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The relativistic mass in physics is a different concept from the invariant mass as known in the layperson language. It is a standard concept that usually brings many doubts in learners' minds. Nonetheless, the theory of special relativity is favorable there to understand the relativistic mass in various frames. The relative change in mass is when the body is in motion. This concept is called relativistic mass. Mass increase happens when the object is in motion when compared to length contraction and time dilation of a thing.This article brings the explanation of relativistic mass in simple terms. With that, it also draws attention to the special relativity, relativistic speed, invariant mass, the mass of light, and relativistic mass formula.
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Key takeaways: Relativistic mass, speed of light, invariant mass, rest mass, special theory of relativity, mass, relativistic mass formula.
Concept of Relativistic Mass
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The mass that is appointed to a body when it is moving or is in motion. It is known as Relativistic mass. It was present in the firstest days of the subject, and its importance has only grown in physics and natural sciences over the centuries. The definition of relativistic mass starts from the times of Galileo and Newton, who basically gave the meaning of mass as the property of a body that governs its velocity while acting on a force.
Relativistic Mass
This definition and concept of mass were accepted and applied for centuries till the time Einstein came up with his theory of special relativity in relation to the motion. In 1905, Albert Einstein discovered that mass can be transformed into energy and the opposite is also true. The formula of special relativity that Einstein gave was E=mc2. At first sight, the theory of Einstein seems to make the concept of mass even more complicated. However, the reality is not so.
The definition of mass as given by Galileo and Newton is still relevant while calculating the mass for a body at rest. Therefore, it is called rest mass. It is also known as invariant mass. As the force-acceleration relationship of that body is constant.
On the other hand, when a body is in motion, the force–acceleration relationship of that body is no longer constant. It depends on two quantities: the speed of the body and the angle between its direction of motion and the applied force. Hence, Relativistic mass refers to the mass of a body that varies with the speed of the body as this speed moves closer to the speed of light. This mass increases with velocity and tends to infinity when the velocity approaches the speed of light.
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Difference between Relativistic Mass and Invariant Mass
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In special relativity, the term 'Mass' has two forms; one is invariant mass and the other is relativistic mass. Whereas, invariant mass is a quantity that remains the same for all bystanders. Meanwhile, relativistic mass differs according to the velocity of the observer. The most important special theory of relativity represents a lot more about relativistic mass when there are comparable measurements of length and time in two different frames. The relative change in the mass is there when the body is in motion. Now, after understanding both relativistic mass and rest mass. We will move forward with the formula of relativistic mass.
Formula for Relativistic Mass
Invariant Mass
The Formula of Relativistic Mass
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The formula of relativistic mass, in words, is expressed as,
Relativistic mass = rest mass / squared root [one minus (velocity/speed of light) squared]
The equation is:
mr = m0 / sqrt (1 – v2 / c2)
There, mr refers to relativistic mass, m0 means rest mass (invariant mass), v is a symbol for the velocity of the body in motion, and c is for speed of light.
Speed of Light
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The speed of light is regarded as a fundamental constant of nature. Its significance is far broader in the special relativity theory of mass than one can realize. It is the constant speed of light that links energy with mass in the famous formula of Albert Einstein.
In the special relativity formula, E = mc2, the speed of light (c) works as a constant of proportionality, connecting the previously disparate concepts of energy (E) and mass(m).
Things to Remember
- Relativistic mass and invariant mass are not the same. They are two different concepts.
- Mass is also known as invariant mass and rest mass.
- Fact: The speed or velocity of light is c = 3x108 m/s. According to special theory in physics, it remains constant no matter who is measuring it.
- Physical Laws are equal in any two reference frames if they move at a constant speed with respect to each other.
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Sample Questions
Ques: The mass of an object in motion is 12 kg. It travels in the air with a velocity of 0.82. What would be its invariant mass? (5 marks)
Ans: The relativistic mass and the velocity are already provided. The relativistic mass of the object, in that case, is 12 kg, and the velocity is 0.82. Now, we simply need to find the invariant mass (rest mass).
To start with, we will write the formula and put the given values in it.
The formula for the relativistic mass is:
mr = m0 / sqrt (1 – v2 / c2 )
Where,
mr refers to relativistic mass, m0 means rest mass (invariant mass), v is a symbol for the velocity of the body in motion, and c is for speed of light.
mr (relativistic mass)= 12kg, v (velocity)= 0.82c, m0=?
Just make sure that you are putting the right values in the formula.
Known: c (speed of light)= 3 × 1088 m/s22(always constant).
Therefore,
mr = m0 / sqrt (1 – v2 / c2 )
12kg= m0/sqrt [{1-(0.82)2}/(c)2]
Invariant mass (Rest mass) mo = 7.2 kg (approximately)
Therefore, the invariant mass (rest mass) of the given object is 7.2 kg.
Ques: A body of mass 1.67 × 10−24 kg travels with a speed of 0.65. Calculate its rest mass? (4 marks)
Ans: In this question also, the relativistic mass and the velocity are already provided. The relativistic mass of the object, in that case, is 1.67 × 10−24 kg and the velocity is 0.65. Now, we clearly need to find the invariant mass (rest mass).
Given: mass m =1.67 × 10−24 kg, v = 0.65c, c = 3 × 108 m/s2.
To start with, we will write the formula and put the given values in it.
The formula for the relativistic mass is:
mr = m0 / sqrt (1 – v2 / c2 )
mr (relativistic mass)= 1.67 × 10−24 kg, v (velocity)= 0.65c, C= 3 × 108 m/s2 (already known and always constant), m0=?.
Just make sure that you are putting the right values in the formula.
Putting the values in the relativistic mass formula…
mr = m0 / sqrt (1 – v2 / c2 )
1.67 × 10−24 kg= m0/sqrt [{1-(0.65)2}/(c)2]
Invariant mass, m0 = 1.26 × 10−24 kg (approximately).
Therefore, the invariant mass of the body is 1.26 × 10−24 kg.
Ques: An electron has an invariant mass of 9.11 x 10 -31 kg. In a detector, the same electron has a mass of 12.55 x 10-31 kg. How fast is the electron moving relative to the detector? (4 marks)
Ans: In this question, the relativistic mass and the invariant mass (rest mass) are already provided. The relativistic mass of the electron, in that case, is 12.55 x 10-31 kg and the invariant mass is 9.11 x 10 -31 kg. As clear, we need to find the velocity of the electron now.
Given: m0= 9.11 x 10 -31 kg, mr= 12.55 x 10-31 kg, c = 3 × 108 m/s2,
v=?.
We can write the relativistic mass formula in the specificity of velocity like this as well.
v = c √(1 – (m0 / mr)2
Now, we need to carefully put the values in this formula...
v = (3.00 x 108 m/s) √(1 – 9.11 x 10-31 kg / 12.55 x 10-31 kg)
v = 2.06 x 108 m/s
Therefore, the electron is moving with a velocity of 2.06 x 108 m/s in relation to the detector.
Ques: The rest mass of an electron is 9.1 x 10-31 kg and it moves with a speed of 4.5 x 105 m/s. Compute the relativistic mass. (4 marks)
Ans: In this question, the velocity and the invariant mass (rest mass) of the body are already provided. The invariant mass of the electron, in that case, is 9.1 x 10-31 kg and the velocity is 4.5 x 105 m/s. As clear, we need to find the relativistic mass of the electron now.
Given: m0= 9.1 x 10-31 kg, v = 4.5 x 105 m/s, c = 3 × 108 m/s2.
To start with, we will write the formula and put the given values in it.
The formula for the relativistic mass is:
mr = m0 / sqrt (1 – v2 / c2 )
mr=?
Just make sure that you are putting the right values in the formula.
Putting the values in the relativistic mass formula…
mr = m0 / sqrt (1 – v2 / c2 )
mr = 9.1 x 10 -31 kg/sqrt (1 – (4.5 x 107 m/s / 3.0 x 108 m/s)2)
mr = 9.8 x 10-31 kg.
Therefore, the relativistic mass of that electron is 9.8 x 10-31 kg.
Ques: The relativistic mass of a proton moving with a speed of 2.4 × 108 m/s (rest mass of proton = 1.67 × 10-27 kg and c = 3.0 × 108 m/s) ? (4 marks)
Ans: In this question, the velocity and the invariant mass (rest mass) of the body are already provided. The invariant mass of the proton, in that case, is 1.67x 10-27 kg and the velocity is 2.4x 108 m/s. As clear, we need to find the relativistic mass of the proton now.
Given: m0= 1.67 x 10-27kg, v = 2.4 x 108 m/s, c = 3 × 108 m/s2.
To start with, we will write the formula and put the given values in it.
The formula for the relativistic mass is:
mr = m0 / sqrt (1 – v2 / c2 )
mr=?
Just make sure that you are putting the right values in the formula.
Putting the values in the relativistic mass formula…
mr = m0 / sqrt (1 – v2 / c2 )
mr = 1.67 x 10 -27kg/sqrt (1 – (2.4 x 108 m/s / 3.0 x 108 m/s)2)
mr = 2.78 × 10-27 kg
Therefore, the relativistic mass of that proton is 2.78 × 10-27 kg.
Ques: Special relativity theory in physics relates 'moving mass' mr to the 'rest mass' m0 of a particle in terms of its speed v and the speed of light, c. A girl remembers the relation almost correctly but mistakes and forgets where to put the constant c.
She writes: (2 marks)
mr = m0 / sqrt (1 – v2 / c2 )
Guess where to put the missing c.
Ans: As explained in the article, C is the speed of light which is always constant in its natural form. On the other hand, quantities inside functions, in this case, square root, should be dimensionless. Therefore, the term (1-v2) will remain positive and the velocity of the body will be related to the speed of light. Thus, the correct formula is where C is there with V.
Formula: mr = m0 / sqrt (1 – v2 / c2 )
The missing C will be kept after velocity or v.
Ques: Rest mass of a photon is? (1 mark)
(a) hv
(b) 0
(c) hv/c2
(d) hv/c
Ans: According to the theory of Einstein light propagates in the form of packets. In other words, quanta of energy, which is called a photon.
The rest mass of photons is zero. It can be shown, according to the relativity theory of light.
According to the relativistic theory equation, the mass of the photon is computed as:
Formula: mr = m0 / sqrt (1 – v2 / c2 )
When v is zero, therefore, m0 is also zero.
And m0 is the rest mass of the Photon.
Ques: What is the value of constant C in the relativistic mass formula. (1 mark)
Formula: mr = m0 / sqrt (1 – v2 / c2 )
Ans: The speed of light, i.e, C is 3 × 108 m/s.
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