Determination of the Acceleration due to Gravity on the Earth’s Surface

I will be investigating the acceleration due to gravity on the Earth’s surface using an experiment based on this effect. The acceleration due to gravity is constant for all objects, as it is defined by the strength of the gravitational field strength of the object, and not the mass of the object it is pulling.Therefore, in a vacuum, all objects will accelerate at the same rate due to gravity. This only applies in a vacuum, as air resistance plays a significant role as a frictional force.

The Earth’s acceleration due to gravity (also referred to as Standard Gravity, or simply g) is 9.80616ms-2 (according to http://medical-dictionary.thefreedictionary.

com/Standard+gravity), or 9.81ms-2 to three significant figures.The weight of an object is its downwards force, which is given by the equation:F = mgWhere F is the downwards force, or weight; m is the mass of the object; and g is the gravitational field strength of the body attracting it (this is simply g on Earth; standard gravity).When forces are taken into account, frictional forces play a part in determining the resultant force. The downwards force of the body is constant, as the mass does not change as the body falls. However, air resistance increases with velocity, in the case of freefall under gravity.

This means that the acceleration will steadily decrease as the body moves faster, since the resultant downwards force would be less.In Figure 1, as the velocity increases, the air resistance arrow would grow, showing that the air resistance would increase as well. Eventually, it will equal the weight arrow and the air resistance will equal the weight.

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This would mean there would be no more acceleration, and the body will have reached a terminal velocity.In a vacuum, there would be no air resistance, so the body will not decrease its acceleration. Therefore, this would give the most accurate value for standard gravity.I predict that the g value I will receive from my experiment will be lower than 9.81ms-2, since air resistance will affect the results taken. The actual effect will depend on the weight, and therefore, the mass of the object which will be used to measure g.Several methods can be used to measure the value of standard gravity.PendulumThis method involves allowing a pendulum to oscillate freely, recording the time taken for each oscillation.

The length of the string used to make the pendulum, and the angle of the drop will also have to be measured, in order to calculate the value of g.This equation (http://en.wikipedia.org/wiki/Pendulum) is used to determine g from the data we can measure. Here, l is the length of the pendulum, and T is the time taken for one oscillation.Equipment* Stopwatch* Pendulum, of a fixed mass and a light stringThe human error that could be involved in timing the oscillations means that to be more accurate, about 12 oscillations would need to occur for each timing.

This would reduce the human error, as the longer period will reduce the percentage error of reaction time. Other inaccuracies would be found in that, due to air resistance, the distance travelled by the pendulum with each oscillation would decrease each time. Therefore, for 12 oscillations, the time taken would be less than what it should be.Ticker Tape TimerThis method involves dropping a length of ticker tape through a ticker tape timer, using a mass attached to the end of it. As the tape accelerates through, the dots made on the tape space further and further apart, with the increasing speed. A dot is made every 0.

02 seconds with the timer, and as the dots space out, the distance travelled by the tape increases with each 0.02 seconds. This distance can be measured, and using the equation:Velocity = displacementtimethe velocity can also be calculated. This can in turn be used to calculate the acceleration of the mass and the tape, using the equation:Acceleration = change in velocitytimeEquipment* Ticker tape timer* Ticker tape* Known mass* Power supplyGroups of five dots would be measured, in order to improve the accuracy, as the larger range would mean the error in measuring the distances would be reduced. Parallax error whilst reading the metre ruler would also have a less significant effect.The experiment itself is flawed, as friction on the tape passing through the timer, and air resistance, would both decrease the acceleration of the tape through the timer.

This would result in a lower value for g.Light GateThis method involves linking a light gate sensor to a computer and dropping a length of card through the beam. When the card interrupts the beam, a signal is sent to the computer, and when the beam is restored, a second signal is sent. If the length of the card is known, the computer can calculate how fast the card dropped through the beam. Using this, and the time taken for it to pass the beam, the acceleration can be calculated.In theory, this would allow for the height of the drop to be varied, as g should remain the same.

Equipment* Light gate sensor* Computer* Power supply/ies* Card of known lengthThis method gives random errors, as there is no way of regulating the angle or tilt that the card is at when it passes through the sensor. This has an effect on the reading given, as the length of the card interrupt will change due to this tilt. Also, air resistance will again have an effect, reducing the value of g. However, the tilt of the card could increase the g value also, meaning that this experiment produces unreliable, inaccurate resultsI chose to use the ticker tape timer method, as it is easily set up for repeats of the experiment. If the mass used is large enough, it will reduce the effect of air resistance and general friction significantly, allowing for a reasonable value for standard gravity.

Trial RunFor my trial run, I used a mass of 100g.Distance(x10-2m)Time(s)Speed(ms-1)Acceleration(ms-2)14.20.11.42n/a18.

60.11.864.426.50.12.

657.939.50.

13.951347.50.

14.758.2554.00.15.406.

25My results give an average acceleration of 7.96ms-2, which is quite satisfactory considering the effects of friction and air resistance. In the actual run, however, I will use a 200g mass, as this should further reduce the effects of these frictional forces. Therefore, the acceleration given should be closer to g.Accuracy and ReliabilityTo improve the accuracy of my results, I will use a mechanical timer, which will remove human error from this process. Also, I will measure the dot-to-dot distance with a metre ruler with millimetre increments, to increase the accuracy of that process.

This should give an error of �1mm.The time increments, height of the drop and the gravitational field strength are all constant, so the accuracy of the readings should be fairly high.To improve reliability, I will repeat the experiment three times. The same scenario will used in each, keeping height, gravity, and time increments the same.

If there are no anomalous results, then my results should be quite reliable.Equipment* A.C Power supply* 200g mass* Clamp stand* Ticker tape timer* Ticker tape* Sellotape* Metre ruler* Connecting wiresRisk AssessmentRiskCautionary ProcedureRisk of electric shock if connecting wires are unsafe or bareCheck to ensure that electrical equipment is kept in safe conditionMasses dropping from heightBe aware, and raise awareness of others, that there are suspended massesSources Used* Sang, D., Gibbs, K. ; Hutchings, R.

(ed.) 2004, Physics 1,o Cambridge University Press, Cambridge, UK* Wikipedia 2007, Standard gravity, viewed 27 March 2007,o ; http://en.wikipedia.org/wiki/Standard_gravity;* Wikipedia 2007, Pendulum, viewed 27 March 2007,o ; http://en.wikipedia.org/wiki/Pendulum;* Saunders 2007, Standard gravity – definition of Standard gravity in the Medical dictionary – by the Free Online Medical Dictionary, Thesaurus and Encyclopedia., viewed 27 March 2007,o ;http://medical-dictionary.thefreedictionary.

com/Standard+gravity;ResultsFirst RunDistance(x10-2m)Time(s)Speed(ms-1)Acceleration(ms-2)6.300.10.

63n/a16.20.11.629.9024.50.12.458.

3034.70.13.

4710.243.20.14.328.

7054.80.15.4811.6Second RunDistance(x10-2m)Time(s)Speed(ms-1)Acceleration(ms-2)9.200.

10.92n/a19.00.11.909.

8027.70.12.778.

7037.80.13.7810.

147.20.14.

729.4057.50.15.7510.3Third RunDistance(x10-2m)Time(s)Speed(ms-1)Acceleration(ms-2)8.

500.10.85n/a18.50.11.8510.027.

70.12.779.2037.

70.13.7710.047.20.14.729.

5057.40.15.7510.2AnalysisThe dot-to-dot spacing on the tape increased, showing that there was acceleration. The graphs that were drawn show that the acceleration was fairly constant; the gradients show that this acceleration was close to 9.8ms-2.

GraphAcceleration(gradient – ms-2)18.629.439.47Average9.17This value is much like I expected, as the book value for g is 9.81ms-2, but only in a vacuum.

Taking friction and air resistance into account, this value is quite accurate.Referring to figure 1 in the plan, air resistance can be calculated using forces equations and Newton’s laws of motion. The second law statesF = maThis means I can calculate the effect of friction on the body as it fell.The ideal downwards force, taking a as g, which is 9.81ms-2. Mass is in kilograms.

F1 = ma= 0.2 x 9.81= 1.962NIn the experiment, however, I found the acceleration to equal 9.17ms-2. This makes the downward force equal to:F1 = ma= 0.2 x 9.

17= 1.834NThe force difference must equal the overall friction acting.Fr = F1 – F2= 0.128NTherefore, there must have been a 0.128N friction opposing the motion of the mass.EvaluationThere were several flaws and limitations of the experiment, when it came to the actual procedure.The tape may have run through the timer at an angle, which would have affected the vertical fall of the mass, and therefore, the distance between the dots. This would have changed the value for g that resulted.

The timer may also have been irregularly marking the tape, which would severely negate the accuracy and reliability of my results. This would have made the timing inconsistent, and the time is crucial to calculating the value of the acceleration. There was no way of knowing which dots may or may not have been anomalous, so this compounded the problem.The tape may have had kinks or small rips in it, which would have altered its path and added to friction. This would change the position or regularity of the dots on the tape.

The reliability of the results is questionable, as the latter two graphs showed an acceleration of within 0.07ms-2 of each other, but the first graph shows a reading of 8.4ms-2, which is a whole 1ms-2 out from the others. This made a definite difference to the average value of standard gravity.Percentage error = (highest result – lowest result) x 100Average result= 1.07 x 1009.17= 11.7% errorThis is a fairly large percentage error.The error bars on my graphs, blue for maximum and black for minimum, also show how wide the possible error for my results is.

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