Simple Harmonic Motion – Pendulum Experiment Report
Abstract
In this experiment, we have obtained the spring value k and period T of a
simple harmonic pendulum motion, which is based on Hooke’s law.
In order to obtain our target data, pendulum motion must be observed and
measured via changing the mass of a small ball, which is fixed at the end of
pendulum. Therefore, two control experiments are prepared and we have
recorded 9 sets of data such as L, which is the lengths of the pendulum and
other related datum.
The value of gravitational acceleration g we have used in this experiment is
9.81m/s2. And eventually, we have found that the change of period T only
depends on the change of L, but has nothing to do with the mass of the ball.
Furthermore, the period T is proportional to the length of L within a certain
error range.
Introduction
A simple pendulum is an idealized physical model, which consists of a
connector, a link and a relatively small object with mass. As an idealized
physical model, the connector is regarded as a frictionless connection with the
outside world. The link here is regarded as a thin line with negligible mass.
And the small object such as a ball, whose radius is much smaller than the
length of the pendulum itself, has a mass value mostly represents the mass
value of the whole pendulum system.
The simple pendulum mentioned above is shown in Fig 1 below.
Fig 1. Simple physical pendulum
In order to make the expression of the pendulum period simplified and error
controlled, the maximum swing angle of the pendulum must be less than 10°.
The expression of the pendulum period can be simplified as shown in Fig 2
below.
Fig 2. Simplified expression of the pendulum period
It can be seen that the period of the pendulum is independent from the mass
of the pendulum system. From the perspective of stress, the restoring force of
the pendulum, which is along the tangent direction of the trajectory arc and
pointing to its equilibrium position can be seen as a portion of gravity, which
is mg*sin(θ). The greater the swing angle is, the greater the restoring force and
vertical acceleration are generated as we can see from Fig 3 below.
Fig 3. Simple pendulum force analysis
Therefore, the period value of the pendulum merely depends on the length
of its link.
Method
Simple harmonic motion (SHM) is a special type of vibration, the amplitude
of its force is directly proportional to the displacement of the equilibrium
position when the system is still, and the direction is opposite to the direction
of displacement. But unfortunately, for special reasons during the outbreak, we
were unable to use the laboratory.
Therefore, the whole experiment was done with software. In this software
(simulation software called Virtual Physics Laboratory, VPL) is used to carry
out the whole experiment. VPL software interface can be seen in Fig 4 as
follows:
Fig 4. Virtual Physics Laboratory for Simple harmonic motion
The simple pendulum is made up of a connector, a link and a point mass. In
SHM, we are interested in its period of oscillation. Another variable we care
about is gravity g, and then we are able to change the equation from T to g as
follows:
𝐿
𝑇 = 2𝜋√ (Equation 1)
𝑔
𝐿
4𝜋 2 2
𝑇
𝑔=
(Equation 2)
In the first experiment, the length of the link and the point mass were set as
10cm and 10kg separately and the data were collected afterwards.
In the control group, the length of the link stayed unchanged, but the mass
of the point mass was increased from 10kg to 14kg.
In each experiment, totally 9 sets of experimental datum were collected as
the length of the pendulum increased.
Result
The following data shown in Table1 and 2 were obtained after the two
experiments mentioned above.
Table 1 Datum obtained from 1st experiment
1st Experiment
Conditions
Amplitude=10cm, mass=10kg
Time of
Length/m
Number of oscillations
±0.01mm
N
10.00
0.10
15.00
9.50
0.60
15.00
0.15
10.00
7.90
0.80
20.00
0.20
10.00
9.10
0.90
25.00
0.25
9.00
9.10
1.00
30.00
0.30
8.00
8.80
1.10
35.00
0.35
7.00
8.30
1.20
40.00
0.40
7.00
8.90
1.30
45.00
0.45
6.00
8.10
1.40
50.00
0.50
7.00
9.80
1.40
Pendulum
length/cm
oscillations
Time period oscillations
±0.1s
T/s
Table 2 Datum obtained from 2nd experiment
2nd Experiment
Conditions
Amplitude=10cm, mass=14kg
Time of
Length/m
Number of oscillations
±0.01mm
N
10.00
0.10
11.00
7.00
0.60
15.00
0.15
11.00
8.60
0.80
20.00
0.20
10.00
9.10
0.90
25.00
0.25
9.00
9.00
1.00
30.00
0.30
8.00
8.80
1.10
35.00
0.35
8.00
9.40
1.20
40.00
0.40
8.00
10.00
1.30
45.00
0.45
7.00
9.50
1.40
50.00
0.50
7.00
9.90
1.40
Pendulum
length/cm
oscillations
T/s
Time period oscillations
±0.1s
Table 3 Processed datum from two experiments
Processed datum from two experiments
Length/m
±0.01mm
√(L)
±
0.01mm
Time period in
Time period in
Average time period /s
1st experiment /s
2nd experiment /s
±0.1s
0.10
0.32
0.60
0.60
0.60
0.15
0.39
0.80
0.80
0.80
0.20
0.45
0.90
0.90
0.90
0.25
0.50
1.00
1.00
1.00
0.30
0.55
1.10
1.10
1.10
0.35
0.59
1.20
1.20
1.20
0.40
0.63
1.30
1.30
1.30
0.45
0.67
1.40
1.40
1.40
0.50
0.71
1.40
1.40
1.40
The results show that the period T values obtained by the two groups of
experiments are the same when the length is equal, which indicates that the
period of simple pendulum is affected by the length and gravity acceleration g,
but not affected by the mass value.
Because the experimental datum are obtained from simulation software,
there is almost no error except the recording error in the collecting process and
not like practical experiment, the errors can come out from the string material,
ball shape, stopwatch used to record time and so on. In the preliminary
estimation of the experiment, it is considered that the mass of suspended solids
may affect the period. However, the result eventually shows that the period T
only varies with the length L and the acceleration g of gravity, and has nothing
to do with the mass value. So the conclusion is the same as the expression as
follows:
1
𝑇 = 2𝜋√ ∗ √𝐿 (Equation 3)
𝑔
We then transform the equation to a linear equation so that we can tell its
spring value k directly from slope of the linear equation, which is 2.11 for this
pendulum. And the R-squared value of 0.99 shows that the fitting accuracy is
quite high.
T vs √(L)
Time period /s
2.00
y = 2.11 x - 0.05
1.50
R² = 0.99
1.00
0.50
0.00
0.00
0.10
0.20
0.30
0.40
0.50
0.60
0.70
0.80
√(L) /m½
Fig 5. Period T vs root value of length √(L)
Discussion
It can be concluded that the experimental results are different from those
expected. Although the experimental formula and simple harmonic motion of
the simple pendulum have been studied before, the mass of the link will still
bring out errors to the system in practical experiment, so we could further
investigate the equations that can take the mass of the link and the friction loss
from the connector into consideration to further reduce the error.
Reference
Simulation software called The Virtual Physics Laboratory(VPL) was used to carry out the whole
experiment. http://vplab.ndo.co.uk/
Mingyuan Ji
P272200