[NOTES/ME-08007]-Equality of Inertial and Gravitational masses

1. Equality of inertial and gravitational masses

An important consequence of equality of gravitational and inertial masses is that, by doing any mechanical experiment, it is impossible to distinguish between the following two options.

1. [(a)] The frame of reference is a non inertial frame moving with with some acceleration \(\vec{a}\);
2. [(b)] The frame of reference is an inertial frame and there is an extra gravitational field \(-m\vec{a}\) acting on bodies inside the elevator.

This is most easily seen by writing equation of motion in the two cases above. Let us consider ball suspended with a spring inside an elevator. The elevator in turn is accelerating upwards with constant acceleration \(a\).

As seen in inertial frame outside elevator

Viewed by an observer outside the elevator and, positioned in a inertial frame, the acceleration of the ball is \(a\) upwards. The equation of motion of the ball using Newton's laws in the inertial frame takes the form. \begin{equation} \text{Mass}\times \text{Accn} = \text{Force} \Longrightarrow Ma = k \Delta\ell -Mg \end{equation}\text{Mass}\times \text{Accn} = \text{Force} \Longrightarrow Ma = k \Delta\ell -Mg \end{equation} The amount \(\Delta\ell\) is the change in length when the ball hangs in equilibrium in the elevator's frame of reference.

View from elevator's frame :: Two interpretations

Viewed from the frame of elevator the body is at rest. Therefore the sum of all external forces and pseudo forces must add to zero. The EOM is for the body as written by applying Newton's laws in the inertial frame is (previous page) \begin{equation} \text{Mass}\times \text{Accn} = \text{Force} \Longrightarrow Ma = k \Delta\ell -Mg \end{equation}\text{Mass}\times \text{Accn} = \text{Force} \Longrightarrow Ma = k \Delta\ell -Mg \end{equation}

First Interpretation

The elevator has acceleration \(a\) Thus pseudoforce is \(Ma\) and external forces are \(mg\) and \(k\Delta \ell\). Sum of all these, with appropriate signs of course, must vanish. This gives \begin{equation} k\Delta \ell -Mg- {Ma} =0 \end{equation}

Second Interpretation

In the second view we can assume that the gravitational field is absent and there is acceleration (\(a+g\) upwards. This give a pseudoforce \(M(a+g)\) downwards and we get \begin{equation} k\Delta \ell - {M(a+g)} =0 \end{equation}  is pseudo force in the two cases
In other words

As viewed from a noninertial frame, {EQ01} can be interpreted as if there is an extra downwards (pseudo) force equal to \(Ma\) which in turn can be seen to be equivalent to a downward gravitational pull of \(Ma\). As a result for an observer in the frame of the elevator, it becomes impossible to distinguish between the following two interpretations of the equation of motion. One view is that the elevator is accelerated and the pseudo force have is to be included in the equation of motion. The other view is that the elevator is not accelerated, but there is a gravitational pull \(W^\prime=ma\) in the direction opposite acceleration in addition to the usual weight of the body \(W=Mg\).

Equivalence principle

No mechanical experiment done entirely within the elevator can differentiate between the above two possibilities. In other words effects of gravitational fields can be mimicked by going to an accelerated frame and having no gravitational field. This is possible only in small regions of space time. This is essence of the equivalence principle. In the form stated here it applies to mechanical systems. A stronger version of equivalence principle states that all laws of Physics respect the equivalence principle. The equivalence principle played an important role in development of general relativity. For a historical introduction see\cite{Wein}

References

[1] Weinberg S., {\it Gravitation and Cosmology: Principles and Applications of the General Theory of Relativity }, John Wiley and Sons Inc, New York (1972)