Determination Of Gravitational Acceleration

Historical and Theoretical Basis

Discuss about the Determination of Gravitational Acceleration.

Historical and Theoretical Basis behind the Determination of Gravitational Acceleration

Aristotle, the great Greek philosopher, believed that every motion or effect has a cause. The downward movement of heavy earthly elements is related to their nature which makes them to move downwards towards the center of the earth. Similarly, light objects are caused by their nature to move upwards to the inner spheres of the moon. In other words, Aristotle believed that heavy objects do not move towards center of the earth because of external force of gravity but rather because of their heaviness. However, Brahmagupta (an Indian mathematician and astronomer) held that because of the spherical shape of the earth, it attracts objects towards its center. Disputing Aristotle, Galileo discovered that all objects accelerate equally when falling. His findings were based on the assumption that if air resistance is negligible, all objects are assumed to fall at the same acceleration (Garg Kalimullah, Arun & Lima, 2007). In the same century, Robert Hooke suggested that there is a force of gravity that inversely varies with the square of their distances. Later, Kepler came up with three laws that govern the orbital movement of planets. His first law states that all planets move in an ellipse with the sun at one focal point, balanced from the center. Secondly, objects near the sun move faster than those farther away. And lastly, the planet’s distance from the sun determines the time it takes to revolve around the sun. Based on Kepler’s laws on the motion of planets, Isaac Newton sought to investigate the motion of all objects. He discovered that all falling objects follow the same principles as outlined in Kepler’s Laws. 

Apart from the laws of motion, Newton also suggested that all matter exerts a force called gravity that pulls matter towards its center and this force is dependent on the mass of the object and also wanes with distance. For instance, the Sun has higher gravity than the earth and the earth has more gravity than an apple. Newton’s Law of gravity explains the earth’s movement around the sun. Ideally, the earth would move straight throughout the universe. However, this is not the case because the sun exerts its pull on the earth that forces the planet to move in elliptical orbit.  Newton’s theory on gravity has made it possible to explain the rise and fall of the ocean waters that occurs because earth’s gravitational pull on the moon.

Determining Gravitational Acceleration

The force of gravity is responsible for the shape and structure of the earth. It is one of the fundamental parameters in physics that governs the motion of objects on the surface of the earth. In simple terms, earth’s gravity can be defined as the force with which the earth attracts objects towards its centers. It is also the rate at which a falling object increases in speed as it falls. A free falling object is able to move with a specific value of acceleration, known as gravitational acceleration and is denoted by letter ‘g’. Theoretically, this value was determined by Newton to be 9.8m/s. There are several ways that has been used by physicists to measure the acceleration due to gravity. Some of these ways have produce errors that compromise the final results. Recently, there have been new developments on the instruments used for measuring gravitational acceleration. The objective of these new developments is to reduce the errors that are common with scientific experiments (Cook 1957, pp. 34).

According to Newton’s principle, gravitational acceleration can be determined by computing the time t for a falling object through a given vertical distance by using the formula

Y= ut +1/2gt²……………………………. .... .. . .. . . . .. . .. . .1

Where u= initial velocity of the object at the start of timing. If the objects falls from a stationary point then u=o and the equation reduces to

y =1/2gt² . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . 2

In such experiments that involve free fall objects, determining the accurate time for the flight is a major challenge since there is no precise means of measuring the start time as one hold a stop watch on one hand and watching the falling object. As a result, the scale and clock does not provide the precision required in this experiments. Application of electronic devices seems to have solved the problem of timing. With electronic timer, the correct start and stop times can be obtained thus increasing accuracy and precision.

In simple terms, Newton’s law on force stipulates that force on an object depends on its mass and acceleration. In other words, force is a product of mass and acceleration (F = ma). Thus, for an object to experience acceleration, a force has to be applied to it. This is what Newton’s Universal law of gravity states. The general formula as set by Isaac Newton for determining the force between two objects is given by

Newton's Universal Law of Gravity

                                                   F = GM₁M₂/ r² . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

Where

G= gravitational constant

M₁ and M2 are the masses of the two objects, r = the distance between the two objects. In other words, force between two objects is obtained by multiplying the product of their masses with a constant, G and dividing by the square of the distance between them. When an object is near the earth’s surface, then the distance is presumed to be the radius of the earth. As a result, the above equation reduces to

F = GmM_e/r_e² . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

In this case, the force depends purely on the mass of the object, since mass and radius of the earth are constant and it corresponds to the weight of the object on the earth’s surface. Therefore,

Ma= GmM_e/r_e²=mg . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5

Rearranging eq. 5

g= GM_e/r_e². . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

From equation 6, it is evident that gravitational acceleration is a constant since it depends on constant values. This equation was first determined by Galileo upon dropping cannonballs of different masses and realizing that they landed at the same time. His explanation was based on the fact that they experienced the same acceleration (acceleration due to gravity). Similarly, when a coin and a feather are both dropped simultaneously in a vacuum where there is no air resistance they land together. Overall, determining acceleration due to gravity we measure the time of travel from a stationary point and the distance travelled as shown in equation 2.

Different Methods to Determine Gravitational Acceleration

Objective: To determine acceleration due to gravity using three different methods; ticker tape timing, stroboscopic analysis, and electronic timing.  

First Method: Ticker Tape Timing

In this experiment,  a length of ticker-tape is threaded in a ticker timer and attached to a trolley at one end. The trolley is pulled by a constant force along the bench. The elastic cord is looped on one end around a rod which is attached to the trolley. To keep the force constant, the cord is stretched by the same amount of force with which the trolley is pulled. The experiment is repeated several to improve on precision. This method is limited in accuracy because of friction of the tape as it passes through the ticker timer which reduces the rate of acceleration. This can be reduced by using larger mass but it also needs investigation.

The experiment was done three times per student in the group. The timer runs at 50 dots per second. Thus, the period was 0.02s.

Tape Number	Initial position	Final position	Results (m/s2)
1	0.015	0.066	7.1875
2	0.0145	0.068	7.4306
3	0.02	0.082	7.75

Second method: Stroboscopic Analysis

• A camera was set up on a tripod directly in front of the stand with the electromagnet at the top of the frame. The position of the camera was adjusted appropriately so that at least one meter of fall was photographed. The strobe light was positioned so that the falling object would be illuminated from the side.
• With the apparatus fully set the flash rate of the strobe was adjusted to 25 flashes per second.
• After connecting the electromagnet to a power supply, the voltage was adjusted to hold the ball bearing.
• With the room lights off, the Camera aperture was opened wide to give a sharp view through viewfinder
• As the strobe light flashes, the cable release is depressed to open the shutter, the ball bearing was released by switching off the power supply. The shutter is then closed when the ball bearing finishes the fall.                     

Trial	C(m/s2)	g(m/s2)
1	4.798	9.596
2	4.899	9.798
3	4.903	9.806
4	4.910	9.820

Third method: Electronic Timing

• The stand was set with electromagnet and then connected to power supply
• The voltage of the power supply was adjusted so that it held the steel strip vertically.
• At the base of the strip, the light beam was mounted just below where it could interrupt the beam (the adjustment is done cautiously so as not to interfere with initial velocity)
• The timer was reset to zero and the steel strip released by switching off the power supply to the electromagnet and timing recorded
• The experiment was repeated 4 times and each time results recorded
• The length of the strip was measured with much focus on accuracy 

Distance s (metres)	Time(1/2t2)	g
2.00	0.209	9.789
1.8	0.183	9.810
1.6	0.163	9.815
1.4	0.143	9.859
1.2	0.122	9.802
1.0	0.102	9.803

When using the ticker timer method, we noticed a few errors. First, the timer would start before the ball enter free fall causing smaller long elapse in time and subsequent low values of g. Secondly, the tape was not in parallel with the floor which caused a random deviation in elapse time. As a result, the value of g was way below the theoretical value. However, with a photogate the precision was highly increased with value nearing the theoretical value. In trial 4, the value was slightly higher possibly because the ball was dropped at an angle that reduced the distance, hence the high value. The electronic timer is the best of the three methods in precision since it provides a percentage error of 0.0009%

Conclusion

From experimental values, it is evident that gravitational acceleration is a constant since it depends on constant values. This support the preposition of Galileo that cannonballs of different masses falls at the same rate since they experienced the same acceleration (acceleration due to gravity). Similarly, when a coin and a feather are dropped in a vacuum where there is no air resistance they land together. This implies that determining acceleration due to gravity we measure the time of travel from a stationary point and the distance travelled which requires accuracy and precision in the instruments. Some errors noted in the values resulted from instruments.

Reference List

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Bill C 1990 Measurements of Acceleration Due to Gravity, Physics Teacher, v28 n5 p291-95

Cook AH 1957. Recent development in the absolute measurement of gravity. Bulletin Geodesique 34(1) 34-59.

 Fredrick JB 1975 Introduction to Physics for Scientists and Engineers, USA: Von Hoffmann Press. 

Garg M, Kalimullah, Arun P and Lima FMS 2007, An accurate measurement of position and velocity of a falling object. American Journal of Physics 75(3) 254–258.

Gerald JH and Brush, SG 2001. Physics, the human adventure: from Copernicus to Einstein and beyond. NY: Rutgers University Press. 

Jewess M 2010 Optimizing the Acceleration Due To Gravity on a Planet's Surface. The Mathematical Gazette, Volume 94, Issue 530July 2010, pp. 203-215

Khongiang, L, Dkhar, A and Lato, S 2015. Accurate Determination of Acceleration Due To Gravity, G in Shillong Using Electronic Timer. Online International Journal available at https://www.cibtech.org/jgee.htm

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Marson, I and Faller, JE 1986 G-The Acceleration Of Gravity: Its Measurement and Its Importance. Journal of Physics E: Scientific Instruments, Vol. 19, No. 1.

Peters, A et al 1999 Measurement of gravitational acceleration by dropping atoms. Nature 400, 849-852

Preston-Thomas, H et al 1960 an absolute measurement of the acceleration due to gravity at Ottawa. Canadian Journal of Physics, 1960, 38(6): 824-852

Wick, K and Ruddick, K 1999. An accurate measurement of g using falling balls. American Journal of Physics, 67(11) 962–965.

https://www.phy.gonzaga.edu/downloads/pdf/accel.pdf