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Physics Controlled Assessment – report
To what extent can we accurately measure the force of gravity using classroom equipment?

by Oliver McDonald

Table of contents * Introduction * History * Theory * Initial method * Improvements to methodology * Final results * Analysis

Our aim was to see how accurately we could measure the force of gravity using a number of methods to reduce the percentage error, getting it to the lowest percentage we could, using classroom equipment.
What is a pendulum?
A pendulum is something that hangs from a fixed point known as a pivot and, when it is pulled back then released, it swings by the force of gravity acting upon it which is given by the restoring force and the pendulum’s mass, which in turn causes it to oscillate about the equilibrium position. The weight that hangs from the pendulum is called a “bob”. Inertia describes the fact that something in motion will stay in motion and, thus, something that is at rest shall stay at rest. This applies to a pendulum and so by those means the pendulum continues to oscillate.
History
The following link shows how the Foucalt pendulum works - http://upload.wikimedia.org/wikipedia/commons/a/a1/Foucault_pendulum_animated.gif
Initial method
The time it takes for one oscillation to occur is known as the time period. As the pendulum swings, the time period of an oscillation is given by the equation:
T=2π√Lg.
So if the time period (T) is known and the length (L) is known, we get an eventual equation that we will use to calculate the force of gravity. This equation is: g=4π²T² Where did we derive this equation from?
There are a number of steps in which this equation is derived from which include the rearrangement of two equations and then the combination of these equations to come out with the equation above.
Step 1:
F=ma
-gsinθ=a
Step 2: sinθ=xL -gxL=a

Step 3: a=ω²x Step 4: a=2πT Step 5: g=lw2 g=l4π²T²

* F = ma (Force = mass * acceleration) * a = -w²x (negative omega squared * x) * W = 2π/T (2 * pi/ time) * g = 4 π^2L/T^2 (4 * pi squared * length/ time squared) * g = w^2L (omega squared * length)
Here is a diagram that outlines how these variables are derived.

Therefore all that I need to change in this experiment is the length and the equipment used to measure the time period. Also the angle at which we drop the bob from the pivot is another variable that will be taken into account to maximise the accuracy of our results.

To get the best results we could and reduce any uncertainties as much as we could, we kept devising improvements to each method as we went along.
Our methods are outlined as follows:
Method 1:
Before using a pendulum, we wanted to see how well using a G-ball would be in calculating the force of gravity. We dropped the ball from different heights off the ground; 1m, 80cm, 60cm and 40cm. the uncertainty increased as the height decreased. We also took into account the uncertainty in the time recorded on the stop clock which depended on the person who was taking the time’s reaction time, which was 0.19s. As the time decreased due to the height decreasing, the uncertainty increased exponentially up to 82.6% at the lowest height. Using the value of time^2 and putting it into the equation used to get the value for gravity, we end up with percentage errors for the value of gravity of up to nearly 182% which, of course, is tremendously high and thus the results we obtained using the G-ball could not be used as the uncertainty is far too high.

Length (m) | Time (s) | Time (s) ^2 | Gravity(N) | ± | 0.0005 | ± | ± | 0.19 | | ± | ± | | ± | ± | 1 | 0.050% | 0.0005 | 0.49 | 38.8% | 0.240 | 78% | 0.186 | 164.4249 | 77.601% | 127.60 | 0.8 | 0.063% | 0.0005 | 0.37 | 51.4% | 0.137 | 103% | 0.141 | 230.6993 | 102.765% | 237.08 | 0.6 | 0.083% | 0.0005 | 0.23 | 82.6% | 0.053 | 165% | 0.087 | 447.7703 | 165.301% | 740.17 | 0.4 | 0.125% | 0.0005 | 0.21 | 90.5% | 0.044 | 181% | 0.080 | 358.0809 | 181.077% | 648.40 |

Method 2:
After seeing how much of a failure using the G-ball proved to be, we realised we needed to use a pendulum to have a more constant motion. We hung a string with a metal bob at the end from a clamp-stand and swung it back and forth once, measuring the time period…...

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