Department of Physics and Materials Sciences
Course: Experimental General Physics for the Engineering I - PHYS192
Fall 2024
Lab report of the Experiment: Free fall Acceleration
Name of Student:
CRN No.
ID:
Lab Section:
Name of Instructor: Mohammed Aslam Vallikkaparambil
Experiment
Date
1
2
3
Name of lab partners:
Objective:
This experiment aims to measure the free-fall acceleration due to the Earth's gravity.
Theoretical background
An object dropped near the Earth's surface will fall with an acceleration of g, independent of its mass. This
was the famous observation of Galileo. This is true specifically when air resistance effects are negligible.
Thus in a vacuum, a feather falls as quickly as a stone. Near the earth surface, the gravity force is constant.
This means a free-falling object will have a constant acceleration 𝑔𝑔. It can be shown that in such a situation
(i.e. constant acceleration 𝑔𝑔) the vertical position of the falling object in the function of time 𝑡𝑡 is given by
1
𝑦𝑦(𝑡𝑡) − 𝑦𝑦0 = 𝑣𝑣0 𝑡𝑡 + 𝑔𝑔𝑡𝑡 2
2
Where 𝑦𝑦0 and 𝑣𝑣0 are the position and the speed of the object at 𝑡𝑡 = 0, respectively. The above equation can
be written in the following form:
1
𝑦𝑦(𝑡𝑡) − 𝑦𝑦0
= 𝑣𝑣0 + 𝑔𝑔𝑡𝑡
2
𝑡𝑡
As the ball is dropped, the initial velocity is assumed to be zero, the equation becomes
∆𝑦𝑦 1
= 𝑔𝑔𝑡𝑡
𝑡𝑡
2
This is the equation that will be used in our experiment. In the linear form, this equation denotes the velocity
of the free-falling object, which is proportional to the time required to fall. The other concept obtained from
the equation is the object's acceleration remains constant throughout the motion.
Procedure and Equipment’s:
The free-fall experiment apparatus consists of an electromagnet, timer and contact plate, as shown in figure
1. The steel spherical ball is dropped from the magnet at different distances.
•
•
•
•
•
We set a distance of nearly 20 cm from the steel ball to
the contact plate, then dropped the ball.
We did this measurement three times.
We repeated the step for the other distances like 30, 40,
50, 60 cm (or nearly so).
When measuring the nominated distances, we tried to
estimate the errors.
We calculated the average time of falling for each
distance and then calculated the velocities.
Fig 1. Experimental setup
Data table
Distance
dropping, Y
(𝑚𝑚)
of Time of dropping the ball,𝑡𝑡 (𝑠𝑠)
0.129
0.125 0.118
0.124
𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉𝑉
𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷𝐷
=
𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴𝐴 𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡
(m/s)
1.61
0.30
0.160
0.162 0.162
0.161
1.85
0.40
0.201
0.200 0.194
0.198
2.02
0.50
0.236
0.246 0.233
0.238
2.10
0.60
0.261
0.260 0.260
0.260
2.31
0.20
𝑡𝑡1(𝑠𝑠)
𝑡𝑡2(𝑠𝑠)
Average
time ,𝑡𝑡avg
(𝑠𝑠)
𝑡𝑡3(𝑠𝑠)
Table 1. Experiment data
In calculating the last two columns, we kept four figures whenever rounding was necessary. The rounding
rules in our case suggest keeping only three digits, and we kept four digits because these columns will be
used in later calculations.
Graph :
A graph is plotted using Microsoft excel between the velocity of the falling ball and the time average, as
shown in figure 2. The chart confirms the relationship between the position, velocity and acceleration of an
object moving under the influence of gravitational force and became a linear graph. The R2 obtained in the
chart is closer to the linearity parameter of R2=1, which also indicates the data accuracy of the experiment.
Velocity- Time graph
y = 505.45x + 100.72
R² = 0.985
250
Velocity (cm/s)
200
150
Figure 2. Velocity –
Average time Graph
100
50
0
0
0.05
0.1
0.15
Time average (s)
0.2
0.25
0.3
We get the error on the slope and intercept using the Excel linest function (we notice that the slope and
intercept are the same given the Excel graph above):
5.05 m/s2
0.36 m/s2
Slope
Error of the slope
Intercept
Error of the intercept
1.00 m/s
0.07 m/s
Table 2. Linest function
Error Analysis
The errors in the measured physical quantities are obtained by different methods based on the principles of
the experiment.
a. We estimated the error on Y to 1𝑚𝑚𝑚𝑚 (i.e. ∆(Y) = 1𝑚𝑚𝑚𝑚).
b. To estimate the error on 𝑡𝑡avg we apply the standard deviation of the average method on the last
(fifth) row, which seems to be the one that would give us the most significant error (and the safest
to quote):
i. From the table, the average for this row is 𝑡𝑡avg = 0.238 𝑠𝑠
ii.
iii.
iv.
c.
s
𝜎𝜎𝑡𝑡
𝜎𝜎𝑡𝑡avg
Our estimate for the error on 𝑡𝑡𝑚𝑚 is then 𝜎𝜎𝑡𝑡𝑡𝑡 = 0.004 𝑠𝑠
Estimation of the error on
∆𝑉𝑉 = ∆ �𝑡𝑡
𝑌𝑌
𝑎𝑎𝑎𝑎𝑎𝑎
𝜕𝜕𝜕𝜕
∆𝑌𝑌
𝑡𝑡
𝜕𝜕𝜕𝜕
� = �(𝜕𝜕𝜕𝜕 . ∆𝑌𝑌)2 + (𝜕𝜕𝑡𝑡
𝑎𝑎𝑎𝑎𝑎𝑎
. ∆𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎 )2 = �(𝑡𝑡
1
𝑎𝑎𝑎𝑎𝑎𝑎
. ∆𝑌𝑌)2 + (𝑡𝑡
−𝑌𝑌
𝑎𝑎𝑎𝑎𝑎𝑎
2
. ∆𝑡𝑡𝑎𝑎𝑎𝑎𝑎𝑎 )2
Using still the fifth row,∆V =0.04𝑚𝑚/𝑠𝑠
Calculation and Result
From the theoretical background of this experiment , we have
∆𝑦𝑦 1
= 𝑔𝑔𝑡𝑡
𝑡𝑡
2
So, from the graph Slope of the plot Velocity-Average Time, the Acceleration due to gravity of
Earth = 2 Slope ,
and the Uncertainty on the Acceleration due to gravity of Earth , ∆𝑔𝑔 = 2 ∆𝑆𝑆lope
𝑔𝑔 = 2x5.05 m/s2 = 10.1m/s2
∆𝑔𝑔= 2x 0.36 m/s2 = 0.721 m/s2
So the final result of the experiment
Acceleration due to gravity of Earth, 𝑔𝑔 = (10.1 ± 0.7) m/s2
By using Chi-Square method, it is easier to make a comparison between the theoretical and experimental
values of acceleration due to gravity:
(9.8−10.1)2
𝜒𝜒 2 = (0.1)2
+(0.721)2
= 0.17 < 1
As it is seen here, the value of Chi-Square is smaller than one, which means that the experimental value of
acceleration due to gravity is close to the theoretical value of it.
The error percentage between the theoretical and experimental value is given by:
%𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 =
9.8−10.1
Percentage error =�
9.8
𝑇𝑇ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣 − 𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸𝐸 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣
× 100%
𝑇𝑇ℎ𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒𝑒 𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣𝑣
�= 3%
Discussion and conclusion
We have measured the free-fall acceleration due to earth gravity and obtained a value of 𝑔𝑔 = (10.1±0.72)
m/s2. This value is compatible with the theoretical value of 𝑔𝑔 = (9.8 ± 0.1) m/s2. The percentage error obtained
is very small, indicating the accuracy of the experimental data. In addition, the value of 𝑅𝑅 2 of the graph is
0.985, indicating that the error is systematic as all points are very close to the trend line, but the slope is
affected very much. This systematic error is related to the instrument, which comes from the
faulty working of the experimental setup. During the experiment, random errors have also occurred while
measuring the distance of dropping the ball and adjusting the electromagnet.
In conclusion, the acceleration due to Earth's gravity has been measured using the concept of a free-falling
object, and the obtained results are very close to the world accepted value. The errors during the experiment
can be avoidable by using more sophisticated pieces of equipment and modern devices which comes with
accurate sensors.
References:
1. PHYS 192 Lab manual
2. Physics for Scientists and Engineers with Modern Physics,10 Edition
by Raymond A.Serway and John W. Jewett