The equivalence principle is one of the fundamental laws of physics, which states that gravitational and inertial forces are similar in nature and often indistinguishable.
We know that gravitational mass is the charge to which gravity couples, while inertial mass is a measure of how fast an object accelerates-providing the same force.
It is stated as:
F = ma
It implies that in increasing the inertial mass, acceleration decreases.
We know from the equivalence principle that the gravitational mass and the inertial mass are the same things.
Then, the gravitational force is directly proportional to the inertial mass, and this proportionality is independent of the type of matter.
This implies the universality of free fall (UFF) that is in a free-fall all objects fall with the same acceleration, i.e., 9.8 ms⁻².
History of Equivalence Principle
Let’s understand the concept of equivalence.
Let us take two objects of different masses.
Here, consider one object like an apple and the box of mass 1 kg hanging through the string.
How Do We Determine the Mass of an Apple?
We may use a beam balance to figure out its mass or compare its mass with the mass of the box, or we can apply a force to the apple so that it starts accelerating.
Here, the scientists found that the masses are equivalent.
The first person to do such measurement was Galileo, followed by Newton.
Galileo did his experimentation on the inclined planes to measure the inertial mass and the gravitational mass and found that they are equivalent.
Newton stated that inertial mass is strictly proportional to the gravitational mass, where he did experiments on the pendulum of a variety of substances to prove the same.
Einstein thought it was interesting that two masses (apple and box) were equivalent, and he figured out the theory of relativity.
He explained that mass could be inertial mass or gravitational mass.
Inertial mass: When we apply a force to the body and measure its acceleration, we find that the inertial mass tries to oppose the acceleration produced in the body.
However, gravitational mass is taking a mass and putting it inside a gravitational field and measuring the force of gravity on this mass.
So, whatever method we choose to find the masses of two different objects will come out to be the same number. This led to the discovery of the general theory of relativity.
Einstein Equivalence Principle
Einstein considered two situations, where he imagined a rocket landed with a man sitting inside it.
So, what’s the force acting on this man?
This force is the force of gravity pulling down on this man.
Suppose a universal or constant acceleration is offered to the rocket, and it starts traveling through space.
Let's say the acceleration of the rocket equals the acceleration due to gravity.
However, Einstein said that there is no way to discriminate between the above two events.
For example, A ball will freely fall under gravity for a boy playing with the ball inside the rocket and on the earth. Therefore, these two events will be similar.
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What is Effective Mass?
According to Newton's second law of motion, when a force is applied to the body, it starts accelerating.
The mass of the body is a property that is measured in terms of obstruction it creates to the acceleration under the influence of an external force.
The effective mass is a term used in Einstein’s general theory of relativity that talks about the inertial mass and the gravitational mass.
Gravity: We know that gravity is the force of attraction between any two bodies in the universe. So, the gravitational force is given by,
F = GMmgravity/r² ..(1)
Where G is the universal gravitational constant
M is the mass of a larger body.
m = The mass of another body.
r = Distance between m and M.
Inertia: A body tends to resist the change in its current state of rest or motion unless an external force is applied to make the changes that are,
F = mInertia x a..(2)
Where mInertia is the inertial mass of the object.
a = acceleration
We know that inertial and the gravitational forces are equivalents (the principle of equivalence).
Now, equation (1) = (2)
mInertia x a = GMmgravity/r²
On arranging the terms, we get:
a = [mgravit/ mInertia ] x GM/r²
Hence, we proved the equivalence mathematically.
Example of Equivalence Principle
It’s often difficult to differentiate between the two terms, inertial mass and the gravitational mass as they are the same things.
Let’s consider a scenario where an elevator in space is being accelerated in one direction. A man inside the elevator would feel as if there was gravity pulling him in the opposite direction. The same happens with a person in a stationary elevator that is located in the earth’s gravitational field.
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