If energy , velocity and force be taken as fundamental quantities, then what are the dimensions of mass?
Step 1: Given
The energy , velocity and force are taken as a fundamental quantity.
Step 2. Concept to be used
The Einstein mass-energy relation states that mass and energy are related to each other and every substance has energy because of its mass also.
The energy that exists due to the motion of an object is known as kinetic energy.
The formula of kinetic energy is,
Where, is the kinetic energy, is the mass of the object and is the velocity of the object.
Since is another form of , they have the same units.
The velocity is the ratio of displacement by time.
The formula of velocity is,
where is the displacement, and is the time taken.
The formula for force is,
where is the acceleration of mass due to force .
Step 3: Find the dimensions of force, velocity and energy
The unit of the velocity is , as the velocity is the ratio of displacement by time given as,
The dimensional formula of this above unit is,
The unit of the force is .
As the force is the product of the mass and acceleration.
So, the dimensional formula of this above unit is,
The unit of the energy is ,
As the energy is the product of mass and the velocity of light in the air, that is,
The dimensional formula of this unit is,
Therefore, the dimensional formulae,
For velocity is .
For force is .
For energy is
Step 4: Express dimension of mass in terms of new units
The SI unit of the mass is .
So, the dimensional formula of mass is,
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Let the dimensional formula of the mass in terms of the energy, force, and velocity be,
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Putting the dimensional formulae of the energy, force, and velocity in the above equation, we get,
Removing the brackets and adding the like terms, we get,
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Comparing the equations and , so we can write,
Now, compare and equate the components accordingly.
So, we can write,
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Step 5: Solve the equations
Now let us consider the equation again,
So,
Substituting equation in the above equation, we get,
Now let us consider the equation
We can break the as,
Substitute the values of and in the above equation, we get,
Now considering the equation again, we can substitute the value of and get,
Now put the values of and in the equation , we get,
Hence, the dimension of mass when energy, velocity, and force are taken as fundamental units is .