Length Of The Pendulum To The Period example essay topic

Investigating factors which affect the period time of a simple pendulum Planning Definitions: Oscillation: Repeated motion of pendulum (to and for) Period (T): Time taken for one full oscillation In this investigation, I am going to experimentally determine a factor which will affect the period of a simple pendulum and the mathematical relationship of this factor. This type of pendulum will consist of a mass hanging on a length of string. Factors which affect the period (T) of a pendulum: -Length (L) of pendulum -Angle of amplitude -Gravitational field strength (g) -Mass of bob I predict that the period will be affected by the length of the pendulum. An increase in length will produce an increase in time.

I based by prediction on the scientific theory I found in a physics text book: The pendulum is able to work when the bob is raised to an angle larger than the point at which it is vertically suspended at rest. By raising the bob, the pendulum gains Gravitation Potential Energy or GPE, as in being raised, it is held above this point of natural suspension and so therefore is acting against the natural gravitational force. Once the bob is released, this gravitational force is able to act on it, thus moving it downwards towards its original hanging point. We can say therefore, that as it is released, the GPE is converted into Kinetic Energy (KE) needed for the pendulum to swing. Once the bob returns to its original point of suspension, the GPE has been totally converted into KE, causing the bob to continue moving past its pivot point and up to a height equidistant from its pivot as its starting point. The same factors affect the pendulum on its reverse swing.

GPE gained after reaching its highest point in its swing is converted into KE needed for it to return back to its natural point of vertical suspension. Due to this continuous motion, the bob creates an arc shaped swing. The movement of the pendulum is repeated until an external force acts on it, causing it to cease in movement. The pendulum never looses any energy, it is simply converted from one form to another and back again. I am therefore going to experimentally determine the relationship between the length of the pendulum and the period. In the scientific theory, I found a formula relating the length of the pendulum to the period.

It stated that: P = 2 L g P = The period g = Gravitational Field Strength L = Length of string This formula shows that L is the only variable that when altered will affect the value of P, as all the other values are constants. The formula: P = 2 L g can be rearranged to produce the formula: P = 4 L g and therefore: P = 4 L g As 4 and g are both constants, this means that P must be directly proportional to L. I can now say that the length of the pendulum does have an affect on the period, and as the length of the pendulum increases, the length of the period will also increase. I will draw a graph of P against L. As they are directly proportional to each other, the predicted graph should show a straight line through the origin: Method -I will firstly set up a clamp stand with a piece of string 50 cm long attached to it. -A mass of 50 g will be attached securely to the end of the string -The mass will be held to one side at an angle of 45 degrees (measured with a protractor), and then released. -A stop clock will be used to time taken for one full oscillation -This will be repeated a number of times, each time shortening the length of string by 10 cm -The length of the pendulum will be plotted against the period on a graph. NB.

The final length of string and mass will be decided after my preliminary investigation. Apparatus: -Meter ruler -Protractor -Clamp stand -G-clamp -Stop clock -String -Mass Diagram: The following factors will be considered when providing a fair test: -The mass will be a constant of 50 g throughout the experiment -Angle of amplitude shall be a constant of 45 degrees. This will ensure that there is no variation of the forces acting on the pendulum. -The value of gravitational field strength will inevitably remain constant, helping me to provide a fair test. -The intervals between the string lengths will increase by 10 cm each time. This will help me to identify a clear pattern in my results.

-If any anomalous results are identified, readings will be repeated. This will ensure that all readings are sufficiently accurate. -To ensure that the velocity is not affected, I will ensure that there are no obstructions to the swing of the pendulum. The following factors will be considered when providing a safe test: -Care will be taken not to let the bob come into contact with anything whilst swinging the pendulum, as the weight is relatively heavy (50 g) -The clamp stand will be firmly secured to the bench with a G-clamp so that the clamp stand will not move, affecting the results.

-Excessively large swings will be avoided (angle of amplitude will be 45 degrees Results of preliminary investigation: Length of string (cm) Period (secs) 502.58 402.31 302.11 201.78 101.39 My preliminary investigation was successful. The results from my table back up my prediction that, as the length of the pendulum increases, the period increases. I learned from my preliminary investigation that my proposed method may not give me sufficiently accurate results. These results may be inaccurate due to a slight error of measurement in time, height or length. Although this experiment produced no anomalies, I will take three readings of each value during my final experiment and take an average. I will also measure the time taken for 5 oscillations rather that 1 and then divide the result by 5.

These two changes will hopefully help me to identify and eliminate anomalies, should they occur. They should also add to the accuracy of my results. Obtaining Evidence I used the method proposed in my plan, taking three readings of each value and measuring the time taken for 5 oscillations rather than for 1. During the experiment, I observed that each oscillation for the same length of string seemed to be equal. This showed that the pendulum did not slow down as the number of oscillations increased. I took the safety measures described in my original plan.

During the experiment I was careful to use accurate measurements in order to obtain sufficiently accurate results, for example: -The string was measured with a meter ruler, to the nearest mm, to ensure that each measurement had a difference of exactly 10 cm. -The angle of amplitude will be measured with a protractor to the nearest degree to ensure that the angle remains constant throughout the experiment. -A stop clock will be used to measure the period accurately. The period was measured in seconds, with the stop clock measuring to the degree of two decimal places of a second.

However, I have rounded up each time to the nearest second to give appropriate results. -The mass was measured using five 10 g masses, to ensure that the mass remained constant throughout the experiment. I then divided by 5 to get the average reading for one oscillation. This again should influence the accuracy of my results. Table of averages: Length of string (cm) Period (secs) 501.45 401.28 301.13 200.91 100.61 Using the formula, T = 2 L g found in the Scientific Theory, I calculated the perfect results that should have been obtained, had my experiment followed the formula exactly: Length of string (cm) Period (secs) 501.44 401.25 301.07 200.91 100.64 Using my averaged results, I squared P to show the relationship between P and L: Length of string (cm) Period (secs) 502.1 401.64 301.28 200.83 100.37 As all my results were accurate, I had no need to repeat any of them. However, had there been an anomalous result, or had I come across any problems, I would have repeated my results to identify the cause and eliminate anomalies.

Analysing evidence and concluding Using the results from my table, I drew a graph to show what had been obtained from the experiment (see graph A). The graph clearly shows a smooth curve with a positive gradient. This indicates that as the length of the pendulum is increased, the period will increase. Although my second graph (see graph B), does not show a perfect straight line through the origin, a line of best fit can be drawn to show this. This backs up the theory in my scientific knowledge, that P is directly proportional to L, i.e. if the length of string was doubled, the period would be doubled.

My table of results drawn from my experiment was extremely similar to the results produced from the scientific formula, showing that my experiment was successful. My two graphs showed resemblance to my predicted graphs, indicating that my results were sufficiently accurate and therefore, my proposed method was reliable for this experiment. My findings indicate that the time period varies directly with the length of the string when all other factors remain constant. Evaluating The evidence obtained from my experiment supported my prediction that as the length of the pendulum increases, the period increases. This is also shown in Graph A, as the graph displays a smooth curve with a positive gradient. My method in squaring P was successful, as I discovered that T was directly proportional to L, providing all other values remain constant.

This was shown by a straight line going through the origin (Graph B). These results were encouraging and led me to believe that my proposed method was sufficient for the experiment. Some of the results were not accurate, as they did not match the results produced by the formula. This could have been due to human error. However, the majority of my results were no more than a decimal place away from the formula results and, therefore, quite reliable. Had there been any anomalous results, I would have repeated my readings.

Factors which may have affected the accuracy of my results include: -Error in measurement of angle of altitude. This angle proved difficult to measure and it was hard to get the exact same angle for each result. To improve the accuracy of this measurement, I could have attached the protractor to the clamp stand so that it was in a fixed position. -Error in measurement of string.

To improve the accuracy of this, I could have marked off the points with a pen to ensure they were as accurately measured as possible. -Human reaction time. Depending on human reaction time, the measurement period time could have been measured inaccurately, due to slow reactions when setting the stop-clock etc. This could have been improved by involving another person to aid me with my experiment, for a quicker reaction time. The procedure was relatively reliable, excluding human error, and so I can conclude that my evidence is sufficient to support a firm conclusion that: The only factor which affects the period of a simple pendulum is its length. As the length increases, so does the period.

If I were to extend my investigation, I would investigate to provide additional evidence to back up my conclusion, for example, changing the mass or angle of altitude. The results gained would hopefully aid me further in supporting my Scientific Theory. It would also be interesting to investigate how the factors are affected when the Gravitational Field Strength is different, i. e... not 9.8 Newtons..