To carry out an experiment of simple harmonic motion using a simple pendulum and determine the acceleration due to gravity, to verify the equation T = 2?? (l/g) and show the relationship between time period and length. Method The apparatus is set up, as above, the string must be measured carefully with a ruler to minimise any error; the length should start off fairly short at about 0. 2m for example. The bob should be secure. A table should be made for results, this should include length, time (twice), average time and time period (= average time/20).

The pendulum is set into motion by a gentle push, some practice in doing this and also counting and timing the oscillations beforehand may help to achieve greater accuracy. The angle of amplitude should be kept similar if not the same for each experiment, to make sure the forces acting on it are the same. It is easier to count the oscillations from the equilibrium position as this is where the pendulum has no energy. The number of oscillations counted should be about 20 as it is easier to count a larger number. The reaction time needed to count one single oscillation would be so small that errors could easily occur.

The timing for 20 oscillations should be repeated to calculate an average time. The string is now adjusted and measured for a longer length, about an extra 10cm. The experiment is repeated 6 more times, with the length of string being increased by 0. 1m. The force of gravity is constant, whereas the length and the time period are variable. Therefore the equation for the time period T = 2(((L/g) can be re arranged and compared to the equation of a graph y = m x + c. Where: T = Time period L = Length g = gravity y and x are always the variables, and m and c are the constants.

Since gravity is the constant, a value for the acceleration due to gravity can be determined from the gradient. T = 2 x L g ( ( ( ( y = m x + c Where m = gradient of graph c = intercept y = vertical axis time period T/s x = horizontal axis length (L/m( An extra column will be needed in the results for the values of (L as the graph will be time against length. The gradient of the graph can be used in the time period equation to find the acceleration due to gravity.

Results Length/m Time/s Time/s Av. time/s T/s (L/m( 0. 2 17. 60 18. 76 18. 18 0. 91 0. 45 0. 3 22. 13 22. 03 22. 08 1. 10 0. 55 0. 4 25. 1 25. 31 25. 21 1. 26 0. 63 0. 5 27. 94 28. 00 27. 97 1. 40 0. 71 0. 6 30. 44 30. 47 30. 455 1. 52 0. 77 0. 7 32. 44 32. 56 32. 5 1. 63 0. 84 0. 8 34. 91 34. 88 34. 89 1. 74 0. 90 Analysis of Results The gradient of the graph can be calculated from and is equal to 2( ?? (g = 1. 18 = 1. 98 ?? 0. 595 1. 98 = 2( = 6. 2 (g (g (g = 6. 2 1. 98 = (3. 13 g = 3. 13i?? g = 9. 8ms(Conclusion The value I obtained for the acceleration of the simple pendulum due to gravity is 9. 8ms(i?? , which is good.

This shows that the experiment was accurate and verifies the equation for the time period. The value is slightly less than the accepted value of 9. 81ms(i?? for the acceleration of gravity which could be due to a number of uncertainties and errors that may have occurred throughout the experiment despite efforts to keep them at a minimum. Timing for the 20 oscillations of each length was repeated because although it should take the same time, as amplitude does not affect the time period, we knew errors would occur and calculating an average time from 2 sets would be more efficient.

The main uncertainty in the experiment was human reaction time for starting and stopping the stopwatch which must be fairly fast because it is does not take long at all for one oscillation and errors could occur very easily. We began timing the oscillations from the centre instead of the ends because this is where speed is at its maximum but acceleration is at its minimum which would make the results more accurate. The amplitude was kept small so as not to move too fast and for ease of counting.

We also tried to keep the amplitude at a similar size each time because although amplitude does not affect the time period, changing the length of the pendulum does. So if the amplitude were greatly increased for one length the results would not follow any kind of pattern that you would expect if the amplitude were kept the same for each length. We also counted 20 oscillations rather than just one because the error that arises through being unable to start and stop the watch when the pendulum is exactly in the intended position is greatly reduced.

The measurement of the length of the pendulum could have a degree of uncertainty in it from its initial measurement to then being set into motion where it may be slightly stretched, however if there was an error in the measurement it would be very small. This error could have been minimised by using a pen to mark the string where the length starts. In the same way it is possible that air resistance could cause an error but because the ball was quite small I believe the error would be minuscule.

When the ball was set into motion it did, on occasion, have a tendency to move in a circular path so we would begin the experiment again. The process of finding my value was made easier by converting the results into a graph, where the advantages outweigh those of calculating each set of results, one being time consumption. The use of a graph with a line of best fit enables you to see relationships between the results, constants, and the possibility of predicting the behaviour of results, showing new values. Instantaneously the line of best fit can reveal results that are inaccurate.

A graph is easier to understand and make sense of results. My graph has a straight line through the origin which shows that length and time period are directly proportional. The length of the string affects the time period. If the length is increased, the time period increases also. All my results are either very close or on the line of best fit showing that there were no serious errors in the experiment. Task 2 Pendulums provide good time keeping because they perform simple harmonic motion and therefore can always have the same time period irrelevant of their mass.

Pendulums dont lose energy it is simply converted from one form to another and then back again, so the motion is continuous until an external force acts on it. The energies used are gravitational potential energy that becomes kinetic energy and vice versa. Grandfather clocks have a time period of 2 seconds. The length of the pendulum needed, can be obtained from the following equation.

3 The acceleration of the spring must be less than or equal to the gravitational force for the mass to not lose contact with the tray. a < g Therefore the maximum acceleration when contact is not lost must be 9. 8ms(i??. With this information it is then possible to calculate the maximum amplitude the spring can have without losing contact, by first finding the time period. where T = time period, m = mass, k = spring constant, f = frequency, a = acceleration and x = displacement. T = 2 (m/k).

T = 2 (0. 13/15) T = 0. 58s Then the frequency needs to be calculated f = 1/T f = 1/0. 58 f = 1. 72Hz And finally the displacement can be calculated from the equation for SHM. a = (2? f) x x = a/(2 f)i?? x = 9. 81/(2? 1. 72)i?? x = 9. 81/116. 8 x = 0. 084 x = 8. 4cm If the mass were to lose contact with the tray, it would happen when the acceleration is greater than the force of gravity, and would happen at the highest point of displacement when the tray is at its maximum acceleration on the way back down. A > g.

The pendulum is set into motion by a gentle push, some practice in doing this and also counting and timing the oscillations beforehand may help to achieve greater accuracy. The angle of amplitude should be kept similar if not the same for each experiment, to make sure the forces acting on it are the same. It is easier to count the oscillations from the equilibrium position as this is where the pendulum has no energy. The number of oscillations counted should be about 20 as it is easier to count a larger number. The reaction time needed to count one single oscillation would be so small that errors could easily occur.

The timing for 20 oscillations should be repeated to calculate an average time. The string is now adjusted and measured for a longer length, about an extra 10cm. The experiment is repeated 6 more times, with the length of string being increased by 0. 1m. The force of gravity is constant, whereas the length and the time period are variable. Therefore the equation for the time period T = 2(((L/g) can be re arranged and compared to the equation of a graph y = m x + c. Where: T = Time period L = Length g = gravity y and x are always the variables, and m and c are the constants.

Since gravity is the constant, a value for the acceleration due to gravity can be determined from the gradient. T = 2 x L g ( ( ( ( y = m x + c Where m = gradient of graph c = intercept y = vertical axis time period T/s x = horizontal axis length (L/m( An extra column will be needed in the results for the values of (L as the graph will be time against length. The gradient of the graph can be used in the time period equation to find the acceleration due to gravity.

Results Length/m Time/s Time/s Av. time/s T/s (L/m( 0. 2 17. 60 18. 76 18. 18 0. 91 0. 45 0. 3 22. 13 22. 03 22. 08 1. 10 0. 55 0. 4 25. 1 25. 31 25. 21 1. 26 0. 63 0. 5 27. 94 28. 00 27. 97 1. 40 0. 71 0. 6 30. 44 30. 47 30. 455 1. 52 0. 77 0. 7 32. 44 32. 56 32. 5 1. 63 0. 84 0. 8 34. 91 34. 88 34. 89 1. 74 0. 90 Analysis of Results The gradient of the graph can be calculated from and is equal to 2( ?? (g = 1. 18 = 1. 98 ?? 0. 595 1. 98 = 2( = 6. 2 (g (g (g = 6. 2 1. 98 = (3. 13 g = 3. 13i?? g = 9. 8ms(Conclusion The value I obtained for the acceleration of the simple pendulum due to gravity is 9. 8ms(i?? , which is good.

This shows that the experiment was accurate and verifies the equation for the time period. The value is slightly less than the accepted value of 9. 81ms(i?? for the acceleration of gravity which could be due to a number of uncertainties and errors that may have occurred throughout the experiment despite efforts to keep them at a minimum. Timing for the 20 oscillations of each length was repeated because although it should take the same time, as amplitude does not affect the time period, we knew errors would occur and calculating an average time from 2 sets would be more efficient.

The main uncertainty in the experiment was human reaction time for starting and stopping the stopwatch which must be fairly fast because it is does not take long at all for one oscillation and errors could occur very easily. We began timing the oscillations from the centre instead of the ends because this is where speed is at its maximum but acceleration is at its minimum which would make the results more accurate. The amplitude was kept small so as not to move too fast and for ease of counting.

We also tried to keep the amplitude at a similar size each time because although amplitude does not affect the time period, changing the length of the pendulum does. So if the amplitude were greatly increased for one length the results would not follow any kind of pattern that you would expect if the amplitude were kept the same for each length. We also counted 20 oscillations rather than just one because the error that arises through being unable to start and stop the watch when the pendulum is exactly in the intended position is greatly reduced.

The measurement of the length of the pendulum could have a degree of uncertainty in it from its initial measurement to then being set into motion where it may be slightly stretched, however if there was an error in the measurement it would be very small. This error could have been minimised by using a pen to mark the string where the length starts. In the same way it is possible that air resistance could cause an error but because the ball was quite small I believe the error would be minuscule.

When the ball was set into motion it did, on occasion, have a tendency to move in a circular path so we would begin the experiment again. The process of finding my value was made easier by converting the results into a graph, where the advantages outweigh those of calculating each set of results, one being time consumption. The use of a graph with a line of best fit enables you to see relationships between the results, constants, and the possibility of predicting the behaviour of results, showing new values. Instantaneously the line of best fit can reveal results that are inaccurate.

A graph is easier to understand and make sense of results. My graph has a straight line through the origin which shows that length and time period are directly proportional. The length of the string affects the time period. If the length is increased, the time period increases also. All my results are either very close or on the line of best fit showing that there were no serious errors in the experiment. Task 2 Pendulums provide good time keeping because they perform simple harmonic motion and therefore can always have the same time period irrelevant of their mass.

Pendulums dont lose energy it is simply converted from one form to another and then back again, so the motion is continuous until an external force acts on it. The energies used are gravitational potential energy that becomes kinetic energy and vice versa. Grandfather clocks have a time period of 2 seconds. The length of the pendulum needed, can be obtained from the following equation.

3 The acceleration of the spring must be less than or equal to the gravitational force for the mass to not lose contact with the tray. a < g Therefore the maximum acceleration when contact is not lost must be 9. 8ms(i??. With this information it is then possible to calculate the maximum amplitude the spring can have without losing contact, by first finding the time period. where T = time period, m = mass, k = spring constant, f = frequency, a = acceleration and x = displacement. T = 2 (m/k).

T = 2 (0. 13/15) T = 0. 58s Then the frequency needs to be calculated f = 1/T f = 1/0. 58 f = 1. 72Hz And finally the displacement can be calculated from the equation for SHM. a = (2? f) x x = a/(2 f)i?? x = 9. 81/(2? 1. 72)i?? x = 9. 81/116. 8 x = 0. 084 x = 8. 4cm If the mass were to lose contact with the tray, it would happen when the acceleration is greater than the force of gravity, and would happen at the highest point of displacement when the tray is at its maximum acceleration on the way back down. A > g.