What's the difference between the five masses: inertial mass, gravitational mass, rest mass, invariant mass and relativistic mass?

I have learned in my physics classes about five different types of masses and I am confused about the differences between them.

What's the difference between the five masses:

1. 
   

   inertial mass,

   
   


2. 
   

   gravitational mass,

   
   



3. 
   

   rest mass,

   
   


4. 
   

   invariant mass,

   
   


5. 
   

   relativistic mass?

   
   

Answer

Let us define the inertial mass, gravitational mass and rest mass of a particle.

To every particle in nature we can associate a real number with it so that the value of the number gives the measure of inertia (the amount of resistance of the particle to accelerate for a definite force applied on it) of the particle.

Using Newton's laws of motion,

$$m_i = \frac{F}{a}$$

(This is defined using Newton's law of universal gravitation i.e. the gravitational force between any two particle a definite distance apart is proportional the product of the gravitational masses of the two particles.) To every particle in nature we can associate a real number with it so that the value of the number gives the measure of the response of the particle to the gravitational force.

$$F = \frac{Gm_{G1}m_{G2}}{R^2}$$

All experiments carried out till date have shown that $m_G = m_i$

This is the reason why the acceleration due to gravity is independent of the inertial or gravitational mass of the particle.

$$m_ia = \frac{Gm_{G1}m_{G2}}{R^2}$$

If $m_{G1} = m_i$ then $$a = \frac{Gm_{G2}}{R^2}$$

That is acceleration due to gravity of the particle is independent of its inertial or gravitational mass.

This is simply called the mass and is defined as the inertial mass of a particle as measured by an observer, with respect to whom, the particle is at rest.

There was an obsolete term called relativistic mass which is the inertial mass as measured by an observer, with respect to whom, the particle is at motion. The relation between the rest mass and the relativistic mass is given as

$$m = \frac{m_0}{\sqrt{1-v^2/c^2}}$$

where $v$ is the speed of the particle and $c$ is the speed of light, $m$ is the relativistic mass and $m_0$ is the rest mass.