PASCO ME-9833 User Manual

Instruction Manual

Manual No. 012-10461A

Physical Pendulum

Set

Model No. ME-9833

Model No. ME-9833 Physical Pendulum Set

Equipment ................................................................................................... 3

Introduction................................................................................................. 4

Basic Setups ............................................................................................... 4

Background Information............................................................................ 5

Experiment 1: Parallel Axis Theorem ....................................................... 7

Experiment 2: Minimum Period for an Oscillating Bar ......................... 11

Experiment 3: Moment of Inertia Based on Period of Oscillation........ 15

Experiment 4: Use a Physical Pendulum to Measure ‘g’ ...................... 19

Appendix A: Specifications ......................................................................23

Appendix B: DataStudio Instructions......................................................23

Appendix C: Technical Support, Copyright, Warranty ..........................24

Appendix D: Teacher’s Notes...................................................................25

Physical Pendulum Set Model No. ME-9833

Physical Pendulum Set

Model No. ME-9833

Equipment

Included Equipment Recommended Equipment

1. Pendulum Bar, 28 cm Rotary Motion Sensor (PS-2120 or CI-6538)

2. Offset Hole Base and Support Rod (see PASCO catalog)

3. Solid Disk Super Pulley with Clamp (ME-9448A)

4. Thin Ring Mass and Hanger Set (ME-8979)

5. Thick Ring PASCO Computer Interface (see PASCO catalog)

6. Irregular Shape Vernier Caliper (SF-9711)

7. Mounting Screws (qty. 6) (not shown) String or Thread (see PASCO catalog)

Balance (see PASCO catalog

Model No. ME-9833 Physical Pendulum Set

Introduction

The Physical Pendulum Set consists of six parts: pendulum bar (28 cm), offset hole, solid disk, thin ring, thick ring, and irregular shape. This set of objects allows the study of physical pendula, moments of inertia, and the parallel axis theorem. It includes six mounting screws for attaching the physical pendula to a Rotary Motion Sensor in order to measure the object’s acceleration due to an applied torque, or the object’s period of oscillation when the pendulum swings freely.

Basic Setups

Using the Rotary Motion Sensor to Measure Moment of Inertia

You can use the Rotary Motion Sensor (RMS) to measure the motion of a rotating object as it is accelerated by a net torque. The ratio of net torque to angular acceleration is the object’s moment of inertia.

• Mount the Rotary Motion Sensor horizontally on a support rod with the three step pulley on top.

• Arrange the three step pulley on the shaft on top of the sensor so that the largest diameter step is nearest to the sensor.

• Mount a Super Pulley with Clamp to the platform area at the

Figure 1: Top view - Adjust the angle of the Super Pulley.

end of the sensor opposite to the support rod.

• Connect a string to one of the steps on the three step pulley and drape the string over the Super Pulley. Adjust the angle of the Super Pulley with Clamp so that the string is tangent to the step on the three step pulley (see Fig. 2).

• Put the object to be measured onto the shaft of the sensor above the three step pulley. Secure it in place with one of the mounting screws.

• Attach a mass hanger to the end of the string that is draped

Figure 2: Moment of Inertia setup

over the Super Pulley.

Using the Rotary Motion Sensor to Measure Period of Oscillation

You can use the Rotary Motion Sensor to measure the period of oscillation of the objects in the Physical Pendulum Set.

• Mount the Rotary Motion Sensor horizontally on a support rod so that the three step pulley is on one side (oriented vertically).

• Remove the mounting screw and the three step pulley.

• Use a mounting screw to attach the object to be measured to the shaft of the sensor.

Object

Figure 3: Period of oscillation setup

Physical Pendulum Set Model No. ME-9833

Setting Up the Rotary Motion Sensor and Interface

Connect the Rotary Motion Sensor to the interface and connect the interface to a computer. For the PASPORT Rotary Motion Sensor, connect the sensor’s plug to a compatible PASPORT interface (e.g., USB Link, PowerLink, Xplorer, Xplorer GLX). For the ScienceWorkshop Rotary Motion Sensor, connect the yellow plug to digital channel 1 and the black plug to digital channel 2 of a compatible ScienceWorkshop interface (e.g., ScienceWorkshop 500 or ScienceWorkshop 750).

In general, the set up of the DataStudio program will be for measuring angular acceleration or for measuring period. To determine angular acceleration, find the slope of the plot of data on an angular velocity versus time graph display. To determine period, use the Smart Cursor to find the time for ten oscillations of the physical pendulum and then divide the total time for ten oscillations by ten.

Figure 4: Smart Cursor

For more information about setting up the DataStudio program, refer to the Appendix.

Background Information

Pendulum Period

If a body is suspended from a fixed point other than its center of mass and set in motion, it has a periodic motion that is very nearly simple harmonic motion. The period of the angular motion depends on the pull of gravity and the moment of inertia of the body.

For that reason, the physical pendulum is a useful device for determining the acceleration due to gravity and the moment of inertia. The analytical relationships for the period, T, are as follows:

for a simple pendulum:

T2π

and for a physical pendulum: where

----=

T2π

------------------=

MgL

T is the period L is the length of the simple pendulum g is the acceleration due to gravity I is the moment of inertia about an axis through the pivot point M is the mass of the pendulum, and

is the distance from the pivot point to the center of gravity, cg

The first equation is seen to be a special case of the second if ML

formulas give good results if the angular amplitude is small.

Torque and Moment of Inertia

mgsinθ

Figure 5: Physical pendulum

is substituted for I. These

For an object accelerating about an axis, the torque is the product of the force and the moment arm (the distance from the pivot point perpendicular to the line of action of the force):

Model No. ME-9833 Physical Pendulum Set

τ Iα r ⊥F==

For the basic setup for measuring moment of inertia shown in Figure 2, where T is the

τ rT=

tension in the string and r is the radius of the step pulley to which the string is attached. The net

ΣFmgT– ma==

force is the difference between the tension, T, and the weight, mg, of the hanging mass. Therefore, the tension, is where “a” is related to the angular acceleration, α, by

Tmgma–=

a = rα. The torque becomes:

τ rT rmg rma– rmg rm rα()–== =

τ rm g rα–()=

where the angular acceleration is measured by the Rotary Motion Sensor.

Parallel Axis Theorem

For a physical pendulum, the moment of inertia about an axis through the point of suspension can

be found using the parallel axis theorem, where I

pivotIcg

2 cg

is the moment of inertia

pivot

about the point of suspension, Icg is the moment of inertia about the center of mass, m is the mass of the pendulum, and Lcg is the perpendicular distance from the pivot point to the center of mass. In other words, the moment of inertia about a parallel axis is the moment of inertia about the center of

mass plus the moment of inertia of the entire object treated as a point mass at the center of gravity.

Description of the Experiments

1.Parallel axis theorem: Measure the moment of inertia, Icg, around the center of gravity of an object, and the moment of inertia, I

center of mass. Confirm that I

pivot

, around a pivot point of the object that is not through the

pivot

= Icg + ML

where m is the mass of the object and Lcg is the

distance from the pivot point to the center of gravity.

2.Minimum period of a bar: Given a bar of length, L, calculate Lcg, the distance from the pivot point to the center of gravity that would give the minimum period of oscillation for the bar.

Measure the distance that gives the minimum period and compare it to the calculated result.

3.Determine moment of inertia: Use the period, T, of a physical pendulum to calculate the moment of inertia, I. Compare the calculated value to the theoretical value for I.

4.Acceleration due to gravity: Measure the acceleration due to gravity, g, using period, T, moment of inertia, I, mass, m, and the distance from the pivot point to the center of gravity, Lcg.

Physical Pendulum Set Model No. ME-9833

Experiment 1: Parallel Axis Theorem

Equipment

Item

PASCO Interface and DataStudio software Base and Support Rod

Rotary Motion Sensor String or thread

Irregular Shape from Physical Pendulum Set Mass and Hanger Set

Super Pulley with Clamp Vernier calipers

Balance

Item

Theory

The moment of inertia about a parallel axis, I

, is the moment of inertia about the center of

pivot

gravity, Icm, plus the moment of inertia of the entire object treated as a point mass at the center of gravity:

pivotIcm

where L is the perpendicular distance from the center of gravity (cog) to the pivot

point.

Procedure

Use the Rotary Motion Sensor to measure the angular acceleration of a rotating irregular shaped object when it rotates about its center of gravity due to a net torque, and also when it rotates about a parallel axis a distance “L” away from the center of gravity. Use the angular acceleration and the known applied torque to determine the moment of inertia about the center of gravity and the moment of inertia about the parallel axis.

Equipment Setup: Center of Gravity

1. Set up the Rotary Motion Sensor horizontally on a support rod so that the three step pulley is

on top. Start the DataStudio program on the computer and connect the sensor to the computer interface.

2. Mount a Super Pulley with Clamp to the platform area at the end of the sensor opposite to the support rod.

3. Connect a string to one of the steps on the three step pulley and drape the string over the Super Pulley. Adjust the angle of the Super Pulley with Clamp so that the string is tangent to the step on the three step pulley.

4. Use calipers to carefully measure the diameter of the step to which you attached the string. Calculate the radius of the step and record it in the data table (Table 1).

5. Measure and record the mass, M, of the irregular shape object to be measured.

Model No. ME-9833 Physical Pendulum Set

6. Put the object to be measured onto the shaft of the sensor above the three step pulley. Secure it in place with one of the mounting screws.

7. Measure and record the mass, m, of a mass hanger.

8. Attach the mass hanger to the end of the string that is draped over the Super Pulley.

Data Recording: Center of Gravity

1. Wind the string about the step pulley so that

the mass hanger is just below the Super Pulley. Hold the irregular shape in place.

2. Start recording data and then release the irregular shape so that it rotates freely.

3. Stop recording data when the mass hanger reaches its lowest point.

Mounting screw

Figure 1-1: Irregular shape mounted on the RMS at the center of gravity

4. Repeat the process a total of three times.

Analysis: Center of Gravity

Find the moment of inertia about the center of gravity.

1. Set up a graph display in the software of angular velocity versus time.

2. Select a region of the first run of angular velocity data and select ‘Linear Fit’ from the

‘Fit’ menu.

3. Record the value of the slope as the first value of angular acceleration around the center of gravity.

4. Repeat the process for the other two runs of data.

5. Find the average angular acceleration, α

, and record it in the data table.

6. Calculate the net torque, τ = rm(g - rα). Calculate the moment of inertia about the center of gravity by dividing the net torque by the average angular acceleration. Record the result

--------

as the moment of inertia about the center of gravity, Icg. ( )

Equipment Setup: Parallel Axis

1. Temporarily remove the irregular shape from the top of the Rotary Motion Sensor.

Physical Pendulum Set Model No. ME-9833

2. Use calipers to carefully measure the distance, Lcg, from the center of gravity to

Measure L

one of the other pivot points on the irregular shape. Record the measurement as Lcg, the

distance from the center of gravity to the pivot point.

3. Replace the irregular shape on the Rotary Motion Sensor by attaching it with one of the mounting screws through the pivot point that you used for the measurement.

Data Recording: Parallel Axis

1. Wind the string about the step pulley so that the

mass hanger is just below the Super Pulley. Hold the irregular shape in place.

Figure 1-2: Irregular shape mounted through a parallel axis

2. Start recording data and then release the irregular shape so that it rotates freely.

3. Stop recording data when the mass hanger reaches its lowest point.

4. Repeat the process a total of three times.

Analysis: Parallel Axis

Find the moment of inertia about the parallel axis.

1. Use the same setup in the software to find the average angular acceleration.

2. Calculate the net torque (τ = rm(g - rα)). Calculate the moment of inertia about the

parallel axis by dividing the net torque by the average angular acceleration the results as the moment of inertia about the parallel axis, I

3. Calculate ML

, the moment of inertia about the parallel axis of the irregular shape as if

pivot

all its mass is concentrated at its center of gravity. Use the distance from the pivot point to the center of gravity as L

. Record

4. Calculate the sum of Icg and ML parallel axis, I

pivot

5. Find the percent difference between I percent difference. Set up a graph display in the software of angular velocity versus time.

Extensions

. Compare the sum to the moment of inertia about the

and the sum of Icg and ML

pivot

and record the

+ 21 hidden pages

PASCO ME-9833 User Manual

Model No. ME-9833

Table of Contents