[Physics Class Notes] on Relativistic Mass Pdf for Exam

The word “mass” has two implications in special relativity: invariant mass (likewise called the rest mass) that is similar for all observers in all reference frames; while the relativistic mass is reliant on the velocity of the observer. As indicated by the idea of mass-energy relativity, invariant mass is identical to rest energy, while relativistic mass is identical to relativistic energy. 

The expression “relativistic mass” tends not to be utilized in particle and nuclear physics. It is the mass of the body that changes with the change in the body’s speed as its speed approaches the speed of light, i.e., 3 × 108 m/s.

What Will You Learn Here?

This page will discuss the mass of light, a relativistic mass, relativistic speed, rest mass formula, and the relativistic mass formula.

Rest Mass

According to the theory of special relativity, the rest mass is called the invariant mass that remains invariant for all the observers in various frames of reference. The invariant mass is another name for the rest mass of particles inside the system. In contrast, the rest mass is preferred over the rest energy (E). 

Hence, invariant mass is a characteristic unit of mass utilized for systems that are being seen from their focal point/centre of mass (COM frame). 

Point To Note:

The idea of invariant mass doesn’t need bound systems of particles. In such a case, it might likewise be applied to systems of unbound particles in high-speed relative motion. 

Along these lines, it is regularly utilized in particle physics for systems that comprise broadly isolated high-energy particles. 

Assuming such frameworks were obtained from a solitary particle, the calculation of the invariant mass of such frameworks, which is a never-changing quantity, will give the rest mass of the parent particle (since it is preserved over a long period).

Rest Mass Formula

It is helpful in computation that the invariant mass of a system is the total energy of the system divided by c2 in the COM frame (where, by definition, the momentum of the system is zero).

The rest mass is denoted by mo.

So,  the rest mass formula is:

mo  = E/c2

Rest Mass is a Conserved Mass

The invariant mass of any system is likewise a similar quantity in every single inertial frame, it is a quantity determined from the total energy in the COM frame. 

Following the calculation of the rest mass in the aforementioned method, the rest mass is also used to compute system energies and momenta in other frames where the momenta are not zero, where the total energy will fundamentally be unexpectedly different as compared to the COM frame.

Similarly, as with energy and momentum, the invariant mass of a system (having multiple particles) can’t be varied or changed, and it is in this manner monitored that the rest mass of the system cannot be destroyed or changed; thus, it remains conserved, as long as the system is not prone to all influences.

Relativistic Mass

From the above text, we understand that relativistic mass is studied under Einstein’s special theory of relativity. It is a mass that is associated with the body in motion.

Mass, in physics, is a quantitative proportion of idleness/inertia, an essential property of all matter. It is, as a result, the obstruction that a body of matter offers to an adjustment of its speed or position upon the action of force.

The quantifiable inertia and the distortion of spacetime by a body in a given frame of reference are controlled by its relativistic mass.

Relativistic Speed

Relativistic speed is the speed at which relativistic impacts become important to the ideal exactness of estimation of the phenomenon being noticed.

[]  

Do You Know?

When any closed system (for instance a container of hot gas) is measured, which necessitates that the estimation is taken in the center of the momentum frame where the system has no net energy. In such a case, the invariant mass becomes equal to the relativistic mass, which is the total energy of the system divided by c2.

So, the relativistic mass is the sum total quantity of energy in a body/system divided by c2. 

Mathematically, the relativistic mass formula is:

E = mREL c2

For a particle possessing finite rest mass “mo” moving at a speed “v”, i.e., relative to the observer, one finds the following relativistic mass formula:

 mREL  =  [frac{m_{0}}{sqrt{1-frac{v^{2}}{c^{2}}}}]

Where,

 mREL   = mass of an object in motion

 mo   = rest mass

Point To Note:

In the center of the momentum frame, the speed “v = 0”, and the relativistic mass equals the rest mass, while in other frames, the relativistic mass of a body or system of bodies includes a contribution from the “total” kinetic energy of the body, i.e., the kinetic energy of the center of mass of the body, and is larger the faster the body/system of bodies move. 

Therefore, unlike the invariant mass, the relativistic mass relies on the frame of reference of the observer. 

However, for a given single frame of reference and isolated systems, the relativistic mass also remains a conserved quantity. The relativistic mass is also the proportionality factor between velocity and momentum, which is given as per Newton’s second law of motion:

P   =  mRELv 

Fun Fact

Do You Know What the Mass of Light is?

Well! Light is a bundle of photons. Photons (mass of light) have  “mo = 0, “ i.e., zero rest mass; however, they do contribute to the inertia and possess weight in a gravitational field of any system comprising them.