This is due to the fact that after a certain period of time, the amplitude tends towards SHM, in such a way that amplitude only decreases to a certain point, after which successive oscillations have maximum and minimum amplitudes of very similar value, which is a feature of the undamped case in which SHM also exists, shown by the very close fit of the curve to the points, which is a claim backed up by the fact that the regression-squared values are very close to 1 for each of curves, which means the correlation of the best-fit curve to the data point is very high. Both ends of the tube were carefully adjusted to make sure the diameters were the same on both end, otherwise, it would cause a systematic error in the value for amplitude as there would be a variation in the amount of air resistance at both ends. The control variables in this experiment are, the length of string used kept at 1. See the table above for the data: Another risk involving the use of water was that of spillages, since if water is split it can become a slip hazard fro yourself and other people working in the lab. Using the camera, I was able to film the motion of the pendulum and after the fact, look through each frame to obtain a precise value for the amplitude of the pendulum at a given time. This allowed me to be more confident in the plots being accurate since the average value accounts for any anomalous results, which I discarded if found, and thus reduces the random error in the value for k which I obtain, as it will actually be the average value for k, as the decay constant.

In the future I would test a variety of springs with different spring constants, in order to decide on the spring which would allow me to gain the widest range of data possible, as my results were limited to an added mass of only g total, which in hindsight did not affect the spring much, and it was fairly stable when this amount of mass was attached, so I could have gone beyond g to perhaps 1 kg, in order to gain a wider set of data and thus a more precise value for the spring constant at larger loads. From here I added the mass in order to stretch the string and hence adjust the length accordingly in order to obtain the required length. This was a problem for me, since I required the posing to oscillate upwards and downwards in a straight line, as opposed to when resonating, the spring oscillates side to side due to the conversion of energy in the spring between kinetic and gravitational potential. Despite there being a slightly rotten end to one side of the ruler, I used this same ruler throughout the entire time in the laboratory, as this made my results more consistent and reliable than if I had changed to a different ruler part way through my research. This allowed easy alterations to the length of the string, which was necessary to gain the results I needed for this experiment. I decided to use this to measure the diameter as opposed to an ordinary measuring device such as a ruler, because the calliper gives a more precise reading for the diameter, and can be more easily adjusted to fit the tube being used.

Notice that there is no point at the 0,0 origin, since when there no mass attached to the end of the spring it pendulu not oscillate, so no time period can be recorded.

A zero famped is an example of a systematic error. The mass hanger was attached to one end of the spring, upon which the yellow 50 g masses were added one by one in order to see the effect of additional mass on the extension of the spring.

Simple and Damped Harmonic Motion

Next, I had to measure a first diameter of tube which I would use to gain some data. The general trend of the line fits the data extremely well with an R2 value of 0.

For forced oscillations also known as driven oscillations you cannot usually solve the position of the oscillator as a function of time dampd in steady state without using more advanced techniques with differential equations.

Prediction Since this experiment is an improvement on Experiment 2, the actual quantities being measured remained the same. For an exponential, this ratio should change with respect to the constant of the exponential function as shown for each curve.

Note the spring is out of shot here, but was attached through one the its ends, onto the clamp stand arm so that dampev coils were parallel to the ruler, in order to make observing the change in extension of the spring easier. To improve coufsework, I would use a taller clamp stand which would allow me to start at a greater height from the ground and hence allow the use of longer lengths of strings to further test my hypothesis.

Limitations and Evaluation Both these sub-experiments were very successful as they allowed me to test my hypothesis using knowledge of physics and practical methods, which were reliable and carefully controlled throughout in order to obtain the most accurate value for the spring constant.

An example of a systematic error would be air resistance, especially for experiments 1, 2 and 3. The experiments, with respective laboratory hours in brackets planned before experimenting began for effective time managementare as follows: Below are the graphs for each of the 6 diameters: This measurement was taken 3 times, then averaged in order to reduce any random error in the equilibrium position which may have been caused by the spring oscillating whilst taking the reading, which could cause an inaccuracy in my reading for the equilibrium position form the ruler.

Having found values for k using these two methods, I will calculate a theoretical k value, in order to compare which is the more accurate method in calculating a value for k which is closest to the true value.

Additionally it may due to the value for force, not taking into account the value of ams for the spring, so the force value at the origin is inaccurate and therefore the accurate value for force should be 0.

Damped Oscillation.

Interestingly, the negative percentage coursewoek for the last area of dampef used, shows the experimental value was actually higher than its theoretical value which may have been due to a systematic error in my observation of the amplitude or in the apparatus used. This shows the curve used may not be the best fit to the points after all. By squaring both sides of the equation for period, the following equation is obtained: An additional control variable suing this improved setup, is the position of card above the mass, using the wooden swing to sit the card on.

From knowing this, and with the data I had collected, I was able to graph period squared vs length to obtain an equation for my data, where the value of m gradient could be used to calculate damled value of accretion due to gravity, g, using my data.

Voursework decided to begin with a mm by mm card and then reduce this by mm by mm each time, with my final area being mm x mm, due to many errors found in the experiment which meant, I had to ind ways to reduce these errors and improve the experiment beforeI moving on. Although it was very useful because it allowed me to courswork and precisely obtain values for amplitude over a longer period of time than would be possible just using the naked eye, it involved a lot of preliminary work to make sure it was setup correctly and positioned in an ample position to easily see the markings on the protractor.

Damped Oscillation. – GCSE Science – Marked by

This is due to the fact that springs are not very efficient at storing and releasing energy. I decided upon using rectangular card, as I felt it would have a substantial effect on the dzmped of the pendulum — indeed it did, however the motion of the pendulum was no longer straight back and forth as before, it was now slightly elliptical which meant the period of the damped motion was slightly longer than the undamped case in experiment 1, whereas theory states that since the period is unaffected by the amplitude, any change in the amplitude should not affect the value for period of an oscillation.

Next, once the tube had been attached to the ruler as shown, I had to decide upon the mass to be attached to the spring. This is a very simple case. Therefore, I measured the spring on its own using the electronic balance, making sure to tear it To zero prior to weighing the spring to reduce any systematic error caused penxulum possible dirt or dust particulates present on the balance pan.

Simple and Damped Harmonic Motion – UBC Wiki

We can solve the damped harmonic oscillator equation by using techniques that you will learn if you take a differential equaitons course. Using this formula, with the value for k and m found, I calculated the period to: From there, the calliper gives digital reading of the diameter in millimetres, which I recorded in the table for each diameter being tested.

As discussed previously, this was most likely due to an angled rather than straight release of the mass. Using these values for force applied and the spring constant calculated from method 1 to be The time taken to make one complete oscillation is defined as the period, T. Despite the improvements and very credible data obtained, there were a still a few limitations which were noticeable in this experiment. This risk assessment spans all the experiment I conducted and takes into account any parts which could cause harm or injury to people whilst taking place.