The Simple Pendulum Experiment
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General Plan The first thing I am going to do is outline a general plan for this experiment. In this experiment, I am going to be measuring the effect of two variables on the time of one oscillation of a simple pendulum. The two variables that I have chosen are the length of the pendulum's string and the mass of the pendulum's bob. I will vary these two items and record results for the time of one oscillation of a pendulum with different mass/length of string. I must also be able to determine the value for acceleration due to gravity (hereby referred to as g) To do this, I must be able to find some equation that links length of a string of a pendulum or the mass of a pendulum's bob with time. For this I will need to carry out research. Aim The aim of this experiment is to determine the effects of two factors on the time of one oscillation (or swing) of the simple pendulum, and also to determine a value for g (acceleration due to gravity) Design Before producing a plan I will conduct a preliminary experiment this will help me find and basic flaws in the setup of my experiment, and will also allow me to find room for improvement on my actual experiment. It will also allow me to experiment with different values for my variables, to find suitable limits to my measurements, and to find a suitable interval between my measurements of these variables. After gaining the results of this experiment, I will be able to plan my main experiment more thoroughly. Preliminary Experiment The diagram below shows a brief setup of the simple pendulum experiment The pendulum (3) will be held by a clamp stand (1) which will be placed on a work bench table (2) Fig. 1: Simple Pendulum Experiment Setup [IMAGE] Figure 2 [IMAGE] I will use a digital chronometer (4), which is accurate to within How to Cite this Page
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±0.005s to measure the time of the oscillation, and a ruler (accurate
to within ±0.0005m) to measure the length of the string. To record the time of one oscillation, I will hold the pendulum with the string taut at an angle to it's original position (as shown in Fig. 2) I will release the pendulum without applying any force to it, so as not to influence the swing, and allow the pendulum to swing due to the force on it from gravitational acceleration. I will take the time of one oscillation to be the time it takes for the pendulum to return to it's original position. I will start the chronometer as I release the pendulum, and stop it after one oscillation. Preliminary Results Length of String: 100.3cm (~ 1m) Time for one oscillation: 1.74 s Preliminary Evaluation From my preliminary results, I have found that there are many flaws in the design of this experiment. However, most of these can be resolved. First of all, I must address the safety of the experiment. As the mass is swinging from the pendulum, the clamp stand has the tendency to tip because of the laws of moments. This can be remedied by attaching the Clamp stand to the table with the aid of a GClamp I also found out that the table was too low for me to increase the length of the pendulum's string by large amounts. Therefore I have decided to secure the clamp stand to a stool, and then place this on the table, thus increasing the height of the pendulum's pivotal point. A very obvious problem is the inaccuracy of my time keeping. As I am judging when the pendulum's oscillation stops and starts, I am introducing a "reaction time" into the equation, because I am not able to react fast enough to press the button on the chronometer exactly when the pendulum starts/stops it's oscillation. To rectify this, I can measure my reaction speed using a reaction speed ruler, (see Reaction Time Experiment later) and deduct this twice from any time readings I make (twice because I have to deduct for the start error, and the end error) Even after doing this, my timing will still be inaccurate, as I am only measuring one oscillation. Instead of doing this, I will measure the time of 30 oscillations, and then use this figure to work out an average of the time taken for one oscillation. To make this easier to understand I can write it as follows [IMAGE] Where T is the time for one oscillation, A is the time for 30 oscillations, and R is my reaction time. Reaction Time Experiment. The purpose of this experiment is to determine the time t takes me to stop/start the digital chronometer, so I can use this to determine a time period for one oscillation of the simple pendulum using the equation stated on the previous page. To do this experiment, I have 2 choices, the first of which is to see how long it takes me to start ad stop the digital chronometer. However, doing this beings into account all other sorts of possible errors, therefore I have chosen the second option, which is to use a piece of equipment known as a reaction time ruler. This ruler has a set of marks measured in time, and this is done because we know the acceleration due to gravity (Approximately 9.8 m s 2). Therefore, we can work out the reaction time by holding the ruler above the hand of the person whose reaction time is to be measured, and dropping it. When the person catches the ruler, we can read off the time that it took that person to catch the ruler, therefore determining the reaction time of the person. I will carry out this experiment 3 times and take an average of the results to determine my reaction speed. The reaction time ruler is accurate to within ±0.005s. Reaction time Experiment Results Measurement 1 = 0.26s Measurement 2 = 0.16s Measurement 2 = 0.18s Average = (0.26+0.16+0.18) ÷ 3 Average Reaction Time = 0.2s Length Experiment The setup of the apparatus for this experiment is included on a Page 7 The pendulum is held in place by a clamp stand, which is secured to a stool by GClamp. The stool is placed on a table/workbench. The mass of the pendulum is kept constant during this experiment The length of the pendulum's string is variable for this experiment I will use a ruler to measure my pendulum's string, and ensure that it is at my start value, which I have chosen to be 80cm (0.8m). I have chosen this value because lengths of greater than 2m will be too long and the pendulum will hit the floor. If the pendulum's string is too short then the time for an oscillation will be minute, and hard to measure. Also, 80cm is easy to measure with a metre rule. The ruler is accurate to within ±0.0005m The Digital Chronometer is accurate to within ±0.005s I will then setup my experiment as shown in Fig 3 on Page 7. I will then use the digital stopwatch to measure the time for 30 oscillations, and record my results in a table. I will repeat the above, but increasing the length of the pendulum's swing by 10cm each time (up to 150cm) using a ruler to make sure the length of the string is accurate, until the final length (1.5m) has been reached. I have chosen the final length to be 1.5 m, as this gives me 8 results, which is enough to plot a decent graph of my results and to spot any "stray" results. I have chosen to increase the length of the string in 10cm intervals, as this is easy to measure with a metre rule. 1cm increments would make the difficult to implement, and increments of 20cm would make the experiment very hard to do, as it would either mean increasing the final length to 2.2m, making the pendulum hit the floor, or only recording 4 results. When I release the pendulum, I will not start the stopwatch straight away, as I will let the pendulum settle into a steady oscillation, ad I may have accidentally pushed the pendulum when releasing it, and doing this will allow any extra force to dissipate. Once I have obtained my results, I will then process these using the equation (as stated above) [IMAGE] To find the time of one oscillation of the pendulum (for each length of string). Once I have obtained my results, I will plot a graph of the time of one oscillation against the length of the string, to help me observe the effect that modifying the length of the pendulum's string has on the time of oscillation of the pendulum. Apparatus Setup Figure 3: Apparatus Setup Apparatus List · Metre Rule (accurate to within ±0.0005m, ranges from 0m  1m) · Top Pan Balance (accurate to within ±0.005g, ranges from 1000g1500g) · Digital Chronometer (accurate to within ±0.005s, ranges from 0s to 600s) · Clamp Stand · GClamp · Plasticine · Stool · Workbench/Table · Reaction time ruler · Simple Pendulum Mass Experiment. The setup of this experiment is shown on page 7. As before, the pendulum is held in place by a clamp stand, which is secured to a stool by GClamp. The stool is placed on a table/workbench. The length of the pendulum's string is kept constant during this experiment The mass of the pendulum is variable for this experiment I will use a ruler (accurate to ±0.0005m) to make sure the length of the pendulum's string for this experiment is 1m. I have chosen this length, as it is very easy to setup with a 1m rule. I will then measure and record the mass of the pendulum's bob using the toppan balance (accurate to ±0.005g). Next, I will record the time for 30 oscillations of the pendulum, and record this data in a table. I will then repeat the above, but adding 1gram of plasticine in a uniform ring around the pendulum's bob each time, taking 8 readings in all (so I have enough to plot a decent graph) The Bob of my pendulum weighs 30g, therefore this will be my start value, and my end value will be 37g. The Toppan balance will measure the mass of the plasticine to be added. Once I have recorded my results, I will use the formula (as stated above) [IMAGE] to work out the time of one oscillation of the pendulum. I will then plot a graph of mass against time to help me see the effects of changing the mass of the pendulum on the time of the pendulum's oscillation and proceed to analyse my results. Length Experiment results Length (cm) A (s) A2R (s) T (s) 80 56.8 56.4 1.88 90 59.8 59.4 1.98 100 62.8 62.4 2.08 110 65.5 65.1 2.17 120 67.9 67.5 2.25 130 69.1 68.7 2.29 140 71.5 71.1 2.37 150 73.9 73.5 2.45 Mass = 35 grams L = Length of Pendulum's swing A = Time for 30 oscillations R = Reaction Time T = Time for one oscillation Mass Experiment results Mass (g) A (s) A2R (s) T (s) 30 62.2 61.8 2.06 31 62.2 61.8 2.06 32 62.2 61.8 2.06 33 62.2 61.8 2.06 34 62.2 61.8 2.06 35 59.8 59.4 1.98 36 59.8 59.4 1.98 37 62.2 61.8 2.06 Length = 1 metre L = Length of Pendulum's swing A = Time for 30 oscillations R = Reaction Time T = Time for one oscillation Time vs. Length Graph Graph 1: Time of one pendulum oscillation against Length of pendulum's string Time vs. Mass Graph Graph 2: Time of one pendulum oscillation against Mass of pendulum's bob Analysis of Results To help my analysis, I have drawn graphs of the results, which you can find on pages 11 and 12. The first of these graphs show the time of the pendulum's oscillation in comparison to the length of pendulum's string, and the second of which shows the time of one oscillation of the pendulum in comparison with the mass of the pendulum's bob. I have noticed that the mass of the pendulum has no effect on the time of the oscillation of the pendulum. The line is straight apart from a couple of stray results. The graph of the length of string increases so does the time of oscillation I have researched this subject and have found the formula for the simple pendulum. [IMAGE] Formula taken from "Accessible Physics" Page 78 T = time for one oscillation of the pendulum (s) l = Length of pendulum's string (m) g = acceleration due to gravity As you can see, my statement that mass has no effect on the time of the pendulum's oscillation can be proven by this equation, as we see no example of mass being used in this equation I can now work out a value for g using this equation [IMAGE] The equation is now in the form y = mx, where [IMAGE] Therefore, [IMAGE], so if I plot a graph of T2 against length, then I can find the acceleration of gravity by taking the gradient of this line to be m in the above equation ([IMAGE]) L (cm) T2 (s2) (3 s.f.) 80 3.53 90 3.90 100 4.33 110 4.78 120 5.06 130 5.24 140 5.62 150 6.00 T2 = time for one oscillation2 (s2) L = length of string (cm) Graph of T2 vs. l Graph 3: T2 against l Determination of g Gradient 1 = 4.3secs ÷ 1m = 4.3 Gradient 2 = 4.45secs ÷ 1m = 4.45 Gradient 3 = 4.15secs ÷ 1m = 4.15 Average Gradient = (4.3 + 4.45 + 4.15) ÷ 3 = 4.3 g = 4p2 ÷ 4.3 = 9.18102734985056615705534046500107 = 9.18 (3s.f.) ± 0.32 Accepted Value for g = 9.8 ms2 (0.62 difference) Evaluation The only possible sources of error from this experiment are in my reaction time, as at times it may have been higher or lower than my measured reaction time, or from a miscount in the amount of oscillations when recording 30 oscillations. One way of fixing this would to be to take more than one measurement of 30 oscillations, and then take an average of the results. The more measurements taken, the less likely the final results will be erratic. I could also use data logging, positioning a light gate at the centre of the pendulum's swing, which would be linked to a computer, which would time the oscillation, and could also count the number of oscillations. I found there were a few stray results in my implementation of this experiment. I found 2 slightly low results (those of 35g and 36g) for my mass experiment, and also found that my T2 against time graph was not a straight line (as it should have been), which made my calculation of g inaccurate and awkward to do. Overall, this experiment would only have ever shown an approximate value for g, as for sake of easiness, a lot of the lengths and masses were rounded to the nearest cm/half gram. This may account for some stray results. The techniques that I have used for this experiment, for example the reaction time ruler, are not based on general scientific practice, however, given the equipment available to me, and for the purpose of this coursework, the techniques used are suited well to this experiment. Conclusion My results show that the mass of the pendulum's bob has no effect on the time of one oscillation of the pendulum My results also show that as the length of the pendulum's string increases, so does the time of oscillation. I have determined a value for g as 9.18 ms2 (3s.f.) ± 0.32 (difference in gradients) The accepted value for g is 9.8 ms2. My determined value may have been closer if the experiment was operated under stricter conditions 
