General Physics Laboratory Report

EXPERIMENT 2

DETERMINATION OF GRAVITATIONAL
ACCELERATION WITH A REVERSIBLE PENDULUM
Class: CC19 / Group: 01 Lecturer’s comment

Full name:
1. Tăng Gia Bảo - 2152429
2. Lê Dương Khánh Huy - 2153380
3. Nguyễn Lê Anh Tuấn - 2153079

I. Aims/Purposes:

Aims/Purposes:
– Calculate the gravitational acceleration.
Theory:

– Physical pendulum is a solid, mass m, that can oscillate around a fixed horizontal axis
passing point 01that is placed higher than its center of mass G (Figure 1). O1 is called the
suspension of the pendulum.

– Equilibrium position of the pendulum coincides with the vertical of the line O1G. When
pulling the pendulum out of the equilibrium position, with a small angle α, then release
it, the components Pt of the gravity P = mg acting on the pendulum a torque M1 by:
M 1=−Pt . L1=−mg . L 1. sinα (1)
– Where g is the gravitational acceleration, L1 = O1G is the distance from the center O1 to
the center of mass G, (-) indicates torque M1 always pull the pendulum to equilibrium
position, meanwhile rotating a reverse angle α. When α is small, we can consider the
approximate:
M 1 ≈−mg L1 α (2)
– Basic equations for the rotation of the pendulum around an axis across 01:
 M1
β 1= (3)
I1
– With β1 = d2α/dt2 is angular acceleration, I1 is the inertia moment of the pendulum to
which the rotation axis passes through O1. Combine (3) and (2) and substitute ω12 =
mg.L1/I1, we get the harmonic oscillation equation of the pendulum:
d2 α 2
+ ω1 .α =0
d t2

– Root of equation (4):α =α 0 . cos ⁡(ω1 .t +φ).

– with α 0 as amplitude, ω 1 is angular frequency, φ is the initial phase at time t = 0.
2π
– From (5), Period T1 can determined: T 1= =2 π √ ❑ (6)
ω1
– In the physical pendulum, we can find a point O2, located on line through O1 and G so
that when the pendulum swings around a horizontal axis passing through O2, the
oscillation period is equal its period when oscillating around the axis O1. The physical
pendulum is called reversible.
– Indeed, we can easily prove that there exists the hanging points O2, as follows: When the
oscillation around the axis through point O2 (Figure 1), period T2 of the pendulum is the
same calculation above, and we found:
2π
T 2= =2 π √ ❑ (2)
ω2
– G is the distance from the rotation axis passing through O2 to the center of mass G and I2
is the inertia moment of the pendulum to the rotation axis passing through O2.
– IG is called the inertia moment of the pendulum for the rotating axis through the center of
mass G and parallel to the two axis O1 and O2. According to Huygens-Steiner theorem:
2
I 1=I G + m L1 (8)
I 2=I G + m L22 (9)
– If the suspension point O2 satisfies T1 = T2 conditions, substitute (9), (8) to (7), (6) we
find equation to determine O2 locate:
I
L1 . L2= G (10)
m
– On the other hand, from (6), (7) we can identify gravitational acceleration:
2
4 π . ( L1 + L2 ) .( L1 −L2)
g= 2 2 (11)
T 1 . L1−T 2 . L2
– If two point 01, 02 satisfying formula (10), than T1 = T2 = T, and equation to determine
the gravitational acceleration is:

2
4π . L
g= 2
(12)
T
II. Method, Equipment and Procedure:
Method:
– To determine the local gravitational acceleration we need to determine period T1 and T2,
And then apply the equation to calculate the local gravitational acceleration.
 Equipment:
1. Physical pendulum.
2. MC-963A meter
3. Optical infrared port.
4. Pendulum suspension.
5. Ruler 1000mm.
6. Caliper 0-150mm, accuracy 0.1 or 0.05mm.
7. Paper 120x80mm.
Procedure:
In this experiment, two suspension points (two blades O1, O2) fixed, one must locate
Weighted C (i.e. change the position of the center of mass G, so that (10) is satisfied), to let the
pendulum become irreversible. The following procedure:
1. Turn on the Weighted C closed to the weights 4. Use the caliper to measure the distance
x0 between them. In many cases, the pendulum was created so that the Weighted C was
extremely closed to the weights 4 (x0 = 0). Record the value x0 in table 1. Place the
pendulum on the rack in the forward direction (the word "Forward": toward the people
that are doing the experiment), measuring the time of 50 oscillation periods and recorded
in Table 1, below the columns 50T1.
2. Reverse the pendulum ("Reverse direction": toward the people doing the experiment),
and measure the time of the 50 oscillation period, recording the results in Table 1 below
50T2 column.
3. Set the location Weighted C to weights 4 away a distance x' = x0 + 40mm, (using calipers
to check). Measuring the period of 50 cycle and 50 reverse cycle with this position,
recording the results in Table 1.
4. Performance the measurement results on the graph: vertical axis 120mm, performances
50T1 and 50T2 time, horizontal axis 80mm, shows the position x of C. Connect the 50T1
and 50T2 points together by straight lines, their communication is the approximate
location of x for T1 = T2 (H3).
5. Use calipers to place the weight C on the right position x1. Measured 50T1 and 50T2.
Record results in table 1.
6. Adjust the weight C to the right position: The graph in Figure 4 shows the line 50 T1
slope than the T2, which means to the left of the crossover point than 50T2> 50T1, and
the right of the crossover point than 50T1> 50T2. From the results of measurements 5, at
x1 we can conclude that to obtain the best results, we must shift direction the weight C so
that 50T1 = 50T2.
7. Finally, when the best location of Weighted C has been identified, we measured each
direction 3-5 times to get random error, Record results in Table 2.
8. Use a ruler (1000 m) to measure the distance L between the two blades O1, O2. Record
in Table 1. (Only once carefully measured, taking the error ΔL = ±1mm).
9. To complete the experiment, turn off the MC-963 meter and unplug it from the power of
~ 220V.
III. Equations:
Calculating the period T:
1 (50 T 1 +50 T 2)
T= . .
50 2
∆ T 1 (∆ 50T 1 +∆ 50 T 2)
⇒ = .
T 50 2
Error T system:
∆T clock
∆ T sys =
50

Error T total:
∆ T =∆ T sys+ ∆ T

Gravitational acceleration
4 π2. L
g=
T2

∆g ∆ π ∆L ∆T
¿> δ= =2 + +2
g π L T

¿> ∆ g=δ . g

IV. Experimental Data:

IV.1 Table 1:
L= 700 ± 1 (mm)
Weighted position
50T1 (s) 50T2 (s)
(mm)
x0 = 0 mm 83.35 83.77
x0+40 = 40mm 84.19 84.07
x1 = 30.02 mm 84.04 84.02

IV.2 Raw data:

 IV.3 Table 2: At the best position x1', physical pendulum becomes T1= T2 = T:
Best position x'1 = 30.02 (mm)
Data 50T1 (s) Δ (50T1) 50T2 (s) Δ (50T2)
1 84.04 0.007 84.02 0.003
2 84.05 0.003 84.01 0.007
3 84.05 0.003 84.02 0.003
Mean 84.047 0.004 84.017 0.004
V. Calculations:
V.1Determine the oscillation period of the reversible pendulum:
❖Calculate the mean period T of the reversible pendulum from the values in table 2:
1 (50 T 1 +50 T 2) 1 (84.047+84.017)
T= . = . =1.681( s)
50 2 50 2
1
❖ Random error of T: ∆ T = . ¿ ¿
50
0.01 −4
❖ Systematic error of T: ∆ T sys = =2 ×10 ( s)
50
❖ Absolute error of T: ∆ T =∆ T sys+ ∆ T =2× 10−4 + 8 ×10−5 =0.00028( s)
V.2Calculate the gravitational acceleration:
4 π 2 L 4 π 2 .0.7 m
❖ Calculate the mean value of gravitational acceleration: g= 2
= 2
=9.770( 2 )
T 1.681 s
❖ Calculate the relative error of g:
−3 −4
∆g ∆π ∆ L ∆T 0.005 10 2.8 ×10
❖ =2 + +2 =2. + +2. = 4.946×10−3 = δ
g π L T 3.14 0.7 1.681
−3 m
=> ∆ g=δ . g= 4.946×10 × 9.770=0.048( 2 )
s
VI. Conclusions:
m
g=g ± ∆ g=9.770 ± 0.048( 2 )
s
VII.Questions:
1. What is the same and different between the physical pendulum and mathematical
pendulum?

Mathematical pendulum Physical pendulum

Model A simple pendulum is A physical pendulum is a

considered as an ideal realistic model of the
model because sometimes it pendulum; it has a finite
is difficult to achieve in body and shape.
reality.

Suspension A simple pendulum needs a A physical pendulum does

tread or a string to suspend not need any string for the
from a rigid support. suspension.

Tension There is a tension force As a physical pendulum

acting on the string, which does not need any string for
helps the object to suspend. the suspension, there will be
no tension.

Time period The time period depends on The time depends on the
the square root of the ratio square root of 2/3 ratio of
of length of the pendulum length and the acceleration
and the acceleration due to due to gravity.
gravity.

Mechanical energy Mechanical energy is Here also mechanical

conserved at every energy is conserved at every
oscillation. oscillation.

Oscillation A simple pendulum The angle of oscillation of

oscillates with a large angle the physical pendulum is
in a short interval of time. small.

Gravity Gravity acts at the center of Gravity acts towards the

the bob of the pendulum. center of mass of the
physical pendulum.

Torque When a simple pendulum is The physical pendulum

free to rotate, the torque is rotates freely due to the
due to gravity or angular torque applied to the center
acceleration. of mass.

2. Prove that any physical pendulum hanging given point O1 can be found O2 to let the
pendulum become irreversible.

In the physical pendulum, we can find a point O2, located on line through O1 and G so that when
the pendulum swings around a horizontal axis passing through O2, the oscillation period is equal
to its period when oscillating around the axis O1. The physical pendulum is called reversible.
Indeed, we can easily prove that there exists the hanging points O2, as follows: When the
oscillation around the axis through point O2 (Figure 1), period T2 of the pendulum is the same
calculation above, and we found:

2π
T 2= =2 π √ ❑
ω2

3.Show the way to adjust Weighted C to become a reversible pendulum with two given
suspension points O1, O2.

4. Write expressions identify the oscillation period of a reversible pendulum with small
amplitude.

T =2 π √❑

5. To determine the oscillation period of a reversible pendulum, we must measure many

periods (50 periods for example), but not measure each period? When such a measure,
which error can be overcome? How to calculate these kinds of errors?

When measuring a certain quantity many times, often no matter how careful you are, the results
between the measurements are almost always different. That proves that in the measured results
there is always an error and the result we get is only its approximate value.

There are many causes of errors, but the main ones are as follows:

Due to the lack of precision measuring machines and tools

Due to the low skill level of the person measuring, the ability of the senses is low

Due to external factors affecting, for example, weather changes, rain and wind, unusual heat and
cold, etc.

So that, when we measure it many times, we will collect the approximate value as much as we
can.

∆g ∆π ∆ L ∆T
=2 + +2
g π L T
6. Write a formula to measure the errors by using reversible pendulum? in the formula,
determines the number of ?
∆g ∆π ∆ L ∆T
=2 + +2
g π L T
π≈ 3.14