# Objective Theory

Figure 1 shows a body of irregular shape pivoted so that it can turn without friction about an axis through point ππ. In ... as shown, the weight...

Lab Manual

General Physics Lab (International Campus) Department of PHYSICS YONSEI University

Physical Pendulum / Torsion Pendulum Ver.20170517

[International Campus Lab]

Physical Pendulum, Torsion Pendulum

Objective

Investigate the motions of physical pendulums and torsion pendulums.

Theory

When the body is released, it oscillates about its equilibrium -----------------------------

Reference

-------------------------th

Young & Freedman, University Physics (14 ed.), Pearson, 2016

position. The motion is not simple harmonic because the

14.6 Physical Pendulum (p.475~477)

torque πππ§π§ is proportional to sin ππ rather than to ππ itself.

9.4 Energy in Rotational Motion (p.307~312)

radian. Then the motion is approximately simple harmonic.

However, if ππ is small, we can approximate sin ππ by ππ in

9.5 Parallel-Axis Theorem (p.312~313)

With this approximation,

14.4 Application of SHM β Angular SHM (p.471) πππ§π§ = β(πππ¨π¨ππ)ππ

-----------------------------------------------------------------------------

(2)

1. Physical Pendulum A physical pendulum is any real pendulum that uses an extended body, as contrasted to the idealized simple pendulum 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 ππ. 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 ππ, which we use as a coordinate for the system.

The distance from ππ to the center of gravity is ππ, the moment of inertia of the body about the axis of rotation through ππ is πΌπΌ, and the total mass is ππ. When the body is displaced

as shown, the weight πππ¨π¨ causes a restoring torque πππ§π§ = β(πππ¨π¨)(ππ sin ππ) Lab Office (Intβl Campus)

(1)

Fig. 1

The restoring torque on the body is proportional to sin ππ, not to ππ. However, for small ππ, sin ππ β ππ, so the motion is approximately simple harmonic.

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Physical Pendulum / Torsion Pendulum Ver.20170517

Using the rotational analog of Newtonβs second law for a rigid body, β πππ§π§ = πΌπΌπΌπΌπ§π§ , we find

A body doesnβ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,

β(πππ¨π¨ππ)ππ = πΌπΌπΌπΌπ§π§ = πΌπΌ ππ2 ππ πππ¨π¨ππ =β ππ ππππ 2 πΌπΌ

ππ2 ππ ππππ 2

called the parallel-axis theorem, between πΌπΌcm (moment of

inertia of a body about an axis through its center of mass) (3)

and πΌπΌππ (moment of inertia about any other axis parallel to the

origin axis) (Fig. 3):

Comparing this with the equation for SHM, πππ₯π₯ = β(ππ βππ)π₯π₯,

we see that the role of (ππ βππ) for the spring-mass system is played here by the quantity (πππ¨π¨ππβπΌπΌ ). Thus ππ = οΏ½

πππ¨π¨ππ πΌπΌ

πΌπΌ ππ = 2πποΏ½ πππ¨π¨ππ

πΌπΌππ = πΌπΌcm + ππππ2

(6)

where ππ is the mass of body and ππ is the distance between

two parallel axes. (4)

(5)

From Eqs. (5), (6) (ππ = ππ), πΌπΌcm = (1β12)πππΏπΏ2 , and Fig. 4(a),

the period ππ of a slender rod with length πΏπΏ is πΏπΏ2 + 12ππ2 ππ = 2πποΏ½ 12π¨π¨ππ

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 πΌπΌcm = (1β12)ππ(ππ2 + ππ 2 ) , however, if ππ βͺ ππ(= πΏπΏ)

then it approximately becomes πΌπΌcm = (1β12)πππΏπΏ2 as Fig. 2(a).

(7)

From Eqs. (5), (6) (ππ = ππ), πΌπΌcm = (1β2)ππππ 2 , and Fig. 4(b),

the period ππ of a solid cylinder with radius ππ is ππ2 + 2ππ2 ππ = 2πποΏ½ 2π¨π¨ππ

Fig. 2

Moments of Inertia of Various Bodies

Fig. 3

The parallel-axis theorem.

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Fig. 4

(8)

Various physical pendulums

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Physical Pendulum / Torsion Pendulum Ver.20170517

The balance disk has a moment of inertia πΌπΌ about its axis.

The twisted steel wire exerts a restoring torque πππ§π§ that is

proportional to the angular displacement ππ from the equilibrium position. We write πππ§π§ = βππππ , where ππ is a constant called the torsion constant.

Using the rotational analog of Newtonβs second law for a rigid body, β πππ§π§ = πΌπΌπΌπΌπ§π§ = πΌπΌππ2 ππ/πππ‘π‘ 2 , we find βππππ = πΌπΌπΌπΌ Fig. 5

A graph of the period ππ as a function of distance ππ from the center of mass for a 50cm-length slender rod pendulum.

or

ππ2 ππ ππ = β ππ 2 ππππ πΌπΌ

(9)

This equation is exactly the same as πππ₯π₯ = β(ππ βππ)π₯π₯ for

simple harmonic motion, with π₯π₯ replaced by ππ and ππ βππ

replaced by ππ βπΌπΌ. So we are dealing with a form of angular

simple harmonic motion. The angular frequency ππ and peri-

od ππ are given by ππ = οΏ½ππ βππ and ππ = 2πποΏ½ππβππ , respectively, with the same replacement:

β²ππβ² ππ = οΏ½ πΌπΌ

(10)

πΌπΌ ππ = 2πποΏ½ β²ππβ²

Fig. 6

(11)

A graph of the period ππ as a function of distance ππ from the center of mass for a 10cm-radius solid cylinder pendulum

2. Torsion Pendulum A restoring force on a body undergoing periodic motion originates 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.

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Fig. 7

The steel wire exerts a restoring torque that is proportional to the angular displacement ππ, so the motion is angular SHM.

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Physical Pendulum / Torsion Pendulum Ver.20170517

Equipment

1. List

Item(s)

PC / Software Data Analysis: Capstone

Qty.

1

Description

Records, displays and analyzes the data measured by various sensors.

Data acquisition interface designed for use with various Interface

1

sensors, including power supplies which provide up to 15 watts of power.

Measures the magnitude of force. Force Sensor

1

Range: β50N ~ 50N

Resolution: 0.03N Rotary Motion Sensor (RMS)

Slender Rod (or Long Rectangular Plate)

Spherical Cylinder (or Disk)

Lower Wire Clamp

(in the case)

Balance Disk

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1

Measures rotational or linear position, velocity and acceleration

Length: 500mm

Pivot Point (Holes):

Width: 20mm

60, 100, 144, 190, 230mm from center of gravity

1

Pivot Point (Holes): 30, 50, 70, 90mm from center of gravity

Upper Wire Clamp

Wires (3 ea)

1

1 set

Clamp wires.

Exerts a restoring torque when twisted. 1 set

Material: Steel Diameter: 0.8mm, 1.2mm, 1.6mm

1

Has a moment of inertia πΌπΌ = (1β2)ππππ2 about its axis.

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Physical Pendulum / Torsion Pendulum Ver.20170517

Item(s)

Qty.

A-shaped Base

Description

1 set

Support Rod (600mm)

Ruler

Provide stable support for experiment set-ups.

1

Measures length.

String / Scissors

Shared

Exerts a torque to twist a wire.

Electronic Balance

Shared

Measures mass.

2. Details

(1) Force Sensor It contains a small photogate sensor and an optical code Refer to the βCircular Motion and Centripetal Forceβ lab manual.

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 positioning or timing.

(2) Rotary Motion Sensor

The Rotary Motion Sensor is a bidirectional angle sensor designed to measure rotational or linear position, velocity

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.

and acceleration.

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Lab Manual

Physical Pendulum / Torsion Pendulum Ver.20170517

Procedure

Experiment 1. Physical Pendulum

(3) Run Capstone software.

(Slender Rod or Long Rectangular Plate) β  The interface automatically recognizes the RMS. (1) Set up equipment.

Mount the RMS on the support rod so that the shaft of the sensor is horizontal (parallel to the table).

β‘ Adjust the sample rate of measurement. - [Rotary Motion Sensor]: 100.00 Hz

(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

β’ Add a [Graph], and then select [Time(s)] for the π₯π₯-axis

pivot point is 230mm above the center of gravity.

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(4) Begin recording data.

Physical Pendulum / Torsion Pendulum Ver.20170517

β’ Repeat measuring times for all oscillations and find the period of oscillation. Also, calculate and record the theoretical

Click [Record] and then let the pendulum swing.

period of oscillation based on the length ππ from the pivot point to the center of gravity.

β  With the pendulum on the equilibrium position, click [Recπ‘π‘ππ (s)

ord] to begin recording data. 1

ππ = π‘π‘ππ β π‘π‘ππβ1 (s)

2 3

β‘ Gently start the pendulum swinging with a small amplitude

4 5

β¦

(within 5Β°). β’ After 5~6 oscillations, click [Stop] to end recording data.

(5) Find the period ππ of oscillation. β  Choose any reference point of measurement (for example, peaks or zero up-crossings).

β‘ Use [Show coordinateβ¦] to read off the time of the point.

ππaverage (s) ππtheory (s)

πΏπΏ2 + 12ππ2 ππ = 2πποΏ½ 12π¨π¨ππ

(7)

πΏπΏ = 500mm

ππ = 230, 190, 144, 100, 60mm

(6) Repeat measurement. Repeat steps (4) and (5) for the holes that are ππ = 230mm,

190mm, 144mm, 100mm, and 60mm from the center hole. (7) Plot a ππ-ππ graph.

Using your results in step (6), plot a ππ-ππ graph and com-

pare it with Fig. 5 in Theory section.

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Physical Pendulum / Torsion Pendulum Ver.20170517

Experiment 3. Torsion Constant

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 ππ. Also, use Eq. (7) to calculate ππ under the condition of minimum ππ, 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

β  Slip the lower wire clamp onto the support rod.

β‘ Clamp the RMS at the top of the support rod so that the shaft of the sensor is vertical.

Experiment 2. Physical Pendulum (Solid Cylinder) β’ Align the guide of the upper wire clamp with the slot of the Repeat the procedure of expt. 1 using a disk.

shaft of the RMS. Slide the upper wire clamp onto the shaft and firmly tighten the thumbscrew.

β£ 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.

β€ Connect the sensors to the interface.

ππ2 + 2ππ2 ππ = 2πποΏ½ 2π¨π¨ππ

(8)

ππ = 100 mm

ππ = 90, 70, 50, 30 mm

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β₯ Wind a string around the largest pulley.

Lab Manual

Physical Pendulum / Torsion Pendulum Ver.20170517

(2) Set up Capstone software.

β  Configure the Rotary Motion Sensor. - Click the RMS icon and then click the properties button (βΌ). - 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.

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.

β‘ Configure the Force Sensor. - Click the FS icon and then click the properties button (βΌ).

Caution

- Check [Change Sign]. (The sign of FS data is initially negative for the pulling force.)

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 causes 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.

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Physical Pendulum / Torsion Pendulum Ver.20170517

β’ Configure calculator.

(4) Begin recording data.

Define the torque ππ as below, where β[Force(N)]β is meas-

ured data by the Force Sensor, and βrβ is the radius of the 3

rd

(largest) pulley (= 24mm).

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)-β ).

β£ Add a graph. Select [Rotary Motion Sensor β Angle(rad)] for the π₯π₯-axis

and [ππ(Nm)] (defined in stepβ’) for the π¦π¦-axis.

(5) Analyze your graph. Find the torsion constant ππ. β  Click [Select range(s) β¦] icon and then drag the data (3) Zero the Force Sensor.

range of interest.

NOTE To zero the sensor, press the [Zero] button on it WITH NO FORCE exerted on the sensor hook.

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β‘ Click [Select curve fits β¦ ] and select [Linear: mt+b] to find linear fit for selected data points. The torsion constant ππ

is equal to the slope of the ππ-ππ graph.

Physical Pendulum / Torsion Pendulum Ver.20170517

Use the setup detailed in expt. 3. Remove the string and attach the balance disk to the 3-step pulley with the thumbscrew. (Be careful not to attach the disk directly on the shaft without the pulley.)

(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.

(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 120~180Β° and

release it. Keep recording data for about 5-6 oscillations and stop recording data.

Measure the radius and the mass of the balance disk and calculate the theoretical value of πΌπΌ. (Suppose the balance

disk is a perfect solid cylinder and apply the relationship πΌπΌ = (1β2)ππππ

Determine the time for each period of oscillation and verify Eq. (11).

2 .)

πΌπΌ ππ = 2πποΏ½ ππ

(11)

(5) Change the wire and repeat step (4).

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Result & Discussion

Your TA will inform you of the guidelines for writing the laboratory report during the lecture.

End of LAB Checklist