By experimentation I aim to obtain an accurate value for the gravitational field strength, g (N kg-1), of the Earth. I will carry out an experiment using a Bifilar Pendulum.
The Bifilar Pendulum
I will set a metal rod oscillating by giving it a small angular displacement,q(rads), about the vertical axis when it is fastened onto two strings that are held by corks firmly clamped onto stands. The motion of the Bifilar Pendulum is Simple Harmonic Motion, I will prove this later in the report.
The metal rod is uniform and of mass M (kg) and length L (m). The two strings are of length y (m). I will make sure that both strings are parallel to each other, always perpendicular to the rotating rod and the same distance from the rod’s centre of mass (l/2 d metres). The inclination of the strings to the vertical will be given by a(rads).
I will vary the separation of the two strings, d (m), from d » 0.1m to d » 0.7m and record the time for twenty oscillations of the rod, 20T (s), from this value I will calculate the periodic time, T (s), ie the time taken for one oscillation.
The diagram below shows the setup of the experiment:
I will not model the oscillating rod as a particle, instead I will regard it as a system of connected particles moving in circles of different radii. The same torque applied to different bodies produces diferent angular accelerations, indicating that each body has an individual amount of Rotational Inertia that controls the degree of change in rotation. The measure of a body’s Rotational Inertia is called Moment of Inertia. The more difficult it is to change the angular velocity of a body rotating about a particular axis the greater is its Moment of Inertia about that axis. The Moment of Inertia of a body is a function of the mass of the body, the distribution of that mass, that is its size and shape, and the position of the axis of rotation. Therefore these factors must affect the rotational motion of the metal rod.
Values of Moment of Inertia for rigid bodies can be calculated using calculus. The Moment of Inertia about an axis perpendicular through the centre of the metal rod can be found by:
Mass per unit length = m
Mass of elemental length = mdx
| let I = Moment of Inertia |
| I = å mr2 (for all the particles of the rod) |
| \ I = åmx2dx |
As d x -> 0 Moment of Inertia becomes:
+1/2 L +1/2 L ò-1/2 Lmx2dx = m[1/3 x3]-1/2 L |
| \ I = 1/3m{[1/8 L3 ] - [- 1/8 L3 ]} |
| I = 1/12 mL3 |
| M = mL |
| \ I = 1/12 ML2 |
I will now prove that the motion of the Bifilar Pendulum is in fact Simple Harmonic Motion.
Vertically the metal rod is in equilibrium so the Tensions of the strings that hold the rod are given by:
| T = 1/2 Mg |
So the Tensions of the strings when they are inclined to the vertical by a (rads), are given by:
| T = 1/2 MgCosa |
So the restoring forces, TR, which move the strings back to their original positions are given by:
| TR = 1/2 MgSina |
The angle a (rads) is very small so the restoring forces become:
| TR = 1/2 Mga |
Since both the angle of inclination to the vertical of the strings, a (rads), and the angular displacement of the metal rod, q (rads), are very small and then:
| ya = 1/2qd |
| \ a = qd 2y |
\ TR = Mgqd |
Therefore the restoring Couple, CR, which acts towards the equilibrium position so is negative, is given by:
| CR = -Mgqd2 4y |
Applying Newton’s Second Law for the rotational motion of the rod, which is of constant mass:
Id2q= -Mgqd2 |
| \ d2q= -Mgqd2 dt2 4Iy |
| a = -w2s |
| \ w2 = Mgd2 4Iy |
So because the motion of the metal rod is Simple Harmonic Motion its Time Period, T (s), is given by:
| T = 2p w |
| \ T = 2pÖ4Iy ÖMgd2 |
| \ I = MgT2d2 16p2y |
The Moment of Inertia of the metal rod, as I have previously proven, can also be given by the following equation:
| I = 1/12 ML2 |
| \ ML2 = MgT2d2 12 16p2y |
T2 = 4p2L2y´ 1 |
| T µ1 d |
| let k2 = 4p2L2y 3g |
| \ T = k ´1 d |
This equation is in the form of y = mx + c, so if I plot T (s) against d-1 (m-1) I will obtain a straight line graph through the origin, whose gradient will be found by:
| Gradient = dy dx |
| \ Gradient = T = Td d-1 |
\ Gradient = k |
So when I have a value for k (ms) I will use it to find the gravitational field strength, g (N kg-1), of the Earth as shown below:
g = 4p2L2
3k2
Results
The graph below shows the results for the Bifilar Pendulum experiment:
Now I am able to calculate a value for gravity using the above graph:
Gradient = 3.5 = 0.53 ms
6.6
g = 4p2L2y
3k2
k2 = 0.532 m2s2
L2 = 0.6622 m2
y = 0.478 m
\ g = 9.80 N kg-1
Conclusion
By experimentation I have obtained an accurate value for the gravitational field strength, g (N kg-1), of the Earth. The value of gravity that I have obtained from the Bifilar Pendulum experiment is 9.80 N kg-1. The value of gravity that I have obtained from the light gates experiment is 9.77 N kg-1.
Taking into consideration England's latitude and longitude, our distance from the Earth’s centre of mass and the effects of the rotation of the Earth the approximate value of gravity in England is 9.8143 N kg-1.
So the percentage error in the Bifilar Pendulum experiment is:
Percentage Error = 9.81 - 9.80 x 100%
9.81
T = 2pÖ4Iy
ÖMgd2
If this occurred it would have definitely caused a significant error in the results for this experiment because from the above formulae the gravitational field strength, g (N kg-1), of the Earth was calculated.
The lengths of the two strings may not have been the same, or they may not have been the same distance from the centre of mass of the metal rod (1/2 d metres), so the rod may have been slightly slanted. Therefore the position of the axis of rotation of the rod was not perpendicular through the centre of the rod, this means that its Moment of Inertia is not exactly given by the formula shown below:
I = 1/12 ML2
This may have caused the slight error in the result of this experiment.
If the metal rod that was used was not a uniform rod (of uniform density) then its centre of mass may not have been at the axis of rotation, which was perpendicular through the centre of the rod. Therefore the Moment of Inertia may not exactly be given by the formula shown above. This may have caused the slight error in the result of this experiment.
The error in this experiment may have been caused by the incorrect measurement of the length of the metal rod, L (m), the string separation, d (m), the string length, y (m), or of the time for twenty oscillations, 20T (s). A faulty stopwatch or human error may have caused this.