Discuss the Determination of Gravitational Acceleration.

Introduction

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The Historical and Theoretical Foundations of Gravitational Acceleration

Aristotle, the great Greek philosopher believed that all motion and effect have a cause.

The down movement of heavy earthly substances is caused by their nature. They move down towards the center of earth.

Similar to light objects, their nature causes them to move upwards to reach the inner spheres.

Aristotle believed heavy objects move away from the center of earth not because of the external force of gravity, but rather because they are heavy.

Brahmagupta was an Indian mathematician/astronomer who believed that gravity attracts heavy objects towards the center of the earth because of its spherical form.

Galileo, who was disputing Aristotle’s findings, discovered that all objects accelerate equally when they fall.

These findings were based on the assumption, that if the resistance to air is negligible all objects fall at the identical acceleration (Garg Kalimullah & Arun, 2007).

Robert Hooke, a century earlier, suggested that gravity has an inverse relationship with their distances.

Kepler developed three laws later that govern the orbital motion of planets.

Kepler’s first law stated that all planets move in an “ellipse” with the sun at one central point and balanced from the middle.

Second, objects close to the sun move slower than those further away.

The distance between the sun and a planet determines how much time it takes to turn around.

Isaac Newton, who was guided by Kepler’s laws concerning the motion of the planets, sought to examine the motions all objects.

He discovered that all falling objects are guided by the same principles described in Kepler’s Laws.

Newton proposed that all matter has a force called gravity. This pulls matter towards the center. It is dependent on its mass and decreases with distance.

For example, the Sun has a higher gravity than the Earth and the earth is heavier than an apple.

Newton’s Law of gravity describes the earth’s orbit around the sun.

In an ideal world, the earth would move directly throughout the universe.

This is not possible because the sun exerts its gravitational pull on the Earth, forcing the planet into an elliptical orbit.

Newton’s theory about gravity has allowed us to explain the rise or fall of ocean waters due to the earth’s gravitational force on the moon.

The shape and structure of the Earth is controlled by gravity.

It is one fundamental parameter in physics that governs how objects move on the earth’s surface.

Simply put, gravity refers to the force by which the earth draws objects toward its center.

It also measures the speed at which a falling object accelerates as it falls.

A free falling object can move with a particular acceleration. This acceleration is called gravitational acceleration. The letter ‘g” denotes the value.

Newton calculated this value to be 9.8m/s.

Physics scientists have used many methods to determine the acceleration caused by gravity.

Some of these methods can produce errors that can compromise the final result.

New developments in instruments for measuring gravitational acceleration have been made recently.

These developments are intended to reduce the errors common in scientific experiments (Cook, 1957, pp.

Newton’s principle of gravitational acceleration states that a falling object can be determined at a given vertical distance through the use of the formula.

Where u= initial velocity at the beginning of timing.

If the objects fall from a fixed point, then u=o. The equation decreases to

y =1/2gt2

The accuracy of the flight time is critical in free-fall experiments. It is impossible to measure the start time if you don’t have a stopwatch and are watching the falling object.

The scale and clock do not provide the necessary precision for these experiments.

Electronic devices seem to have solved this problem.

Electronic timers can accurately determine start and end times and increase precision.

Newton’s law about force states, in simple terms: force on an object is dependent on its mass and acceleration.

Force is therefore a combination of acceleration and mass (F = Ma).

For an object to experience acceleration, it must have a force applied to it.

This is Newton’s Universal law on gravity.

Isaac Newton provided the general formula to determine the force between two objects. It is here.

F = GM1M2/ r2

Where

G= gravitational constant

M1 and m2 are the masses of both objects. r refers to the distance between them.

To put it another way, the force between two objects can be obtained by multiplying their masses with G, a constant, and dividing the result by the square distance between them.

If an object is located near the earth’s crust, the distance between them is assumed to be their radius.

This reduces the above equation to

F = GmMe/re2

In this instance, the force does not depend on the mass of an object. Mass and radius of the Earth are constants and the force corresponds to its weight on the earth’s surface.

It is clear that gravitational speed is a constant because it depends on constant values from equation 6.

Galileo discovered the equation when he dropped cannonballs of different mass and realized they were all landing at the same moment.

He explained that they both experienced the same acceleration (acceleration caused by gravity).

The same thing happens when a coin or a feather is dropped simultaneously into a vacuum that has no resistance. They land together.

To determine acceleration due gravity, we use equation 2 to calculate the time taken from a stationary point as well as the distance traveled.

Second Section

Objective: To determine acceleration caused by gravity using three different methods, ticker tape timing (stroboscopic analysis), and electronic timing.

First Method: Timing Ticker Tape

A length of ticker tape is attached to a trolley by attaching it at one end.

The trolley is pulled in a steady direction along the bench.

The trolley is secured to the elastic cord by looping one end of the cord around a rod.

To maintain the force constant the cord is stretched with the same force that the trolley is pulled.

This experiment is repeated multiple times to increase precision.

This method has limited accuracy due to friction as the tape passes through the ticker-timer, which reduces the speed of acceleration.

However, this can be decreased by using a larger mass. This also requires investigation.

Results

Three students in each group performed the experiment.

The timer works at 50 dots per minute.

Therefore, the period was 0.02 seconds.

Tape Number

Initial position

Final position

Results (m/s2)

Stroboscopic Analysis: Second Method

The tripod was used to mount the camera and the electromagnet on top.

The camera was placed in a way that allowed for at least one meter of fall to be photographed.

The strobe lamp was placed so that the falling object could be illuminated from the sides.

After the apparatus was fully adjusted, the flash rate for the strobe could be set to 25 flashes/second.

After connecting an electromagnet to a supply of power, the voltage was adjusted so that the ball bearing would remain in its place.

When the room lights were off, the Camera aperture opened wide to allow a sharp view through the viewfinder.

When the strobe lights flash, the cable release should be depressed to open and close the shutter. The power supply was switched off in order to release the ball bearing.

Once the ball bearing has finished falling, the shutter closes.

Results

Trial

Third method: Electronic Timing

The stand was first set with an electromagnet, and then connected to the power supply.

The voltage of power supply was adjusted so that the steel strip could be held vertically.

The light beam was installed at the bottom of the strip. It was just below the point that it could interrupt the beam.

The timer was reset back to zero. The steel strip was then released by switching off power to the electromagnet. Timings were recorded

The experiment was repeated four more times with different results each time.

Measurements of the length of the strip were made with great care and attention to accuracy

Distance s (meters).

g

Discussion

We found a few mistakes when using the ticker timing method.

The ticker timer method was flawed in two ways. First, it started before the ball entered free fall, resulting in shorter elapsed times and lower values of the g. Secondly, the tape wasn’t parallel with the floor, which created a random deviation from the elapsed period.

Therefore, the theoretical value for g was much lower than actually existed.

However, the precision of a photogate was greatly increased and it even reached the theoretical value.

The value in trial 4 was slightly higher, possibly because the ball had been dropped at an angle that decreased the distance. This could explain the high value.

The electronic timer offers the highest precision, with a 0.0009% error.

Conclusion

Because gravitational acceleration relies on constant values, experimental results show that it is a constant.

This supports Galileo’s assertion that cannonballs weighing different masses fall at the exact same rate because they both experienced the same acceleration (acceleration by gravity).

Similar results can be seen when a coin or feather is dropped into a vacuum that has no resistance. They land together.

This means that to determine acceleration due gravity, we need accuracy and precision in our instruments.

The values may contain errors due to instruments.

Reference List

Bell, H 1916 A New Method of Determining “G”, The Acceleration Due to Gravity.

American Meteorological Society.

Bill C 1990 Measurements for Acceleration due to Gravity Physics Teacher, v28.n5 p291-995

Cook AH 1957.

Recent advances in the absolute measurement gravity.

Fredrick JB 1975 Introduction to Physics, USA: Von Hoffmann Press.

Garg M. Kalimullah. Arun P. Lima FMS 2007. Accurate measurement of velocity and position of falling objects.

American Journal of Physics 75(3) 254-258.

Brush, SG 2001. Gerald JH.

Physics, the human adventure: Copernicus to Einstein, and beyond.

NY: Rutgers University Press.

Jewess 2010 Optimizing Acceleration Due to Gravity on the Planet’s Surface.

The Mathematical Gazette. Volume 94. Issue 530July 2010. pp.

Khongiang L., Dkhar A. and Lato S 2015.

The Electronic Timer allows for the precise determination of acceleration due to gravity, G in Shillong.

Kurtus R 2015 Overview of Force of Gravity.

Marson, I. Faller. JE 1986 G. The Acceleration Of Gravity. Its Measurement And Its Importance.

Journal of Physics E, Scientific Instruments, Vol.

Peters, A. and al. 1999 Measurement gravitational acceleration using dropping atoms.

Nature 400, 849-852

Preston-Thomas, H. et. 1960 an absolute measurement for the acceleration due gravity at Ottawa

Canadian Journal of Physics 1960, 38(6), 824-852

Wick, K. Ruddick K 1999.

Falling balls are used to measure g accurately.

American Journal of Physics 67(11), 962-965.

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