answer

By means of nuclear fusion.

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Question

How do nuclear power plants convert mass into energy?

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Answer

By means of nuclear fission.

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Question

Can energy be used to form particles with mass?

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Answer

Yes, it can.

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Question

The Big Bang is a great example of the conversion between mass and energy, but which was converted into which?

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Answer

Energy was converted into mass.

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