MEASURING THE COEFFICIENT OF RESTITUTION OF BALL-METAL SURFACE COMBINATION USING RC CIRCUIT

1Najoji, S. D., 2Ishaku I, and 3Musa, A. J.

1Department of Basic Science, 2Department of science Laboratory Technology

3Department of Mechanical Engineering Technology

Federal Polytechnic PMB 1006, Damaturu, Yobe State, Nigeria.

ABSTRACT

This paper presents a simple method that allows the determination of the coefficient of restitution of ball-metal surface using RC circuit as a timer, a metre rule and graphical analysis. The experiment emphasizes on the simple model of free fall of vertical non-spinning motion of three different balls which leads to good results. The average values of the coefficient of restitution or loss of kinetic energy in ascending order are 0.3122, 0.6223 and 0.7818 for the rubber squash ball, plastic table tennis ball and rubber lawn tennis ball used in the experiment respectively.

Keywords: Coefficient of restitution, RC circuit, ball-metal surface, collision

INTRODUCTION

The coefficient of restitution (COR) is a measure of how much kinetic energy is lost in a real collision. It determines the nature of the collision as being perfectly elastic (ε = 1), perfectly inelastic (ε =0) and real (0 <ε< 1).The COR is a convenient way of specifying how much energy in a collision is transferred to internal energy. It depends on the nature of the colliding materials but under practical conditions, it is a constant factor representing the energy transfer from kinetic energy to internal energy. Dropping a ball on a concrete surface from a given height is one of the simplest ways of determining the COR of the ball-surface material combination. A host of authors (Aguiar and Laudares, 2003; Stengaard and Lagsgaard, 2001; Smith et al,. 1981; Guercio and Zanetti, 1987; Bemstein, 1977; Cross, 1999; Foong et al., 2004; Bridge, 1998) have shown interest in the measurement of the COR for a collision between a ball and a flat concrete surface using a stopwatch or the sound produced by the impact/collision between the two surfaces. The standard procedure employed by these authorswas allowing the ball to fall down vertically from a known height and make a large number of bounces/collisions with the concrete surface before coming to rest. For example, a recording of the sound of impact of the ball, each time, with the surface is made either with the aid of a pen recorder or computer sensorsor high speed cameras and forces. The use of a stopwatch can be a source of error in the time interval measurement, as the successive bounces do not remain vertical and too much energy is transferred to the surface if the ball is allowed a large number of bounces. Our aim in this work is to describe a new technique for the measurement of the COR which is simpler, cheaper and overcomes the errors due to non-vertical bounces and too much loss of energy to the surface for large number of bounces.

Theory

Consider a ball of mass m dropped from initial height h on a horizontal hard surface. Assuming air resistance is negligible and only vertical motion is considered, using the law of conservation of energy, we can write

1

where v is the velocity of the ball as it strikes the collision surface. From equation (1) the velocity of the ball as it strikes the surface is given by

2

Measuring the Coefficient of Restitution of Ball-Metal Surface Combination Using Rc Circuit

The coefficient of restitution ε for a given pair of colliding bodies for real collisions is the ratio of the velocity vs of separation after a collision to the velocity va of approach before a collision given by

3a

Thus according to this equation, it means after the first impact with the collision surface, the velocity of the ball changes to

3b

The ball again rises to a certain maximum height and falls down to suffer a second impact with the hard surface. The time interval between the first and second collisions is given by (Ajay, 2009)

4

Substituting equation (3b) in equation (4) yields

5

It is worthy to note that the velocity of approach va just before the second impact/collision occurs is of the form of equation (2). Therefore, substituting  in equation (5) we have

6

This is the equation relating the time interval t between the first and second collisions, the initial height, the acceleration due to gravity, and the COR for the ball-surface combination.

Now we know that the initial voltage Vo across a charged capacitor of capacitance C decreases to V as it discharges through a resistor of resistance R in time t is given by

7a

Or                                                            7b

If the series RC circuit is incorporated in the experimental-setup such that the time in equation (7b) is equivalent to the time interval between two successive impacts which can be calculated. Thus, equation (7b) is the expression for the time interval between the first and second impacts for the ball-metal surface combination under the influence of gravity.

The graph of the calculated t values from equation (7b) against the square root of the initial height h according to equation (6) has slope, from which the average COR ε can be calculated.

This method has advantage over use of stopwatch to measure the small time interval taken by free falling objects through short distances. For example, the time t taken for a freely falling objects from rest at a height of 1 m according to the formula  is about 0.45 seconds neglecting air resistance and taking g = 9.81 m/s.  In practice it is difficult to measure such a small time by means of stopwatch because the reaction time of the experimenter to start and stop the stopwatch may be greater than the time interval of interest (Yerima et al., 2008).It is simpler than the cathode ray oscilloscope (CRO) technique (Ajay, 2009) since it does not require trained technicians for its operation. Its components are cheaper and available in our local markets unlike CRO which is sophisticated, expensive and not available in most of our schools.

 ball

METHODOLOGY

 K2

S                                K1

SC                     RC

K3M                           E

Fig. 1 Set up for measuring COR

International Journal of Research and Scientific Innovations    Volume 8, Number 1, 2018

Fig. 1 is the experimental set up used to determine the COR for the ball-metal surface combination. When relay keys K1, K2 and K3were closed the capacitor of capacitance C was charged to a maximum voltage Vo through a resistance R by a battery of emf E. With K1 and K2 opened, the screw S was adjusted in such a way that as the ball was just released to freely fall vertically without spinning, K2 at the same time closed and the capacitor began to discharge.  When the ball just collided with the flat, smooth, hard and massive metal surface M below it, K3 opened and the capacitor stopped discharging simultaneously. The voltage drop V across the capacitor and the vertical height h through which the ball has fallen were determined. The major uncertainties in the experiment come from the release of the ball which may induce unwanted spinning and the quality of the horizontal surface on which the ball impacts. These uncertainties were avoided by adjusting the screw S gently to allow the ball fall freely without spinning before it fell on the firmly fixed flat, hard, massive and smooth metal plate.

RESULTS AND DISCUSSION

The time of fall and height of three different balls on hard smooth metal surface were determined with the aid of an RC circuit and metre rule respectively (Table 1). The results show that for a given height, the time of fall for the rubber lawn tennis ball is highest and least for rubber squash ball.

Table 1 Time intervals between successive collisions for ball-metal surface combinations

 (m) Time, t (ms) Rubber squash ball Plastic table tennis ball Rubber lawn tennis  ball 0.72 285.7 457.1 514.3 0.87 342.9 542.9 628.6 1.02 393.1 628.6 725.7 1.12 441.4 685.7 800.0 1.23 445.7 731.4 86.86 1.32 457.1 788.6 942.9 1.40 485.7 851.4 1000.0

Fig.2Time of free fall of ball

Figure 2 shows the variation of the time of free fall of three different balls with height. Using least squares method the following empirical equations were obtained:

8

9

10

where the subscripts sb, tb and lb refer to squash ball, tennis ball and lawn tennis ball respectively. The slopes of these regression lines and their R2 values from the computer-generated fits using excel are shown in Table 2, along with the average values of COR determined from the slopes. Equations 8-10 clearly show that we have not forced the fits to pass through the origin, as would be expected from the model discussed above. Non-zero intercepts on the time-axis

Measuring the Coefficient of Restitution of Ball-Metal Surface Combination Using Rc Circuit

represent systematic errors that can be minimized by attentive experimenter. Even with these experimental details in mind, the experiment is simple and yields results that are consistent with expectations.

Table 2 Results from least squares method

 Ball Regressionslope (sm-1/2) R2 value Average COR Rubber squash ball 0.282 0.986 0.3122 Plastic table tennis ball 0.562 0.996 0.6223 Rubber lawn tennis ball 0.706 0.999 0.7818

The method is simple and cheap as it does not require sophisticated and expensive equipment such as CRO with amplifier used as sound detector (Ajay, 2009) that require trained technicians to operate it. It has been easy and convenient to determine the small time of fall with maximum error of 9 ms for short distances less than 1.40 m in the case of lawn tennis ball.

CONCLUSION

The experiment was easily performed without the need for special equipment. It emphasizes on careful measurements and linear regression analysis, and uses a simple model to determine the COR of bouncing balls. The values of the COR for three different metal-ball surface have been determined using RC circuit as a timer for comparative studies. The values of the time intervals between two successive impacts lie between 286 ms and 1000 ms and the average values of the COR lie between 0.3122 and 0.7818.

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