The two forces are not in opposite directions. This means there is a resultant force on the bob to the right.
If the bob is moving to the right the force makes it accelerate and the speed increases as it moves towards the centre.
If the bob is moving to the left the force decelerates it and it slows down.
The diagram shows the arcs through which two pendulums swing. The red one is twice the length of the blue one. Notice that the blue arc is always at a steeper angle than the red arc, and always above it.
The blue pendulum has the most gravitational potential energy at the top of the swing because it is higher. This means the kinetic energy and hence speed through the centre will also be greater than for the red pendulum.
From previous experiments I know that for trolleys running freely down a ramp that the bigger the angle of the ramp the bigger the acceleration of the trolley. This same principle can be applied to the falling pendulums. The steeper the arc the bigger the acceleration of the pendulum will be. A bigger acceleration means a shorter time for each swing. Unlike a ramp the arc of swing is not a straight line. The arc has the steepest gradient at the top and is flat when it reaches the middle. The acceleration of the bob will thus decrease from a maximum at the top of the swing to zero at the centre.
For these reasons as the string gets longer the time per swing will get longer.
The cotton is clamped between two small blocks of wood. This ensures that the cotton swings from a single fixed point. A small lump of plasticine is attached to the end of the cotton and the length is adjusted by pulling the cotton through the two blocks. Gravity may be considered to act through the centre of gravity of the bob. For this reason the length of the cotton is measured from the wooden blocks to the centre of the bob.
Timings for twenty complete swings are started and stopped as the pendulum passes through the mid-point. A long pin is set up at the mid-point, at right angles to the plane of swing, to provide an accurate reference point. This is achieved by positioning yourself so that you are looking directly along the line of the pin. As the cotton passes the point the stopwatch is started and counting is started at "0". The pendulum will swing to one side, then back through the centre and to the other side. When it passes the centre again "1" is counted for the first complete swing. This is repeated and on the last swing you must again look along the pin and the timer is stopped as the cotton passes.
Each timing is repeated three times. It is important to check that the pendulum is swinging in a single plane before measurements are started. The size of swing should also be kept small.
cotton, plasticine, metre rulers, digital stopwatch, long pin
There are no safety risks involved.
1. Varying mass of bob
The mass was changed by a factor of five and this had little effect on the time.
2. Varying displacement of swing
The size of the swing was changed by a factor of three and this had little effect on the time.
3. Varying length
The length was changed by a factor of two. The time increased as the length increased but by a factor of 1.4 approximately.
From the trial data it is easily seen that the length of the pendulum is the only variable that has a significant effect on the time of swing.
mass of bob = 10g
size of each swing kept small (maximum displacement about 10cm)
Changes to the plan
Extra results were taken at lengths of 5cm and 10cm. This was done to help extend the range over which the trend in the data could be tested. The number of swings timed was increased from 20 to 50 for these two lengths. This prevented a possible drop in the accuracy if the total recorded time was to become too short.
3. Analysis and Conclusions
Graph of results
(Remember, in a proper writeup the graphs would be printed out A4 sized, not this small)
To show how the time per swing varies with the length
Graph1 shows that the time for each swing increases as the length increases.
The gradient of the graph decreases as the length increases. This shows that the rate of increase of time per swing decreases as the length increases.
To show how the time per swing varies with the square root of the length
Graph2 shows the time for each swing plotted against the square-root of the length. This gives a straight line graph through the origin.
The equation for a straight line through the origin is;
y = mx
The gradient m measured from the graph = 2.5÷12.5 = 0.20
If T is the time for one swing in seconds, and L is the length in centimetres, the equation for the line can be written as;
T = 0.20√L
The time for one complete swing is proportional to the square root of the length. All the points for Graph2 lie on a straight line so the conclusion is very reliable over this range. It seems likely that the same trend would continue if the string was made longer. Shorter lengths look like they would also follow the same pattern although it gets more difficult to take the measurements as the time gets shorter. For very short lengths the trend may not continue and would be very difficult to measure. It is not possible to try lengths shorter than the diameter of the bob, for instance. A much smaller bob on a very fine and lightweight filament could be made and tested.
I was able to predict that the time would increase but can not prove that it is proportional to the square root of the length.
Accuracy of Results
Measuring the length
One difficulty when measuring the length is deciding where the centre of the bob is. The uncertainty in determining this measurement is probably about 1 mm. Measuring the length beyond about 1 metre is more difficult than short lengths because the measurement has to be done in two parts using metre rulers. The total error in measuring the longest length (160 cm) could be 2 or possibly 3 millimetres. For the shortest length the error is not likely to be more than a millimetre.
Adjusting the length of the pendulum to an 'exact' value such as 40.0mm was not as difficult as I thought it would be. The cotton could quite easily be slowly pulled through the wooden blocks a little bit at a time until the measured length was correct. Had this not been possible, or to save a little bit of time, any value close to the required value would have been perfectly acceptable and would not have made any difference to the conclusions.
Measuring the time
The stopwatch measures to one hundredth of a second although the overall accuracy of the time measurements are not that good. The human reaction time to start and stop the watch roughly cancel each other out as the same event is being observed, and reacted to in the same way, each time. Errors are produced by any variability in the reaction time of the individual which could be affected by many things. The following table gives the difference between the maximum and minimum values of time for each length.
The maximum range is 0.13 seconds and the average is .07 seconds. This shows that the timings are repeatable to within a couple of tenths of a second. Taking more time measurements may give a slightly more accurate average for each length, but not by much.
The smooth trend in the graph indicates that the results are accurate and reliable. There are no anomalous results or anomalies to be seen in the trend of the graph.
No significant problems or difficulties were encountered when carrying out this investigation. The accuracy and reliability of the results and conclusions are very good. Within the accuracy of the method used, and for the range of values investigated, it is clear that the time for a complete swing of the pendulum is proportional to the square root of the length.
The procedure used was simple and straightforward and no difficulties were encountered. A small improvement could be made to measuring the length of the pendulum. A longer rule, or piece of wood, could be placed level with the point of suspension, and a set square could be placed along the flat side and just touching the bottom of the pendulum. This distance could then be measured more accurately than trying to guess where the middle of the bob is. The diameter of the bob could be accurately measured with some vernier callipers so that the true length of the pendulum could then be calculated.
The thread used was quite stretchy. If the investigation was repeated I would replace it with something more rigid, such as extra strong button thread.
More repeats could be taken but I don't think this would add much to the accuracy of the conclusions.
Here's what a star student thought of this essay
Quality of writing
His table is clear and plotting his average results on a graph, his results are clearly shown. He plots his independent variable on the y axis and depenent on the x axis as you are meant to. There are very few spelling or grammatical errors.
Level of analysis
The student shows his method well and tries to reduce errors by thinking of how to do his experiment while reducing errors. Which I would recommend always doing as it is best to know what you are doing before you do it, you don't want to be confused as to what to do when you get to the experiment. He takes an appropriate number of repeats making his experiment more reliable. He makes sure to control all variables other than his dependent and independent variables as meant. You must be careful to do this as if you don't you could get incorrect results.
Response to question
The student answers the question well, he at first explains the theory then makes predictions based on theory. I would recommend this as you get your marks for explaining why something will happen, not just stating it will. This student explains why using theory very well. He makes a table which shows that as length increases so does the time period, and a graph. He also goes further and plots a graph of the time period against the square root of length, to prove that the theory is correct and that T is proportional to the square root of length.