Only test masses up to 1kg, as otherwise the spring’s elastic limit could be broken, spoiling the spring.
Hypothesis:
I predict that the time period of the mass-spring system will increase as the mass on the spring increases. Take two masses as an example – a large mass and a small mass. It requires more energy to move the large mass at the same speed as the small mass. However, if the same amount of energy is used to move the larger mass as the smaller mass, then the larger mass will move slower than the smaller mass. Similarly in this experiment, the spring only has the same amount of torsion energy to accelerate the weights each time, so the time period for one oscillation shall increase as the mass on the spring increases.
Previous to this experiment and on a similar level, we performed another oscillation experiment with a length of string and a weight on the end. In this experiment, we clamped a string between two pieces of wood onto a stand. We started with 10 cm of string between the point at which it met the ball mass and the point where it was gripped by the 2 pieces of wood in the clamp. The weight was pulled to one side, and allowed to start oscillating. We started timing when the mass reached one side. We timed 10 oscillations, and repeated it 3 times. Then we increased the length of string by 5 centimetres, and repeated the experiment 3 times. We repeated this up to 50cm.
Each time we increased the length of the string, the time period of one oscillation increased, and because the distance needed to be travelled is proportional to the amount of energy had. This relates to our investigation – the amount of torsion energy is proportional to how far the mass has to travel, so when the mass is increased, the oscillations must be slower.
Results
Analysis
The two graphs above show how the time period of the mass-spring system is increasing as the mass on the spring increases. This proves my hypothesis which states that the time period of the oscillating system will increase as the mass increases. I have noticed that the line on the graphs are curved, showing that the relation between mass and time period is not directly proportional. They seem to be levelling out towards the end showing that the time period for larger masses is more similar.
The graph above clearly shows how the difference between the time periods of the oscillating system decreases as more weight is added on. This could be because the elastic energy contained inside the spring increases as more weight is put on it. This means the total weight to be moved is done so relatively faster, so the time period is more and more similar as more weight is added as the spring is more taught.
When studying the graphs at the beginning of this section, I noticed that the time periods were increasing at first, but then slowing towards the mass limit. This is typical of a graph such as y=√x, so in attempt to give a graph showing direct proportion, I squared the time periods, and came up with the above graph. It shows that the results of the time periods squared all lie extremely close to the line of best fit, showing that there is a direct link between mass and time period in this graph of y=√x.
We have learnt that the elastic energy in a spring increases as it becomes more taught. We know this because the difference in the time periods decreased as we increased the weight, showing that the energy delivered to make the weights oscillate must increase.
My hypothesis that the time periods would increase with more mass was proven correct by my first two graphs, but I did not predict that the time period difference and mass on the spring would not be proportional.
The website states that the following equation is the general formula for the frequency of simple harmonic oscillation –
“The frequency of oscillation, F0, is the number of back-and-forth movements made per second. Since 1/2π0 is a constant, and since the square roots of large numbers are larger than the square roots of smaller numbers, we can see that increasing the stiffness k will increase frequency, while increasing the mass m will decrease frequency.”
It also states that frequency of oscillation is equal to the inverse of time period –
“Frequency is the inverse of period. High-frequency oscillations have short periods, and vice versa.”
Assuming these two equations are correct, we can deduce that the following equation is the formula for the time period of a mass spring oscillating system:
Therefore:
Evaluation
The experiment went well in my opinion, as I obtained a full set of results that were accurate and that I was able to make good use of. It surprised me how similar the three times I obtained for each mass were – on occasion being exactly the same as the last result down to 100th of a second. But although they were very similar to each other, they would probably not be so similar to the actual time periods of the mass-spring system, due to delayed reaction times. Also the results might have been affected if the masses we used were not exactly 100g masses, and sometimes the bench on which I was working on was jogged, which may have slightly disrupted the experiment.
However, all in all, the experiment went well considering the limitations of the apparatus we had. I could possibly have narrowed down the gaps between masses by using a 50g mass to obtain a larger set of results (150g, 200g, 250g etc.) but I believe using 100g pieces gives sufficient results to show how the time period of a mass-spring system changes as mass increases. A better timing system would, however, improve my results to make them more accurate at showing the time period. I could use a laser tripwire system linked to a computer, so when I wanted to start timing, I would start the oscillating system going, then when the wire was tripped for the first time, it would start the timer, and when it was tripped again it would stop the stopwatch. That way, I could accurately find out the time period for a mass-spring system.
To extend this work I could investigate the effect of the stiffness of a spring on the time period, using the same mass each time for different stiffnesses of spring but performing the experiment in the same way.
Bibliography:
