PHY400

( EXPERIMENT 2 )

ɡ from simple pendulum

NAME: SYAFAWANI BINTI CAROL

STUDENT ID: 2019672332

GROUP: AS2533D

LECTURER NAME: DR. ZAKIAH BINTI MOHAMED

OBJECTIVE

 To determine the acceleration due to gravity, g, by means of a simple

pendulum

THEORY

A simple pendulum consists of a bob suspended by a light (massless) string of

length, ‘L’ fixed at its upper end. In an ideal case, when a mass is pulled back and
release, the mass swings through its equilibrium point to a point equal in height to
the release point, and back to the original release point over the same path. The
force that keeps the pendulum bob constantly moving towards its equilibrium position
is the force of gravity acting on the bob. The period, ‘T’, of an object in simple
harmonic motion is defined as the time for one complete cycle. To determine g of the
measured quantities based on equation:

𝐿
T = 2π √
𝑔

Where, T and L are the period and length of the pendulum.

APPARATUS

 Pendulum bob
 String
 Meter rule
 Stopwatch
 Clamp
 Retort stand

PROCEDURE

1. The pendulum bob was attached to the string.

2. The length, L, was set to 1m initially, and then tied to the pendulum

3. The pendulum bob was displaced slightly to the side and then released. The
time, t, was noted for 5 oscillations using a stopwatch

4. The steps 1-3 were repeated to obtain 8 more sets of readings each time with
length, L, decreased by 0.10 m

5. The data have been recorded in appropriate table

6. g of the measured quantities have been determined from an appropriate

𝐿
graph based on the equation , T = 2π √ , where T and L are the period
𝑔
and length of the pendulum. The uncertainty of the g have been determined.
7. The percentage difference between the acceleration due to gravity determine
in this experiment (gexperiment) and the standard acceleration due to gravity at
sea level, gstandard = 9.81 m/s2 have been calculated.
DATA AND RESULT

Period (T) of a pendulum at different length (L)

Length L (m) Time, t, for 5 Period T (s) T2 (s2)

oscillations

90.0  102 9.51 1.90 3.61

80.0  102 8.82 1.76 3.10

70.0  102 8.52 1.70 2.89

60.0  102 7.82 1.56 2.43

49.5  102 7.10 1.42 2.02

40.0  102 6.42 1.28 1.64

30.0  102 5.60 1.12 1.25

20.0  102 4.61 9.22  101 8.50  101

10.0  102 3.37 6.74  101 4.54  101

𝑡
To calculate period, T is: T = where n is the number of oscillation.
𝑛
Period (T) of a pendulum at different length (L) and the gravity acceleration, g

Length L (m) Time, t, for 5 Period T (s) T2 (s2) Gravity

oscillations acceleration,
g, ( m/s2 )

90.0  102 9.51 1.90 3.61 9.84

80.0  102 8.82 1.76 3.10 10.20

70.0  102 8.52 1.70 2.89 9.56

60.0  102 7.82 1.56 2.43 9.73

50.0  102 7.10 1.42 2.02 9.79

40.0  102 6.42 1.28 1.64 9.64

30.0  102 5.60 1.12 1.25 9.44

20.0  102 4.61 9.22  101 8.50  101 9.29

10.0  102 3.37 6.74  101 4.54  101 8.70

𝐿
To calculate the g: T = 2π √𝑔
Period (T) of a pendulum at different length (L) and the gravity acceleration, g and
percentage difference

Length L (m) Time, t, for Period T T2 (s2) Gravity Percentage

5 (s) accelerati difference,
oscillations on, g, ( (%)
m/s2 )

90.0  102 9.51 1.90 3.61 9.84 0.31

80.0  102 8.82 1.76 3.10 10.20 3.98

70.0  102 8.52 1.70 2.89 9.56 2.55

60.0  102 7.82 1.56 2.43 9.73 0.82

50.0  102 7.10 1.42 2.02 9.79 0.20

40.0  102 6.42 1.28 1.64 9.64 1.73

30.0  102 5.60 1.12 1.25 9.44 3.77

20.0  102 4.61 9.22  101 8.50  101 9.29 5.30

10.0  102 3.37 6.74  101 4.54  101 8.70 11.31

( 𝑔 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑−𝑔 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 )
To calculate % differences: × 100 %, where g
𝑔 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑
standard is 9.81 m/s2
Graph period, T against length of pendulum, L in mm

2.5

1.5

0.5

0
0 100 200 300 400 500 600 700 800 900

CONCLUSION

In this experiment, the acceleration due to gravity, g, have been determined by

means of simple pendulum. The possible sources of uncertainties may be due to
human errors comes in when measuring the period using a stopwatch. The reaction
time of the observer plays a significant error when starting the stopwatch and when
stopping it. This error can be minimized by repeating the experiment many times.
Besides, instrument errors - using a digital stopwatch also introduce errors.
Replacing the digital stopwatch by an analog one will introduce more errors. The arc
angle also introduce errors. As much as possible small angles must be used. The
angle of the arc must not exceed 30 angular degrees. The friction between the
swinging bob and the surrounding air is another source of error. This can be
minimized by using a heavier bob than a lighter bob. The shape of the bob must be
spherical to minimize this friction. The friction of the string and its pivotal anchor point
cannot be eliminated. The value of the acceleration due to gravity g in the locality is
not constant and must be obtained from reliable sources.
POST LAB QUESTION

1. Do you think the percentage difference between gexperiment and gstandard is

acceptable in this experiment?

 Yes, it is acceptable in this experiment. The percent difference is a

comparison between a theoretical estimate and an experimental result.

2. Do you think the experiment to determine g by using a pendulum can be

improved? Describe the improvement.

 Yes, it can be improved by making timings by sighting the bob past a fixed
reference point (called a fiducial point) to improve the accuracy of the
measurement.

3. Can you think of any other way to determine g?

 I think another way to determine g is by timing the free fall of an object going
through certain measured vertical distance and then calculate g by using an
appropriate equation of motion.
PRE LAB QUESTION

I. Different g at different latitudes on earth

1. Comparing the values of g at 16 places and cities in the world at various

latitudes, what can you say about the trend of these values?

 By comparing the values of g at 16 places and cities in the world at various

latitudes, I can say about the trend of these values is as the latitudes
decreases, the gravity, g, becomes decreases. This is because the Earth is
not a perfect sphere—it's slightly flattened at the poles and bulges out near
the equator, so points near the equator are farther from the center of mass.
The distance between the centres of mass of two objects affects the
gravitational force between them, so the force of gravity on an object is
smaller at the equator compared to the poles. This effect alone causes the
gravitational acceleration to be about 0.18% less at the equator than at the
poles. Second, the rotation of the Earth causes an apparent centrifugal force
which points away from the axis of rotation, and this force can reduce the
apparent gravitational force (although it doesn't actually affect the attraction
between two masses). The centrifugal force points directly opposite the
gravitational force at the equator, and is zero at the poles. Together, the
centrifugal effect and the center of mass distance reduce g by about 0.53% at
the equator compared to the poles.
2. Can you guess the appropriate values of g for Kuala Lumpur and the
South Pole?

 g = g45 – ½ ( gpoles – gequator ) cos ( 2 lat π/180 )

gpoles = 9.832 m/s2

g45 = 9.806 m/s2
gequator = 9.780 m/s2
lat = latitude

 values of g for Kuala Lumpur

g = 9.806 – ½ ( 9.832 –9.780 ) cos ( 2 ( 3.1 ) π/180 )

g = 9.806 – 0.026
g = 9.780 m/s2

 values of g for South Pole

g = 9.806 – ½ ( 9.832 –9.780 ) cos ( 2 ( 90.0 ) π/180 )

g = 9.806 – 0.026
g = 9.780 m/s2

3. Why g is larger at the poles?

 The different linear velocity of earth rotation at the poles and at the equator.
The linear velocity of the earth is bigger at the equator than at the poles. Next,
The density of the earth’s core which increases as we go deeper into the
centre of the earth. The equatorial bulge, where the poles are flattened and at
a shorter distance to the centre of the earth.
II. A straight line graph and its usage in data analysis

1. What is the meaning of m and c on the straight line equation y = mx + c?

 Equations of straight lines are in the form y = mx + c (m and c are numbers).

m is the gradient of the line and c is the y-intercept (where the graph crosses
the y-axis).

2. (a) How do we transform equation (1) so that we can plot a straight line
graph relating the variable T and L?

 the equation that relates the period of a pendulum to its length is as follows:

𝐿
𝑇 = 2𝜋 √
𝑔

where g is the acceleration due to gravity at the surface of Earth. This

equation can be written to more closely resemble the equation we determined
from the graph:

𝐿
T2 = 4π2 √𝑔

And it can be manipulated even further so that the equation is in the form y =
mx + b:
4𝜋2
T2 = L+0
𝑔

(b) How do we determine the value of g from this graph?

 to find an accurate value for ‘g’, we need to graph T2 versus the length (L) of
the pendulum.
III. Simple Pendulum

1. What is a simple pendulum?

 A simple pendulum performs simple harmonic motion, its periodic motion is

defined by an acceleration that is proportional to its displacement and directed
towards the Centre of motion.

2. How do we set up a simple pendulum in the lab?

 For this investigation, limited resources like, clamps, stands, a metre ruler, a
stopwatch, a metal ball (bob), and some string were used. The experimental
set-up was equal to the diagram, shown in figure. In this investigation, the
length of the pendulum was varied (our independent variable) to observe a
change in the period (our dependent variable). In order to reduce possible
random errors in the time measurements, we repeated the measurement of
the period three times for each of the ten lengths. We also measured the time
for ten successive swings to further reduce the errors. The length of our
original pendulum was set at 100 cm and for each of the following
measurements, we reduced the length by 10 cm.
3. What is the period of a simple pendulum?

 The period of a pendulum is the time it takes the pendulum to make one
full back-and-forth swing

4. What is the best way to determine the period of a simple pendulum?

 Construct your pendulum as desired, simply measure the length of the string
from the point it is tied to a support to the center of mass of the bob. You can
use the formula to calculate the period now. But we can also simply time an
oscillation (or several, and then divide the time you measured by the number
of oscillations you measured) and compare what you measured with what the
formula gave you.

𝑳
5. What is the condition to use the equation 𝑻 = 𝟐𝝅 √ ?
𝒈

 This equation can only be used in small angles condition. The reason for this
comes out from the derivation of the equation of motion. In order to derive this
relationship, it is necessary to apply the small angle approximation to the
function: sine of θ, where θ is the angle of the bob with respect to the lowest
point in its trajectory (usually the stable point at the bottom of the arc it traces
out as it oscillates back and forth.).
6. If we vary the length of a pendulum, the period will change. Make an
appropriate table to record the data of L and T.

Length L, (m) Time for x Period T, (s)

oscillations

7. How do we determine the uncertainty (error) from a data that are plotted
in a straight line graph?

 The line of gradient m is the best-fit line to the points where the two extremes
m1 and m2 show the maximum and minimum possible gradients that still lie
through the error bars of all the points. The percentage uncertainty in the
gradient is given by [m1-m2/m] = [Δm/m] x 100%.

8. How do we calculate the percent error between the value gexperiment and
gstandard ? Take gstandard = 9.81 m/s2.

( 𝑔 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑−𝑔 𝑒𝑥𝑝𝑒𝑟𝑖𝑚𝑒𝑛𝑡 )
 % error = × 100%
𝑔 𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑