[NOTES/ME-08006]-Inertial Mass vs Gravitational Mass

1. Inertial mass

A force \(\vec{F}\) acting on a body produces an acceleration. By Newton's second law we have force proportional to acceleration of the body. Thus \(\vec{F}=m_i\vec{a}\), where \(m_i\) is a property of the body; we call this inertial mass. \subsection{Gravitational mass} It is known that the gravitational pull \(vec{P}\) of a body \(A\) (mass \(M\)) on another body is proportional to the mass \(m_g\) of the second body and we can write \(P \varpropto m_g \). The body \(A\) will exert double gravitational pull on a body of mass \(2m_g\), and so on. Given the two situations above, it turns out that the \(m_i, m_g\) are some properties of the second body appearing in two different contexts. There is no reason to expect a relation between the two inertial and gravitational masses of a body. In order to understand the point being made above, note that the electrical force due to a charge \(Q\) on a second charge is proportional to the charge \(q\) of the second body. Here charge \(q\) plays the role played by \(m_g\) for gravitational force, and that the charge \(q\) is in no way related to the inertial mass. It turns out that the ratio of inertial and gravitational masses is same for all bodies and can be set equal to one. The equality of gravitational and inertial masses has been experimentally verified to a great accuracy. \subsection{Equality of gravitational and inertial masses} That these two masses are different can also be understood by asking how can we measure inertial mass, and how can we measure gravitational mass. A simple arrangement for measurement of inertial masses is shown in figure below. Here \(A,B\) are two cylinders mounted on a stand. A spring attached to the inside the hollow cylinder. A nail holds the two cylinders in place with spring in compressed position. When the nail is removed the two cylinders fly apart in opposite directions. The horizontal components of their momenta,\(p_A,p_B\), will be equal and opposite at all times before hitting the ground, \(P_A=P_B\). Their velocities are related by \(M_A^i V_A=M_B^i V_B\). Here \(M^i_A, M^i_B\) denote the inertial masses of the two bodies. A video clip of the motion of the two cylinders can be made and analyzed with a suitable software and horizontal components of velocities can be measured. Taking mass of one of the bodies as unit, inertial mass of the other body is now determined. One can use a spring balance to determine the gravitational pull on different bodies gives a scheme of measurement of gravitational masses. \subsection{Equality of gravitational and inertial masses} It turns out that the ratio of gravitational and inertial masses is same for all bodies and therefore, with suitable choice of units, the ratio can be taken as unity. Several precise experiments have been done and the ratio of gravitational and inertial masses is known to to be unity with an error of about 1 part in \(10^{12}\).