Measuring g using
a simple pendulum
The aim of this experiment was to find out a value for g (the
gravitational force of the earth at sea level) using a simple pendulum and a
video recording camera. For this experiment a value for g of 10.016N ± 0.5N was
obtained, which is within the range of the accepted value for g (9.81 N in
Edinburgh)1
Background
Physics:
Gravity exerts a force on all objects, this force is directly
proportional to the mass of the object. The forces is measured in Newtons (N)
or Acceleration (ms2). The constant of this force is the
acceleration due to gravity, or “g”. The value for g on Earth has been calculated
to be 9.81N and this value remains fairly constant for a few thousand feet
above the Earth’s surface.
One method that can be used to approximate this value of g is a
simple pendulum. A bob mass attached to the end of a string and pulled to the
side through a very small angle, the bob is let go and oscillates back and
forth. The motion produced is simple harmonic motion, where the mass oscillates
around the centre point with an acceleration proportional to its displacement
and always against the displacement vector.
To start2 we
will define the displacement as arc length “s”, figure 1 shows that the force
on the mass (m) is tangent to s and therefore equals mgsin?. The weight (mg)
has both components mgcos? along the string and mgsin? tangent to arc s. The
tension of the string removes the component mgcos? parallel to the string. This
results in a restoring force towards the equilibrium point at ? = 0.
The displacement (s) is directly proportional to ?. When ? is
expressed in radians, and is small, the arc length of a circle is proportional
to its radius (L in this case) by s = L? (when ? is roughly < 15° in radians) Figure 1(2)
So: ; for small angles (as sin? = and sin? ? for smaller angles, when ? is expressed in
radians);
The resulting force is then expressed as:
This
is in the form F = kx, in which the force constant is given with: and displacement as x =s, from figure 1. For
angles which are less than about 15º, the small angle approximation (sin? ? in radians) can be used, so the restoring
force is directly proportional to the displacement and the pendulum acts as a
simple harmonic oscillator.
With the equation, we can find the period of a pendulum when ? <
15º, giving the equation for the simple pendulum:
, therefore T =
Finally giving the formula: . Which means that the oscillation of a simple
pendulum (when ? is smaller than 15°) is only dependant on the length of the
string used and acceleration due to gravity, meaning g can be calculated using
a known length of string by working out the period of oscillation and using the
rearranged formula:
Procedure:
Apparatus: Meter length of string, clamp stands, video recording
camera, 10x 100g masses, meter stick, laptop with "track" software
Apparatus
Set up as shown:
Method:
The experiment was set up as shown on the previous page, the first
experiment was 500g of mass on 1 meter of string, and the mass was attached to
the string so the length between the strings pivot point on the clamp stand and
the centre of the mass was exactly 1 meter apart. A whiteboard labelled "500g
at 1m" was stuck next to the pendulum for easy referral when looking at the
video later. The video recording camera was started and the mass bob was pulled
up and let it go so it swung from side to side, the angle of this oscillation
was kept small as the small angle approximation is used in the calculation. The
pendulum made at least 30 complete oscillations before the video was stopped.
The experiment was then repeated for 1000g on the same 1m string, as the mass
shouldn't have had an effect on the experiment, and was allowed to make 30 full
oscillations, The whiteboard was updated to say "1kg at 1m" to reflect this.
Once this was completed the experiment was repeated for 0.88m, 0.80m, 0.70m and
0.61m (as it was not possible to achieve exact values for the string length
when cutting due to the string extended once the mass was added). Each length
was experimented on twice using 500g and 1kg. Once all of the recordings where
done the videos where loaded onto a laptop where they were loaded into a
software called "tracker" which allowed the frame the oscillation started and
the frame it ended to be obtained, giving the overall time for 30 oscillations
(As shown on the raw data chart).
Calculations:
Using the raw data from the experiments the average period for
each string length was achieved. This average was then squared for T2
which was then plotted against Length to give the graph L against T2
(Attached on back)
From the formula: (Achieved on the previous 2 pages)
g = 4?² x gradient of graph
So g = 4?² x 0.2537
= 10.016 N (to 5 s.f)
Uncertainties:
Scale Reading uncertainty in Length: The smallest division on the
meter stick was 1mm, so the scale reading uncertainty was ±0.0005, which gives
a percentage uncertainty of ±0.05%
Calibration uncertainty in T: Biggest difference in values for T
was 0.033s (between 1.595s and 1.562s) meaning that the calibration uncertainty
was 0.033/2 = ±0.0165s, (0.0165/0.61) * 100 = ±2.7%
Absolute Uncertainty in g: as T is squared we need to multiply the
uncertainty in T by 2
So the uncertainty in g is which gave us ±5.4%
So: ±5% in g (5/100 * 10.016 = ±0.5008 N so ±0.5N for g)
The graph of L against T² does not cut the xaxis at 0, this shows
us that L is linearly proportional to T², but not directly proportional. This
is most likely due to a systematic error, which the cause of is unknown, but
can be assumed to be errors in measurement of the length of string or other
factors.
Conclusion:
A value for g of 10.1016 ±0.5N was obtained, which was well within
the range of the accepted value (Scotland is around 9.81N on average).
Evaluation:
The experiment could be improved with the use of a nonstretch
string, as the string's length was not constant when different masses were
added, this would have meant that the length values were not as accurate as
they could have been. Another improvement would be to swing the bob mass
through a smaller angle ?, as the small angle approximation gets more accurate
as ? decreases, a small angle of oscillation would also help remove small
factors like air resistance which may have had an effect on the results of the
experiment.
Resources:

http://www.psmsl.org/train_and_info/training/gloss/gb/gb3/abgrav.html Accessed 18/1/18

https://courses.lumenlearning.com/sunyphysics/chapter/164thesimplependulum Accessed
16/1/18
1 http://www.psmsl.org/train_and_info/training/gloss/gb/gb3/abgrav.html
Accessed 18/1/18
2 https://courses.lumenlearning.com/sunyphysics/chapter/164thesimplependulum Accessed 16/1/18