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General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Physical Pendulum / Torsion Pendulum Ver.20170517
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 1 / 12
[International Campus Lab]
Physical Pendulum, Torsion Pendulum
Investigate the motions of physical pendulums and torsion pendulums.
1. Physical Pendulum
A physical pendulum is any real pendulum that uses an
extended body, as contrasted to the idealized simple pendu-
lum with all of its mass concentrated at a point.
Figure 1 shows a body of irregular shape pivoted so that it
can turn without friction about an axis through point 𝑂𝑂. In
equilibrium the center of gravity (cg) is directly below the pivot;
in the position shown, the body is displaced from equilibrium
by an angle 𝜃𝜃, which we use as a coordinate for the system.
The distance from 𝑂𝑂 to the center of gravity is 𝑑𝑑, the mo-
ment of inertia of the body about the axis of rotation through
𝑂𝑂 is 𝐼𝐼, and the total mass is 𝑚𝑚. When the body is displaced
as shown, the weight 𝑚𝑚𝘨𝘨 causes a restoring torque
𝜏𝜏𝑧𝑧 = −(𝑚𝑚𝘨𝘨)(𝑑𝑑 sin𝜃𝜃) (1)
When the body is released, it oscillates about its equilibrium
position. The motion is not simple harmonic because the
torque 𝜏𝜏𝑧𝑧 is proportional to sin𝜃𝜃 rather than to 𝜃𝜃 itself.
However, if 𝜃𝜃 is small, we can approximate sin𝜃𝜃 by 𝜃𝜃 in
radian. Then the motion is approximately simple harmonic.
With this approximation,
𝜏𝜏𝑧𝑧 = −(𝑚𝑚𝘨𝘨𝑑𝑑)𝜃𝜃 (2)
Fig. 1 The restoring torque on the body is proportional to
sin𝜃𝜃, not to 𝜃𝜃. However, for small 𝜃𝜃, sin𝜃𝜃 ≈ 𝜃𝜃, so the motion is approximately simple harmonic.
Objective
Theory
----------------------------- Reference --------------------------
Young & Freedman, University Physics (14th ed.), Pearson, 2016
14.6 Physical Pendulum (p.475~477)
9.4 Energy in Rotational Motion (p.307~312)
9.5 Parallel-Axis Theorem (p.312~313)
14.4 Application of SHM – Angular SHM (p.471)
-----------------------------------------------------------------------------
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Physical Pendulum / Torsion Pendulum Ver.20170517
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 2 / 12
Using the rotational analog of Newton’s second law for a
rigid body, ∑𝜏𝜏𝑧𝑧 = 𝐼𝐼𝛼𝛼𝑧𝑧, we find
−(𝑚𝑚𝘨𝘨𝑑𝑑)𝜃𝜃 = 𝐼𝐼𝛼𝛼𝑧𝑧 = 𝐼𝐼𝑑𝑑2𝜃𝜃𝑑𝑑𝑑𝑑2
𝑑𝑑2𝜃𝜃𝑑𝑑𝑑𝑑2 = −
𝑚𝑚𝘨𝘨𝑑𝑑𝐼𝐼 𝜃𝜃 (3)
Comparing this with the equation for SHM, 𝑎𝑎𝑥𝑥 = −(𝑘𝑘 𝑚𝑚⁄ )𝑥𝑥,
we see that the role of (𝑘𝑘 𝑚𝑚⁄ ) for the spring-mass system is
played here by the quantity (𝑚𝑚𝘨𝘨𝑑𝑑 𝐼𝐼⁄ ). Thus
𝜔𝜔 = �𝑚𝑚𝘨𝘨𝑑𝑑𝐼𝐼 (4)
𝑇𝑇 = 2𝜋𝜋�𝐼𝐼
𝑚𝑚𝘨𝘨𝑑𝑑 (5)
Figure 2 gives moments of inertia for several familiar shapes
in terms of their masses and dimensions. Fig. 2(b) shows that
the moment of inertia of a rectangular plate through center of
mass is 𝐼𝐼cm = (1 12⁄ )𝑀𝑀(𝑎𝑎2 + 𝑏𝑏2) , however, if 𝑎𝑎 ≪ 𝑏𝑏(= 𝐿𝐿)
then it approximately becomes 𝐼𝐼cm = (1 12⁄ )𝑀𝑀𝐿𝐿2 as Fig. 2(a).
Fig. 2 Moments of Inertia of Various Bodies
Fig. 3 The parallel-axis theorem.
A body doesn’t have just one moment of inertia. In fact, it
has infinitely many, because there are infinitely many axes
about which it might rotate. But there is a simple relationship,
called the parallel-axis theorem, between 𝐼𝐼cm (moment of
inertia of a body about an axis through its center of mass)
and 𝐼𝐼𝑃𝑃 (moment of inertia about any other axis parallel to the
origin axis) (Fig. 3):
𝐼𝐼𝑃𝑃 = 𝐼𝐼cm + 𝑀𝑀𝑑𝑑2 (6)
where 𝑀𝑀 is the mass of body and 𝑑𝑑 is the distance between
two parallel axes.
From Eqs. (5), (6) (𝑚𝑚 = 𝑀𝑀), 𝐼𝐼cm = (1 12⁄ )𝑀𝑀𝐿𝐿2, and Fig. 4(a),
the period 𝑇𝑇 of a slender rod with length 𝐿𝐿 is
𝑇𝑇 = 2𝜋𝜋�𝐿𝐿2 + 12𝑑𝑑2
12𝘨𝘨𝑑𝑑 (7)
From Eqs. (5), (6) (𝑚𝑚 = 𝑀𝑀), 𝐼𝐼cm = (1 2⁄ )𝑀𝑀𝑅𝑅2, and Fig. 4(b),
the period 𝑇𝑇 of a solid cylinder with radius 𝑅𝑅 is
𝑇𝑇 = 2𝜋𝜋�𝑅𝑅2 + 2𝑑𝑑2
2𝘨𝘨𝑑𝑑 (8)
Fig. 4 Various physical pendulums
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Physical Pendulum / Torsion Pendulum Ver.20170517
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 3 / 12
Fig. 5 A graph of the period 𝑇𝑇 as a function of distance 𝑑𝑑 from the center of mass for a 50cm-length slender rod pendulum.
Fig. 6 A graph of the period 𝑇𝑇 as a function of distance 𝑑𝑑 from the center of mass for a 10cm-radius solid cylinder pendulum
2. Torsion Pendulum
A restoring force on a body undergoing periodic motion orig-
inates in difference ways in difference situations. Figure 7
shows a kind of torsion pendulum which consists of an elastic
object such as a thin steel wire. When it is twisted, it exerts a
restoring torque in the opposite direction.
The balance disk has a moment of inertia 𝐼𝐼 about its axis.
The twisted steel wire exerts a restoring torque 𝜏𝜏𝑧𝑧 that is
proportional to the angular displacement 𝜃𝜃 from the equilib-
rium position. We write 𝜏𝜏𝑧𝑧 = −𝜅𝜅𝜃𝜃 , where 𝜅𝜅 is a constant
called the torsion constant.
Using the rotational analog of Newton’s second law for a
rigid body, ∑𝜏𝜏𝑧𝑧 = 𝐼𝐼𝛼𝛼𝑧𝑧 = 𝐼𝐼𝑑𝑑2𝜃𝜃/𝑑𝑑𝑑𝑑2, we find
−𝜅𝜅𝜃𝜃 = 𝐼𝐼𝛼𝛼 or 𝑑𝑑2𝜃𝜃𝑑𝑑𝑑𝑑2 = −
𝜅𝜅𝐼𝐼 𝜃𝜃 (9)
This equation is exactly the same as 𝑎𝑎𝑥𝑥 = −(𝑘𝑘 𝑚𝑚⁄ )𝑥𝑥 for
simple harmonic motion, with 𝑥𝑥 replaced by 𝜃𝜃 and 𝑘𝑘 𝑚𝑚⁄
replaced by 𝜅𝜅 𝐼𝐼⁄ . So we are dealing with a form of angular
simple harmonic motion. The angular frequency 𝜔𝜔 and peri-
od 𝑇𝑇 are given by 𝜔𝜔 = �𝑘𝑘 𝑚𝑚⁄ and 𝑇𝑇 = 2𝜋𝜋�𝑚𝑚 𝑘𝑘⁄ , respec-
tively, with the same replacement:
𝜔𝜔 = �′𝜅𝜅′𝐼𝐼 (10)
𝑇𝑇 = 2𝜋𝜋�𝐼𝐼′𝜅𝜅′
(11)
Fig. 7 The steel wire exerts a restoring torque that is pro-
portional to the angular displacement 𝜃𝜃, so the mo-tion is angular SHM.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Physical Pendulum / Torsion Pendulum Ver.20170517
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 4 / 12
1. List
Item(s) Qty. Description
PC / Software Data Analysis: Capstone
1 Records, displays and analyzes the data measured by various sensors.
Interface
1 Data acquisition interface designed for use with various sensors, including power supplies which provide up to 15 watts of power.
Force Sensor
1 Measures the magnitude of force. Range: −50N ~ 50N Resolution: 0.03N
Rotary Motion Sensor (RMS)
1 Measures rotational or linear position, velocity and ac-celeration
Slender Rod (or Long Rectangular Plate)
1 Length: 500mm Width: 20mm Pivot Point (Holes): 60, 100, 144, 190, 230mm from center of gravity
Spherical Cylinder (or Disk)
1 Radius: 100mm Pivot Point (Holes): 30, 50, 70, 90mm from center of gravity
Upper Wire Clamp Lower Wire Clamp
1 set Clamp wires.
Wires (3 ea) (in the case)
1 set Exerts a restoring torque when twisted. Material: Steel Diameter: 0.8mm, 1.2mm, 1.6mm
Balance Disk
1 Has a moment of inertia 𝐼𝐼 = (1 2⁄ )𝑀𝑀𝑅𝑅2 about its axis.
Equipment
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Physical Pendulum / Torsion Pendulum Ver.20170517
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 5 / 12
Item(s) Qty. Description
A-shaped Base Support Rod (600mm)
1 set Provide stable support for experiment set-ups.
Ruler
1 Measures length.
String / Scissors
Shared Exerts a torque to twist a wire.
Electronic Balance
Shared Measures mass.
2. Details
(1) Force Sensor
Refer to the “Circular Motion and Centripetal Force” lab
manual.
(2) Rotary Motion Sensor
The Rotary Motion Sensor is a bidirectional angle sensor
designed to measure rotational or linear position, velocity
and acceleration.
It contains a small photogate sensor and an optical code
wheel on which dark bands are printed in line. As the shaft of
the sensor rotates, the bands block the infrared beam of the
photogate, which provides very accurate signals for position-
ing or timing.
It includes a removable 3-step pulley with 10mm, 29mm,
and 48mm diameters. This allows you to convert a linear
motion into a rotational motion.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Physical Pendulum / Torsion Pendulum Ver.20170517
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 6 / 12
Experiment 1. Physical Pendulum (Slender Rod or Long Rectangular Plate)
(1) Set up equipment.
Mount the RMS on the support rod so that the shaft of the
sensor is horizontal (parallel to the table).
(2) Attach the slender rod to the RMS.
Use the mounting thumbscrew to attach the slender rod to
the shaft of the sensor through the end hole of the rod, so the
pivot point is 230mm above the center of gravity.
(3) Run Capstone software.
① The interface automatically recognizes the RMS.
② Adjust the sample rate of measurement.
- [Rotary Motion Sensor]: 100.00 Hz
③ Add a [Graph], and then select [Time(s)] for the 𝑥𝑥-axis
and [Angle(rad)] for the 𝑦𝑦-axis.
Procedure
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Physical Pendulum / Torsion Pendulum Ver.20170517
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 7 / 12
(4) Begin recording data.
Click [Record] and then let the pendulum swing.
① With the pendulum on the equilibrium position, click [Rec-
ord] to begin recording data.
② Gently start the pendulum swinging with a small amplitude
(within 5°).
③ After 5~6 oscillations, click [Stop] to end recording data.
(5) Find the period 𝑇𝑇 of oscillation.
① Choose any reference point of measurement (for example,
peaks or zero up-crossings).
② Use [Show coordinate…] to read off the time of the point.
③ Repeat measuring times for all oscillations and find the
period of oscillation. Also, calculate and record the theoretical
period of oscillation based on the length 𝑑𝑑 from the pivot
point to the center of gravity.
𝑑𝑑𝑛𝑛 (s) 𝑇𝑇 = 𝑑𝑑𝑛𝑛 − 𝑑𝑑𝑛𝑛−1(s)
1
2
3
4
5
…
𝑇𝑇average(s)
𝑇𝑇theory (s)
𝑇𝑇 = 2𝜋𝜋�𝐿𝐿2 + 12𝑑𝑑2
12𝘨𝘨𝑑𝑑 (7)
𝐿𝐿 = 500mm
𝑑𝑑 = 230, 190, 144, 100, 60mm
(6) Repeat measurement.
Repeat steps (4) and (5) for the holes that are 𝑑𝑑 = 230mm,
190mm, 144mm, 100mm, and 60mm from the center hole.
(7) Plot a 𝑇𝑇-𝑑𝑑 graph.
Using your results in step (6), plot a 𝑇𝑇-𝑑𝑑 graph and com-
pare it with Fig. 5 in Theory section.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Physical Pendulum / Torsion Pendulum Ver.20170517
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 8 / 12
Q
How does the period of this pendulum change when the pivot point moves towards the center of gravity? If it does not steadily increase or decrease, at what pivot point does the pendulum have minimum 𝑇𝑇. Also, use Eq. (7) to calcu-late 𝑑𝑑 under the condition of minimum 𝑇𝑇, and compare the theoretical value with your result.
A
Q When the amplitude of this physical pendulum increases, should its period increase or decrease? Why?
A
Experiment 2. Physical Pendulum (Solid Cylinder)
Repeat the procedure of expt. 1 using a disk.
𝑇𝑇 = 2𝜋𝜋�𝑅𝑅2 + 2𝑑𝑑2
2𝘨𝘨𝑑𝑑 (8)
𝑅𝑅 = 100 mm
𝑑𝑑 = 90, 70, 50, 30 mm
Experiment 3. Torsion Constant
(1) Set up your equipment.
① Slip the lower wire clamp onto the support rod.
② Clamp the RMS at the top of the support rod so that the
shaft of the sensor is vertical.
③ Align the guide of the upper wire clamp with the slot of the
shaft of the RMS. Slide the upper wire clamp onto the shaft
and firmly tighten the thumbscrew.
④ Clamp each ends of the wire under the thumbscrew of the
upper/lower wire clamp. Be sure that the elbow of the bend in
the wire fits snugly against the axle of the thumbscrew.
⑤ Connect the sensors to the interface.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Physical Pendulum / Torsion Pendulum Ver.20170517
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 9 / 12
⑥ Wind a string around the largest pulley.
(2) Set up Capstone software.
① Configure the Rotary Motion Sensor.
- Click the RMS icon and then click the properties button (☼).
- Select [Large Pulley (Groove)] for [Linear Accessory].
- [Change Sign] switches the sign of collected RMS data,
which depends on the setup status or the rotational direction
of the shaft. Check [Change Sign] if required.
② Configure the Force Sensor.
- Click the FS icon and then click the properties button (☼).
- Check [Change Sign].
(The sign of FS data is initially negative for the pulling force.)
Caution
When you slide the 3-step pulley onto the shaft of the
RMS, be sure to align the guide of the pulley with the slot
of the shaft.
Caution
If the retaining ring of the sensor shaft gets entangled in
a string, SLOWLY and CAREFULLY remove the string.
(NEVER apply a firm quick jerk to the string, which caus-
es the retaining ring to warp, and as a result, the sensor
to fail.) If it becomes warped, suspend your experiment
immediately and visit lab office to replace the sensor.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Physical Pendulum / Torsion Pendulum Ver.20170517
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 10 / 12
③ Configure calculator.
Define the torque 𝜏𝜏 as below, where “[Force(N)]” is meas-
ured data by the Force Sensor, and “r” is the radius of the 3rd
(largest) pulley (= 24mm).
④ Add a graph.
Select [Rotary Motion Sensor – Angle(rad)] for the 𝑥𝑥-axis
and [𝜏𝜏(Nm)] (defined in step③) for the 𝑦𝑦-axis.
(3) Zero the Force Sensor.
(4) Begin recording data.
Hold the force sensor parallel to the table at the height of the
largest pulley and slowly pull it straight out. If the angle shows
negative, change the sign of RMS output (see step (2)-①).
(5) Analyze your graph.
Find the torsion constant 𝜅𝜅.
① Click [Select range(s) …] icon and then drag the data
range of interest.
NOTE
To zero the sensor, press the [Zero] button on it WITH
NO FORCE exerted on the sensor hook.
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Physical Pendulum / Torsion Pendulum Ver.20170517
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 11 / 12
② Click [Select curve fits … ] and select [Linear: mt+b] to
find linear fit for selected data points. The torsion constant 𝜅𝜅
is equal to the slope of the 𝜏𝜏-𝜃𝜃 graph.
(6) Repeat measurement for other wires.
Repeat steps (4) to (5) using other wires.
Experiment 4. Torsion Pendulum (Angular SHM)
(1) Calculate the moment of inertia of the balance disk.
Measure the radius and the mass of the balance disk and
calculate the theoretical value of 𝐼𝐼. (Suppose the balance
disk is a perfect solid cylinder and apply the relationship
𝐼𝐼 = (1 2⁄ )𝑀𝑀𝑅𝑅2.)
(2) Set up your equipment.
Use the setup detailed in expt. 3. Remove the string and
attach the balance disk to the 3-step pulley with the thumb-
screw. (Be careful not to attach the disk directly on the shaft
without the pulley.)
(3) Configure Capstone software.
Follow the setup instruction of the experiment 1.
Change [Sample Rate] to 200.00 Hz or 500.00 Hz.
(4) Begin recording data.
Click [Record]. Twist the balance disk about 120~180° and
release it. Keep recording data for about 5-6 oscillations and
stop recording data.
Determine the time for each period of oscillation and verify
Eq. (11).
𝑇𝑇 = 2𝜋𝜋�𝐼𝐼𝜅𝜅 (11)
(5) Change the wire and repeat step (4).
General Physics Lab (International Campus) Department of PHYSICS YONSEI University
Lab Manual
Physical Pendulum / Torsion Pendulum Ver.20170517
Lab Office (Int’l Campus)
Room 301, Building 301 (Libertas Hall B), Yonsei University 85 Songdogwahak-ro, Yeonsu-gu, Incheon 21983, KOREA (☏ +82 32 749 3430) Page 12 / 12
Your TA will inform you of the guidelines for writing the laboratory report during the lecture.
Please put your equipment in order as shown below.
□ Delete your data files from your lab computer.
□ Turn off your lab Computer.
□ Tighten all thumbscrews in position.
□ Put the Wires in the storage case.
□ Leave the Spools of String, Scissors in the basket on the lecture table.
Result & Discussion
End of LAB Checklist