Determining the Acceleration Due to Gravity
Determining the Acceleration Due to Gravity (g)
Apparatus:
* Clamp and Stand
* Light gate
* Power supply
* Data logging terminal
* Computer
* Pendulum bob
* String
* 2x small flat pieces of wood
* Metre rule
* Vernier Scales
* Pen
* Card
* Weights
* Scissors
* Blue tack
Theory:
Gravity is a force of attraction between all objects. This attraction is proportional to the mass of the objects and inversely proportion to the distance between their centres. For a notable force the mass must be large. The earth has a great enough mass to show a measurable force of gravity. This is the force that attracts everything towards the centre of the earth. The force of attraction is greater for each mass, using this equation:
F=mg
Where g is the force of gravity. Force is related to acceleration and mass using this equation:
F=ma
Simultaneously equating these equations gives us:
mg=ma
g=a
Therefore g in newtons is the same value as the acceleration due to gravity in metres per second.
The simplest way to measure g is to drop a mass and measure it's acceleration. We can find acceleration, by finding the change in velocity over a certain period:
a=(v-u)/t
Where velocity is measured over a distance of 5cm, this will be the average velocity. I will consider this velocity to occur in the middle of the 5cm distance. As the object is accelerating, this is unlikely to be the place of average velocity, it is more likely to be after this but it is extremely difficult to determine the exact location. I think this method would be less accurate as it is hard to measure acceleration accurately using just one light gate. This is because the average speeds are used in the equation and not the exact. As the formula we use is change in velocity divided by the time taken for this change. Also this would be affected by air resistance. This will reduce acceleration and give a lower value for g. The air resistance increases as speed increases and so is proportional to the speed. This would mean that the object must be moving slowly to reduce the effects of air resistance.
Another method would be to use a pendulum. Pendulums perform simple harmonic motion (SHM). This is when a mass oscillates freely about a fixed central position. Once it has started to swing there is a restoring force to bring it back to it's equilibrium this restoring force is proportional to the distance from equilibrium, i.e. at the height of it's swing it has the greatest restoring force, and at it's equilibrium position is has no restoring force. Once the mass has reached equilibrium it is travelling at it's greatest speed and so has the greatest momentum, the restoring force is at a minimum so the pendulum will continue past the equilibrium, and so the pendulum will continue to oscillate like this. All energy in a pendulum system is conserved, either in the form of potential energy or kinetic energy. At its highest point (furthest away from equilibrium) it has no kinetic energy, but full potential energy. This potential energy is equal to the energy put into the system (the energy used to start the swing). At the equilibrium position all the energy is kinetic and so the mass is moving at the greatest velocity. The energy is therefore never lost.
energy in an shm system
*from Physics 2 Cambridge
This is for an undampened system, but our system will be slightly damped by air resistance, we are using a small bob and a thin string so this resistance will be negligible. We will not try to measure the air resistance as in theory during SHM, damping will only affect the amplitude and not the frequency.
The forces acting on the pendulum:
A bob A of mass m hangs from a fixed point B with a string of length l. The string makes an angle of ? with the normal. The pendulum oscillates about the equilibrium O. The path of the bob A creates an arc of length x
The forces on A are it's weight mg and tension P. Resolving the weight of A we get a component acting on the tangent to the arc mgsin? this is the restoring force. We also a component acting along the normal to the ...
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The forces acting on the pendulum:
A bob A of mass m hangs from a fixed point B with a string of length l. The string makes an angle of ? with the normal. The pendulum oscillates about the equilibrium O. The path of the bob A creates an arc of length x
The forces on A are it's weight mg and tension P. Resolving the weight of A we get a component acting on the tangent to the arc mgsin? this is the restoring force. We also a component acting along the normal to the arc of mgcos? this force balances the tension P. We know these forces are equal and opposite as the bob A never deviates from the arc. Therefore mgsin? is the angular acceleration:
Therefore by varying the length and recording the period we can determine the acceleration due to gravity (g).
I will plot 1/4?2l on the x axis and 1/T2 on the y. As l increases I would expect T2 to increase proportionally. The result should be a straight line though the origin. The line of best fit should have a gradient equal to that of g. I would expect this to be close to 9.81ms-1 (the known value of g)
Method 1 (using a falling mass)
Variables:
The independent variables will be the weights on the card to see what effects air resistance has on g. The dependent variable will then be the terminal velocity meaning a higher velocity for a higher weight.
The control variable will be the height that the card is dropped from and the card itself must remain the same card.
To measure the acceleration of the mass using one light gate we must find two velocities and the time between them, so we will need to make a special card to enable us to do this. The card has two windows in like so:
The light gate should be set on the stand so that the card can just fall through the light gate i.e. just over the height of the light gate. This stops the card being damaged. Attach the weight to the bottom of the card using blue tack, do not use much blue tack as this will add to air resistance which will act against gravity and so give a lower value of g. A thin weight is used so that the surface area, perpendicular to the plane of movement is lower. This is more streamlined and so reduces air resistance giving a more accurate value for g.
Plug the light gate into the data logging terminal. Set the power supply to the correct voltage of the bulb used in the light gate (3volts). The lights in the room must be turned off for a maximum contrast to the light gate; this will ensure a better graph and greater accuracy. Set the data logging terminal to start recording when a change in light intensity occurs (this is when the card first cuts the light beam. Set the time for as quick as possible, as the card will be moving through very quickly, this would be the most accurate setting. There are different settings for this, using the in built LCD do a few test experiments to make sure you get the whole falling of the card in the time, the graph should look like this:
Once you are happy with the settings you can now just drop the card through the light gate. You must make sure the 'windows' pass through the light beam, you can check this has happened by again examining the graph. The card must be dropped just above the light gate, so it is travelling as slow as possible when entering the light gate, this will help avoid air resistance and so give a better result. Once you have done the experiment a few times you can download the data to the data logging program on the computer and analyse it.
Method 2 (using a pendulum):
Variables:
The independent variable is to be the length of the pendulum. This shall be changed by adjusting the length of the string. This in turn will affect the dependent variable, the period of the system. The length of the pendulum is proportional to the period of the system. The greater the length of the pendulum the longer it will take to complete one oscillation and so the period T will also increase. The same string must be used for each length as a more extendable string may produce different results from a less extendable string, as the length may vary due to speed changes as tension increases, effectively pulling the string longer, thus altering the results.
First set up the apparatus as shown in the diagram, making sure the clamps are secure on the stand and the stand is stable. The two pieces of wood that go in the clamp and grip the string must have a flat bottom, so that the length of the string is always constant. The pendulum bob must be measured using a vernier scale measure this to the nearest millimetre. Measure the length of the string using the metre rule, again to the nearest millimetre. At first this was done whilst the string was clamped. The metre rule couldn't be next to the string and so was at a slight angle giving a reading fractionally too high, so it was decided to measure the string unclamped and mark it off where we should clamp it, this proved easier and more accurate. The length of the pendulum l is the distance between the fixed point B (the bottom of the two flat pieces of wood) and the centre of mass of the bob A. So this distance is the length of the string used, from the mark to the top of the bob and then half the height of the bob. Next the light gate must be set up. This must be positioned so that the pendulum bob, when at rest blocks the light in the light gate. This is at the equilibrium position when the pendulum has the greatest kenetic energy and so is travelling at it's greatest speed. This should produce easier to analyse results. Then plug it in to the data logging terminal. Set the power supply to the correct voltage of the bulb used in the light gate (3volts). The lights in the room must be turned off for a maximum contrast to the light gate; this will ensure a better graph and greater accuracy. When starting the pendulum swinging simply pull it back at right angles to the light gate to ensure it will not knock it during it's oscillations. Do not pull the string back so that it makes an angle greater than ten degrees from the normal. The data logging terminal should be set to a fixed time and should be able to fit a full oscillation in. This is where the bob starts at the midpoint, comes back through the midpoint then ends at the midpoint, so the graph should have at least 3 troughs where the light intensity has dropped significantly. The time should not be set to great, as this would make the readings less accurate. You can check the graphs on the screen of the data logger to see if it is recording correctly. The graph should look like this:
This graph will not be accurate due to the bad resolution of the LCD screen, but can be used to check that the experiment is working properly. Each length of pendulum should be done twice to ensure that they are accurate and repeatable values, if the two results from each measurement don't correspond it may be necessary to do this length again. The lengths of the pendulum should be changed at least 6 times to give enough points to make a good line of best fit on the final graph. There should be a large range in the lengths, but not getting too short as this would make the experiment less accurate. The shorter the pendulum gets the faster it will complete one period and so there this time will be very short, and there would be a high percentage error if this isn't totally accurate. The range of lengths was from 0.6095m down to 0.1095m. We couldn't go any higher as the stand would not allow. Once all the measurements are done the data logger can be connected to the computer via the serial port and then download the data into the data logging program. This will make a table of the data and plot a graph. It also has a very accurate measuring tool. So the period times can be measured on the computer. The graphs can be checked quickly using the formula g=(4?2l)/T2) checking that g is close to 10, if it is far from this value, the experiment for that particular length will need to be done again.
Results:
Method 1 (using a falling mass)
Once the data has been transferred to the data logging program the graphs produced should look like this:
All the measurements must be taken when light intensity is halved, this is when the card is half way across the light gate, and so we are measuring the exact time from when the card, changes to a window, or vice versa. To measure the acceleration we need the change in velocity/time taken:
m
t1
t2
t3
v1
v3
A
kg
s
s
s
Ms-1
Ms-1
Ms-1
+/-0.001
+/-0.001
+/-0.001
+/-0.002
+/-0.002
+/-0.081
0.1
0.0442
0.0288
0.0288
.13
.74
9.26
0.1
0.0441
0.0287
0.0278
.13
.80
0.28
0.2
0.0437
0.0284
0.0278
.14
.80
0.20
0.2
0.0433
0.0281
0.0277
.15
.81
0.23
0.3
0.0432
0.0280
0.0274
.16
.82
0.54
0.3
0.0411
0.0278
0.0274
.22
.82
9.80
0.4
0.0409
0.0277
0.0274
.22
.82
9.74
0.4
0.0408
0.0274
0.0273
.23
.83
9.86
0.5
0.0398
0.0270
0.0268
.26
.87
0.11
0.5
0.0389
0.0268
0.0267
.29
.87
9.85
If I take an average from the table I find g to be 9.99ms-1 +/-0.73. The given value for g is 9.81 giving me a 1.8% error.
Method 2:
The period is measured between points A and B. The initial idea was to measure the time from the bottom of the trough but they are not symmetrical, so it was decided to measure them at the top of the trough, i.e. when the bob has just passed the light gate. We can be sure that this is the correct measurement as it is measured from the same place each time, i.e. when the light intensity just reaches full. The y value in the separation co-ordinates shows zero so we know the two points are at the same level.
After measuring the graphs a table can be created:
l
m
±0.001
T
s
±0.001
Mean T
S
4?2l
m
+/-0.04
T2
S2
+/-0.001
/4?2l
m-1
+/-0.01
/T2
s-2
+/- 0.003
0.091
0.625
0.625
3.591
0.391
0.279
2.560
0.091
0.625
0.190
0.844
0.852
7.513
0.725
0.133
.379
0.190
0.859
0.277
.047
.047
0.918
.096
0.092
0.912
0.277
.047
0.363
.203
.203
4.331
.447
0.070
0.691
0.363
.203
0.440
.328
.321
7.351
.744
0.058
0.573
0.440
.313
0.548
.453
.446
21.614
2.089
0.046
0.479
0.548
.438
+/- 0.35
From graph:
Max: 3.00/0.298= 10.1ms-2
Middle: 2.98/0.300= 9.93ms-2 Mean = 9.97ms-2
Min: 2.96/0.300= 9.87ms-2
I have found the value of g to be 9.83ms-2 using the line of best fit and 9.97ms-2 using the mean of all three values. The exact value of g is 9.81 therefore I have an average percentage error of 1.4%.
Discussion:
Using method one I found g to be 9.99 this is higher than g and I would have thought that may result would be lower due to air resistance. Therefore I think there was a fault in the experiment. For every weight used the value tends to be higher than g, this would suggest a systematic error, either in the timer of the light gate, or maybe the card was not measured accurately enough. If the card didn't travel straight, i.e. went through the light gate slightly tilted would have made the gaps smaller on the vertical plane, so this may have happened. Next time a better window device should be used maybe one made from a heavy metal, that need not have weights attached. This should fall straighter, and also will not get damaged through falls. It could also be measured easily with dividers without being damaged, unlike the soft card. The best thing to improve accuracy would be to have two light gates, this would be a lot easier to measure acceleration. I was going to investigate the effects of mass on acceleration, but the results were too inconsistant to show any correlation. Method two produced more conclusive results. The graph of 1/T2 against 1/4?2l gives a straight line going through the origin, showing that l is proportional to T2 as I stated in my hypothesis. From the equation g=(4?2l)/T2) g must equal the gradient of this graph. I found the gradient of my graph to be 9.97 this is close enough to the g (9.81) to show that this assumption is correct. Because the two values for T of each length of pendulum are similar this shows that the results are quite reliable, in some cases they are exactly the same, which is a very good sign of the reliability of the experiment. My value for g tends to be a bit high throughout my results. I think this could either be down to the data logger not being calibrated perfect, or to bad measurement of the string. As the first experiments results were also slightly high I would be tempter to think that the data logger may not be calibrated exactly. This could be proved by doing the experiment again using a different data logger. When measuring the string on the stand it was difficult to get the ruler in line with the string due to obstructions. So the value when clamped in position was slightly higher than that when it was laid out on the ruler. Though when looking at the string used to see if anything else might have made the results inaccurate, the string appeared to be slightly elastic. Maybe when straightening it on the ruler, the string was stretched a little and so again the measurement would be higher than that of when it was on the clamp. Next time a more inextendable string must be used, maybe some sort of wire. There is one value off the line of best fit when l=0.548, this is most likely an anomaly as this error only occurred once. The most probably cause for this anomaly is when the angle the string was pulled back is greater than 5 degrees to the vertical, as the formula is only intended for small angles. Air resistance will have affected the results, this should however have slowed the acceleration down, but the results are a fraction too high. Although the value for g was a too high and there was one anomaly I think the percentage error of just 1.4% shows that these results are quite reliable and this is a good method to conduct an experiment to find the value of g.
Conclusion
The time gate offered a value of g of 9.99 +/-0.73 of g. The results weren't very reliable so I would say that the second method using the pendulum would be a more reliable experiment. From my graph I have found the value of g to be 9.97ms-2, this is only 1.4% percentage error, if compared to the real value of g, 9.81 ms-2. I think the experiment went well and I have found a close reliable value for g.
Peter Legg Determining g 11/04/2007
