In this investigation, I intend to determine the relationship between the length of a pendulum and the time taken for it to oscillate once. The pendulum itself will, due to limited resources, consist of a mass attached to a length of string. The weight of this mass is irrelevant for my experiment but it is, however, vital that it remains constant so that this possible variable does not alter the results obtained from the ‘length of string’ strand of investigation.Other possible lines of enquiry:* Angle of release* Mass of pendulum* Gravitational forceTheory:The velocity of the pendulum increases as the weight falls and reaches a maximum speed at the bottom of the swing. Here, the kinetic energy is also at a maximum, and the potential energy is at a minimum. Then, as the kinetic energy causes the pendulum to continue rising, the kinetic energy is transferred back into potential energy causing the velocity of the pendulum to decrease and stop as it reaches the top of the swing, before then returning to the bottom of the period and continuing in its swing.The unbalanced forces acting on the pendulum cause the movement and consequent oscillation in this investigation. Gravity attempts to pull the mass towards earth but is resisted by the tension in the string. As there are only two forces acting on the weight, they can only be balanced when they are in opposite directions. This balance only occurs when the pendulum is in the middle of its swing.Basic Procedure:The independent variable I plan to change in my investigation is the length of rope used as the pendulum. This alteration in procedure would be repeated at least 10 times in order to obtain accurate results and be able to plot a concluding graph using enough data to discount any anomalous results.So as to conduct my experiment as fully as possible, an initial preliminary experiment must be used to find the rough parameters of such an investigation. I concluded that a minimum length of 5cm and maximum possible length of 55cm would be relevant as well as conclusive to show if a trend exists.I initially measured the time taken for just 1 oscillation but concluded that, dues to the fact that the time taken with the pendulum at 5cm was so small, (roughly 0.7seconds) it would be impossible to obtain accurate results. Any inaccuracy would be due to any possible human error and/or reaction time taken to stop/ start the stopwatch. I decided that, so as to ensure a fair a test as possible, I would let the pendulum swing 10 oscillations and divide the time by 10, thus discounting, as far as possible, any time inaccuracies as a result of the initial release of the weight.Once started, oscillations gradually die away, i.e. the 10th oscillation will be considerably smaller than the 1st. This will not affect my experiment as I will measure 10 oscillations for each length and, by dividing the results by 10, will establish the average period of each. Comparisons between lengths will therefore be fair.Safety Precautions:The investigation I planned was relatively low-risk as it involved only simple and relatively normal apparatus. However, as with all scientific experiments, certain safety precautions need to be taken such as ensuring that the weight is attached to the string securely and is not likely to come loose in mid-swing and harm and equipment or by-standers. Also it is essential that the clamp and base are secure and are unlikely to move/fall over.Equipment:* Clamp Stand* Clamp* Boss for clamp* Weight with hook (36g)* Stopwatch* Length of string (at least 60cm)* 100cm Ruler* ProtractorPlanning a procedure:The area I have chosen to explore in this investigation is ‘how the time taken for one oscillation is affected by the length of the pendulum’. This dependant variable is expected to alter in relation to the length of the rope (the independent variable) from which the weight is swinging. To begin the experiment I needed to erect a stand with a clamp to which the pendulum was attached. The length of the string would be altered relatively easily by wrapping it around the clamp but would also be secure enough not to be lengthened by any pulling. It is vital, of course, that all lengths are measured and altered accordingly, between each recording.I planned to hang the pendulum from an angle of 900 to the stand (i.e. horizontal) and let the weight swing whilst simultaneously starting the stopwatch. The pendulum would be let to swing 10 complete oscillations before the stopwatch would be paused and a reading taken. This procedure would be repeated 3 times so as to obtain an accurate average which could then be divided by 10 for a single oscillation time.Prediction:I predict that the time taken for each oscillation will be affected by the length of the pendulum. An increase in the length will result in an increase in time taken. I based my prediction on the belief that, by raising the pendulum, the weight gains gravitational potential energy, which results in the obvious action of dropping to its original hanging point. As it is released, the gravitational potential energy is converted into kinetic energy which causes the weight to continue moving past its pivot and up to a height less than that at which it started, before, again, experiencing the force of gravity and falling to its original point of vertical suspension.Reliability:In order to ensure a fair experiment is conducted, certain precautions need to be taken to perform a safe and accurate investigation. As I am altering the length of the pendulum and am measuring the time taken for the pendulum to oscillate once, the difference in time recorded may vary depending on the exact length of string used, reaction time, stopwatch accuracy and the ability to drop a weight from a fixed point at an exact time. For this reason I will be using a single weight, which will remain constant, and 3 readings will be obtained from each length of string, so as to get an average.To conduct a fully accurate procedure, the following possible variables will need to be kept constant:* Mass of weight (36g)* Same string* Angle of Release (90o)Part 2: Obtaining EvidenceReadings:The results tabulated below were taken for each of the 3 experiments conducted using each length of rope. As can be seen, an average reading was then calculated.11 different lengths were used in the investigation, ranging from 5cm to 55cm at intervals of 5cm. A metre-rule was used to measure the length of each string and a stopwatch used to measure the time taken for 10 oscillations.Data:time taken for 10 oscillationslength (cm)test 1 (seconds)test 2 (seconds)test 3 (seconds)average (seconds)average (seconds)/105517.3217.0617.1217.171.7175016.2816.2916.416.321.6324515.5615.5915.5715.571.5574014.6614.6914.6614.671.4673513.7213.9413.5913.751.3753012.6912.7112.6512.681.2682511.7211.7811.6911.731.1732010.4710.5310.5910.531.053159.229.289.169.220.922107.447.637.547.540.75455.625.655.475.580.558Part 3: AnalysisReadings:I plotted the results obtained from the investigation on to a graph to clearly show the changes in oscillation time at each of the 11 different lengths. It is visibly apparent that, as the length of the pendulum increases, the time taken for 1 oscillation increased accordingly.The above graph depicting the change in time taken for 1 oscillation against length of pendulum clearly shows that the 2 factors are directly proportional to each other. The rate of increase evidently becomes less significant as the length of the pendulum increases. This can be seen by the fact that the best-fit line drawn becomes gradually less steep and begins to level out. The evidence displayed in this graph and the subsequent one shows that the increase in time is proportional to the length of rope used, evidently supporting my hypothesis.It is evident from the shape of the curve on the above graph that y = x . And therefore y2 = x. On the graph (on the following page) I have squared the ‘y’ values and plotted the straight line to show that x and y2 are directly proportional.lengthsquare of time5530255025004520254016003512253090025625204001522510100525The straight line, plotted on the above graph, clearly shows that x and y2 are directly proportional as the line of best fit travels directly through the origin.From this investigation is has become clear that the time for one complete oscillation is proportional to the square root of the length. Every point plotted on the above graph lie on a straight line and this demonstrates that we can obtain conclusive evidence from this reliable data. It can be presumed that the same trend would continue if further experiments were to be conducted in which the length of the pendulum continued to be increased.Accuracy:Thanks to the obvious correlation between the oscillation time and length of the rope, I believe that a sufficient number of readings were taken to accurately calculate the results and give conclusive evidence that the increase in time taken is proportional to the length of the pendulum.Part 4: EvaluationThe results obtained clearly help endorse my hypothesis; that gravitational potential energy gained by the weight when the pendulum is raised, increases with length and therefore causes an increase in oscillation size and time.Potential Energy = Mass x Gravitational force x Heighttherefore;Potential Energy = 36g x 10m/s2 x Heightand in the case of the 5cm pendulum;Potential Energy = 0.036kg x 10/m/s2 x 0.05mPotential Energy = 0.018Jin the case of the 55cm pendulum;Potential Energy = 0.036Kg x 10/m/s2 x 0.55mPotential Energy = 0.198JThis proves that a change in Potential Energy = Mass x Gravitational force x Change in Height. Therefore the increase in potential energy is directly proportional to the increase in the length of the rope.The extent of the time taken for 1 oscillation clearly varies according to the length of the pendulum. I conclude that my investigation has been successful.Factors that would need amending if further investigations were to be taken in the topic are as follows:* More accurate method of determining pendulum length. Also less elastic string to be used.* Begin swing of pendulum using a more precise method of release.* Use light gates to obtain completely accurate oscillation periods and avoid any possible human error when operating stopwatches.Possible Further Investigations:In order to explore this topic of oscillation times and the various factors affecting them in more depth, different independent variables could be used, such as change in release angle, weight of pendulum or gravitational force. It could be that more accurate data could be obtained by using a smaller angle (15o) as this would keep a more constant time. Obviously, when investigating another variable, all other possible changes would have to remain constant for every set of experiments in order to conduct a fair and constructive experiment. In the example of ‘weight variation’, each pendulum would have a different mass attached subject while the length of rope used would have to remain constant.Explanation:As the length of the pendulum increases, the affect of the gravitational force on the weight when raised also increases, causing a longer oscillation time. At a smaller length the pendulum takes less time to swing 1 complete oscillation. This is because a gravitational force is acting on the weight, which forces it to counteract the force suspending the weight and cause it to swing to its original starting point.