Experiment No: M5
Experiment Name: Simple Pendulum
Objective: Investigating the relation between length and period of a simple pendulum. Measuring
the gravitational acceleration using a simple pendulum.
Keywords: Simple pendulum, period, simple harmonic motion, gravitational acceleration.
Theoretical Information:
A body tied by a rope and hung from a stationary point (also called pivot) is called a simple pendulum.
Figure 5.1 Simple Pendulum
Equation of motion for a simple pendulum can be derived from several different approaches like
using force, torque and energy arguments. Here we will use the energy approach. We will assume the
gravitational potential energy (U) at the height of the pivot as zero. Then we can write the total energy
of the simple pendulum as follows.
1
𝑚𝑣 2 − 𝑚𝑔𝑙𝑐𝑜𝑠𝜃 = 𝐸𝑡𝑜𝑡𝑎𝑙
2
5.1
While the first term on the left hand side of the equation 5.1 represents the kinetic energy of the
pendulum second term stands for the potential energy. The speed of the pendulum can be written in
terms of its angular velocity with respect to the pivot as 𝑣 = 𝑙
𝑑𝜃
𝑑𝑡
If we substitute this in to the
equation 5.1 we get the following expression:
1 2 𝑑𝜃 2
𝑚𝑙 ( ) − 𝑚𝑔𝑙𝑐𝑜𝑠𝜃 = 𝐸𝑡𝑜𝑡𝑎𝑙
2
𝑑𝑡
5.2
We know from the principle of energy conservation that the total energy of the system should stay
constant. That means it will not be dependent on time. So if we take the derivative of both sides of
equation 5.2 with respect to time it will give us the following expression where the right hand side
simply vanishes.
𝑑𝜃 𝑑2 𝜃
𝑑𝜃
𝑚𝑙 ( ) ( 2 ) + 𝑚𝑔𝑙 ( ) 𝑠𝑖𝑛𝜃 = 0
𝑑𝑡 𝑑𝑡
𝑑𝑡
2
5.3
𝑑𝜃
For further simplification the term 𝑚𝑙2 ( ) can be cancelled out from the left hand side since they
𝑑𝑡
are all different from zero. In this case the equation of motion for the pendulum can be written as
follows:
𝑑2𝜃 𝑔
+ 𝑠𝑖𝑛𝜃 = 0
𝑑𝑡 2 𝑙
5.4
Unfortunately equation 5.4 is a “non-linear” differential equation and it doesn’t have an easy solution.
However if we assume that the oscillations will be “small enough” to be able to use the approximation
𝑠𝑖𝑛𝜃 ≈ 𝜃 conveniently, the equation can be rewritten as follows:
𝑑2𝜃 𝑔
+ 𝜃=0
𝑑𝑡 2 𝑙
5.5
Equation 5.5 is known as the simple harmonic motion equation. It can easily be demonstrated that the
following solution satisfies the equation 5.5.
𝜃 = 𝐴. cos⁡(𝜔𝑡⁡ + ⁡⁡𝜑)
5.6
As it is seen the system makes a periodic motion. Here while 𝐴 ve ∅ are two arbitrary constants which
only depend on the initial conditions of the motion 𝜔 is called the angular frequency of the system
and it is given by the expression:
𝜔=√
𝑔
𝑙
5.7
By definition the relation between the angular frequency and the period of the system is:
𝑇=
2𝜋
𝜔
5.8
If we put equation 5.7 into equation 5.8 we get the following expression for the period of the system.
𝑙
𝑇0 = 2𝜋√
𝑔
5.9
For your report, submit beginning from this page by showing calculations in
reserved blank spaces and plotting graphics to the appropriate places.
T.C.
GEBZE TECHNICAL UNIVERSITY
PHYSICS DEPARTMENT
PHYSICS LABORATORY I
EXPERIMENT REPORT
THE NAME OF THE EXPERIMENT
DATE
:
: ….. / ….. / ……….
PREPARED BY
NAME AND SURNAME:
STUDENT NUMBER
:
GROUP NO
:
DATE OF THE EXPERIMENT: ….. / ….. / ……….
Experimental Procedure:
Figure 5.2: Simple Pendulum Experimental Setup
1. Two steel balls of different size and mass will be used in the experiment.
2. Each balls are attached to a thin rope. The pendulum is created by pinching the rope to the
clamps on the horizontal bar. The length of the pendulum is adjusted by changing the point of
pinching on the rope. When the length is adjusted it is also necessary to change the height of
the clamp to align the ball with the timer. Don’t change the height of the timer.
3. Check the two small holes on the inner faces of the timer, those are the sensor eyes. Height of
the clamp should be aligned such that the ball should be able to pass freely between the sensor
eyes of the timer. The measurement mode of the timer is set with a switch on the timer. It should
always be on the rightmost setting which corresponds to a “3-pass” measurement mode in which
the timer starts when an object passes the first time between the sensor eyes, continues to
measure in the second pass and stops only after the third pass. A typical measurement is
performed by removing the ball from between the sensor eyes, pushing the reset button while
holding the ball and then releasing the ball to swing between the sensor eyes. This way the timer
measures a full period of the pendulum after 3 passes. The period is displayed in second units
up to 4 significant figures. During your measurements always observe and count this 3 passes
to be sure that the timer is measuring the correct time and not an erroneous measurement which
might already be initiated by a passing object like your finger or the ball when setting up the
measurement.
4. You are supposed to measure the periods of the pendulums of 20, 40, 60 and 80 cm approximate
lengths. Those lengths might not be easy to adjust exactly so you are free to adjust to a nearby
approximate value and measure the period. However don’t forget to measure the exact length
(ex. 78,1 cm) and record them to your tables. When measuring the length you should measure
the horizontal distance between the pinching point and the center of the ball.
5. To cope with the random errors measure each period value 4 times. To stay in the “small-angle
approximation” regime you should release the balls as close to the timer as possible. However
also assure that the 3 passes should be possible.
6. Record your measurements to Table 5.1 and Table 5.2 which are for small and big balls
respectively.
Table 5.1 Pendulum length vs. period table for the small ball
Pendulum length
(cm)
Period (s)
Period (s)
Period (s)
Period (s)
(1st measurement)
(2nd measurement)
(3rd measurement)
(4th measurement)
Table 5.2 Pendulum length vs. period table for the big ball.
Pendulum length
(cm)
Period (s)
Period (s)
Period (s)
Period (s)
(1st measurement)
(2nd measurement)
(3rd measurement)
(4th measurement)
Fill in the tables 5.3 and 5.4 by taking the average values of your four consecutive time measurements.
Table 5.3: Pendulum length vs. average period table for big ball
Pendulum length (cm)
Average Period (s)
Table 5.4: Pendulum length vs. average period table for small ball
Pendulum length (cm)
Average Period (s)
You are asked to plot the length vs period (l-T) graph using Table 5.3. Choose the horizontal axis as
l and the vertical axis as T. Represent the data in Table 5.3 as points on your plot. Under the smallangle approximation what type of a curve is expected to pass through those points?
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Plot that curve on your graph paper that fits best to your data points by crude eye estimation.
Graphic 5.1
As it can be seen from your length vs. period graph, the relation between those quantities is not linear.
However it can still be investigated using the linear fitting methods by manipulating the length-period
relationship. We begin by assuming a relationship between period and length of the following form:
𝑇 = 𝐴. 𝑙 𝑎
5.10
If we take the logarithm of Equation 5.10 we get equation 5.11
𝑙𝑜𝑔(𝑇) = 𝑙𝑜𝑔(𝐴) + 𝑎. 𝑙𝑜𝑔⁡(𝑙)
5.11
Note that this is a linear relation of the form 𝑦 = 𝑛 + 𝑚𝑥. So if we take log⁡(𝑙) and log⁡(𝑇) as our
variables, we can determine the values of 𝑎 and 𝑙𝑜𝑔⁡(𝐴) by linear fitting techniques.
In this regard fill in Table 5.5 and Table 5.6 by taking the logarithms of the values of Table 5.3 and
Table 5.4.
Table 5.5: log(l)-log(T) table for the big ball
log(l)
log(T)
Table 5.6: log(l)-log(T) table for the small ball
log(l)
log(T)
Plot log(l)-log(T) graph by taking log(𝑙) as the horizontal axis and log(𝑇) as the vertical axis. Represent
the data in Table 5.5 on your plot as points. Since a line is expected to pass through these points
calculate the slope (𝑎) and the intercept (log⁡(𝐴)) of the line using the linear fitting formulae.
Big ball;
4
∑ 𝑙𝑜𝑔(𝑙𝑖 ) =
𝑖=1
4
∑ 𝑙𝑜𝑔⁡(𝑇𝑖 ) =
𝑖=1
4
∑ 𝑙𝑜𝑔(𝑙𝑖 ). 𝑙𝑜𝑔⁡(𝑇𝑖 ) =
𝑖=1
4
∑(𝑙𝑜𝑔⁡(𝑙𝑖 ))2 =
𝑖=1
𝑎=
4. ∑4𝑖=1 𝑙𝑜𝑔(𝑙𝑖 ). 𝑙𝑜𝑔⁡(𝑇𝑖 ) − ∑4𝑖=1 𝑙𝑜𝑔⁡(𝑇𝑖 ) ∑4𝑖=1 𝑙𝑜𝑔(𝑙𝑖 )
=
4. ∑4𝑖=1(𝑙𝑜𝑔⁡(𝑙𝑖 ))2 − (∑4𝑖=1 𝑙𝑜𝑔(𝑙𝑖 ))2
∑4𝑖=1(𝑙𝑜𝑔⁡(𝑙𝑖 ))2 ∑4𝑖=1 𝑙𝑜𝑔⁡(𝑇𝑖 ) − ∑4𝑖=1 𝑙𝑜𝑔(𝑙𝑖 ). 𝑙𝑜𝑔⁡(𝑇𝑖 ) ∑4𝑖=1 𝑙𝑜𝑔(𝑙𝑖 )
𝑙𝑜𝑔⁡(𝐴) =
=
4. ∑4𝑖=1(𝑙𝑜𝑔⁡(𝑙𝑖 ))2 − (∑4𝑖=1 𝑙𝑜𝑔(𝑙𝑖 ))2
𝑎 =…………;
𝑙𝑜𝑔(𝐴) =…………..
Small ball;
4
∑ 𝑙𝑜𝑔(𝑙𝑖 ) =
𝑖=1
4
∑ 𝑙𝑜𝑔⁡(𝑇𝑖 ) =
𝑖=1
4
∑ 𝑙𝑜𝑔(𝑙𝑖 ). 𝑙𝑜𝑔⁡(𝑇𝑖 ) =
𝑖=1
4
∑(𝑙𝑜𝑔⁡(𝑙𝑖 ))2 =
𝑖=1
4. ∑4𝑖=1 𝑙𝑜𝑔(𝑙𝑖 ). 𝑙𝑜𝑔⁡(𝑇𝑖 ) − ∑4𝑖=1 𝑙𝑜𝑔⁡(𝑇𝑖 ) ∑4𝑖=1 𝑙𝑜𝑔(𝑙𝑖 )
𝛼=
=
4. ∑4𝑖=1(𝑙𝑜𝑔⁡(𝑙𝑖 ))2 − (∑4𝑖=1 𝑙𝑜𝑔(𝑙𝑖 ))2
𝑙𝑜𝑔⁡(𝐴) =
∑4𝑖=1(𝑙𝑜𝑔⁡(𝑙𝑖 ))2 ∑4𝑖=1 𝑙𝑜𝑔⁡(𝑇𝑖 ) − ∑4𝑖=1 𝑙𝑜𝑔(𝑙𝑖 ). 𝑙𝑜𝑔⁡(𝑇𝑖 ) ∑4𝑖=1 𝑙𝑜𝑔(𝑙𝑖 )
=
4. ∑4𝑖=1(𝑙𝑜𝑔⁡(𝑙𝑖 ))2 − (∑4𝑖=1 𝑙𝑜𝑔(𝑙𝑖 ))2
𝑎 =…………;
𝑙𝑜𝑔(𝐴) =…………..
Using these values plot the line which has the equation 5.11 on your graph.
Graphic 5.2
If we assume the small angle approximation is correct what do you expect to find the value for 𝑎?
Staying in the small-angle approximation use log(𝐴) to calculate gravitational acceleration 𝑔:
Compare your 𝛼 values for the small and big balls to the expected value of 𝛼 under the small-angle
approximation. Also compare your 𝑔 values with the expected value of 980 cm/s2.
Discuss the reasons for possible causes of differences.
Conclusion, Comment and Discussion:
(Tips: Give detail explanation about what you've learned in the experiment and also explain the
possible errors and their reasons.)
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Questions:
1) For a simple pendulum with a period of 2 s
a) What length of rope is required?
b) What length of rope is required for a person who wants to find the same period on the Moon?
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2) An hour of simple pendulum is slow. What changes should be made to the mass and length to
correct the situation?
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