MEASURING INERTIAL MASS

                    A brief introduction

                    Inertial mass and gravitational mass are appearing in two independent Newton's laws. In the second law (Eq. 1), the inertial mass mI relates a force F acting on a body and body's acceleration a produced by the force F. This mass clearly presents body's resistance (inertia) against change of status of its motion (acceleration) imposed by the acting force.
(1)mI a = F ;(2) FG = G MG mG r -2.
In the law of gravity formulated for point like or spherically symmetric bodies (Eq. 2). gravity masses MG and mG together with a distance r between centers of these bodies decide about a strenght of a mutual attractive gravity force FG . The constant G there is the universal gravity constant.
                    If a body with much smaller size than Earth is released near Earth surface, then its acceleration is described by the Newton's second law (Eq. 1) with F given by Newton's law of gravity (Eq. 2) where mI , mG and MG stand respectively for inertial and gravitational masses of the body and gravitational mass of Earth. We are disregarding here an air resistance and other small forces like for example Coriolis force. As the result body's acceleration has the following form

(3)                    a = (mG /mI ) G MG r -2.

Galileo's experimental dicsovery of equal accelerations for all free falling bodies demands that the ratio mG /mI must be the same for all bodies. Proper adjustment of G makes this ratio equal to 1, and mG = mI . But this last equation states only that numbers representing both masses are equal, whereas their meanings are different. Consequently the product GMGr -2, with r being Earth's radius, must represent a gravity acceleration g = 9.8 m/s2. Thus, for small bodies near the Earth surface the Newton's law of gravity is reduced to the very popular relation W = mg . Here W stands for a body weight, and it represents a gravity force exerted by Earth on this body, whereas m is de facto a gravity mass of the body. This result clearly shows that weighting bodies means finding their gravitational masses.
                    If we want to measure explicitly body's inertial mass, the Newton's second law must be used. Applying to the body a known constant force and measuring body's acceleration we can calculate from the Newton's second law (Eq. (1)) body's inertial mass. This mass can be compared with gravitational mass obtained by means of weighting to show students that numbers representing both masses with accuracy of a few percent are equal. Such accuracy, of course, is very poor if compared with Eotvos results from the end of nineteenth century, but process of finding a mass of an object without weighting it helps students in better understanding of inertial and gravitational masses.

                    Equipment and its setup

In this experiment we need The experimental setup is shown in Fig.1.

Fig.1

                    Experiment and its results

                    In this particular experiment 0.150 kg glider was used with two 10 g and two 5 g slotted weights (total 30 g) initially placed on its top and hold there with help of the big flat spring paper clip. On the weight hanger an additional single 5 g slotted weight was added. A total 10 g mass there produced enough force to run the glider smoothly through the photogate system with an appreciable acceleration. As a matter of fact this force accelerates a whole system containing the glider, weight hanger with additional weights and pulley. For each next run 5g of mass from the top of the glider is shifted to the hanger increasing the force acting on the system and the system acceleration. This operation keeps the inertial mass of the system constant and allows to increase the accelerating force for each consecutive run. The glider was started every time from the same spot. Such arrangement is not rquired, but it produces non-intersecting lines on a graph velocity vs. time. The results of all seven runs (till all sloted weights are moved from the top of the glider to the hanger) are shown in Fig.2.

Fig.2

                    If we disregard friction forces, the only force causing acceleration of the system is the gravity force acting upon the hanger and sloted weights there. Then, the accelerating force F is given by the product mH g, where mH is a mass of the hanger and sloted weights there. A graph of the force vs. acceleration for our particular experiment is shown in Fig.3.

Fig.3

                    A linearity of this graph verifies the Newton's second law, and the graph slope represents the inertial mass of the system. This mass contains the inertial mass of all weights, inertial mass of the glider, and roughly a half of inertial mass of the pulley. If the pulley were a uniform cylinder then a kinetic energy of its rotational motion could be expressed as mPv2/4, where v is a speed of its edge, which is the same as a speed of the glider. The additional factor 1/2, as compared with a kinetic energy of a point-like mass, appears because an inertia moment of a cylinder with respect to its symmetry axis is given by mCRC2/2, where mC and RC are the mass and radius of the cylinder respectively.
                    Thus, the inertial mass of the system should be compared with an "effective" gravitational mass of the system which contains only a half of the gravitational mass of the pulley. In our case the gravitational masses of the glider (with the fin and paper clip), pulley and sloted weights (including the hanger) were 0.160 kg, 0,004 kg and 0.040 kg respectively. This makes an "effective" gravitational mass of the system to be equal to 0.202 kg, which differs only by a one gramm from the measured inertial mass of the system.

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