objective theory - yonsei universityphylab.yonsei.ac.kr/exp_ref/105_rigidbody_eng.pdfgeneral physics...

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

Lab Manual

The Motion of a Rigid-Body Ver.20180410

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) / 16

[International Campus]

The Motion of a Rigid-Body (Combined Translation and Rotation)

Investigate the motion of various rigid bodies rolling without slipping down an inclined plane.

Assume that a body has a perfectly definite and unchanging

shape and size. We call this idealized model a rigid body.

When we analyze the motion of a rotating body, it is important

to consider how the mass of the body is distributed. Thus we

regard the body as a rigid body.

Fig. 1 A rigid body rotating at an angular speed 𝜔𝜔.

We think of a body as being made up of a large number of

particles, with masses 𝑚𝑚1,𝑚𝑚2,⋯ ,𝑚𝑚𝑖𝑖 at distance 𝑟𝑟1, 𝑟𝑟2, ⋯ , 𝑟𝑟𝑖𝑖

from the axis of rotation (Fig.1). When a rigid body rotates

with angular speed 𝜔𝜔 about a fixed axis, the speed 𝑣𝑣𝑖𝑖 of the

𝑖𝑖th particle is given by

𝑣𝑣𝑖𝑖 = 𝑟𝑟𝑖𝑖𝜔𝜔 (1)

The kinetic energy of the 𝑖𝑖th particle can be expressed as

12𝑚𝑚𝑖𝑖𝑣𝑣𝑖𝑖2 =

12𝑚𝑚𝑖𝑖𝑟𝑟𝑖𝑖2𝜔𝜔2 (2)

The total kinetic energy of the body is the sum of the kinetic

energies of all its particles:

𝐾𝐾 = �12𝑚𝑚𝑖𝑖𝑟𝑟𝑖𝑖2𝜔𝜔2

𝑖𝑖

=12��𝑚𝑚𝑖𝑖𝑟𝑟𝑖𝑖2

𝑖𝑖

�𝜔𝜔2 (3)

The quantity in parentheses is denoted by 𝐼𝐼 and is called

the moment of inertia of the body for this rotation axis.

𝐼𝐼 = �𝑚𝑚𝑖𝑖𝑟𝑟𝑖𝑖2𝑖𝑖

(4)

The SI unit of 𝐼𝐼 is kg ∙ m2.

Objective

Theory

----------------------------- Reference --------------------------

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

9.4 Energy in Rotational Motion

10.3 Rigid-Body Rotation about a moving axis

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


Lab Manual




Eq. (4) shows that the moment of inertia depends on how

the body’s mass is distributed in space. For a body with a

given rotation axis and a given total mass, the greater the

distance from the axis to the particles that make up the body,

the greater the moment of inertia.

As the (inertial) mass represents the tendency of bodies to

keep moving in translational motion, the moment of inertia

represents the tendency of bodies to keep rotating in rota-

tional motion. The greater a body’s (inertial) mass, the harder

it is to change the state of translational motion of the body.

Similarly, the greater a body’s moment of inertia, the harder it

is to start the body rotating. For this reason, 𝐼𝐼 is also called

the rotational inertia.

Figure 2 shows moments of inertia of various bodies.

In term of moment of inertia 𝐼𝐼, the rotational kinetic energy

𝐾𝐾 of a rigid body is

𝐾𝐾 =12 𝐼𝐼𝜔𝜔

2 (5)

To find the kinetic energy of a rigid body that has both trans-

lational and rotational motions, we again imagine the body to

be made up of particle. In Fig. 3, 𝒗𝒗��⃗ 𝑖𝑖 of this particle relative to

an inertial frame is the vector sum of 𝒗𝒗��⃗ cm of the center of

mass and 𝒗𝒗��⃗ 𝑖𝑖′ of the particle relative to the center of mass.

𝒗𝒗��⃗ 𝑖𝑖 = 𝒗𝒗��⃗ cm + 𝒗𝒗��⃗ 𝑖𝑖′ (6)

Fig. 2 Moments of inertia of Various bodies

The kinetic energy of this particle in the inertial frame is

𝐾𝐾𝑖𝑖 = (1 2⁄ )𝑚𝑚𝑖𝑖𝑣𝑣𝑖𝑖2 = (1 2⁄ )𝑚𝑚𝑖𝑖(𝒗𝒗��⃗ 𝑖𝑖 ⋅ 𝒗𝒗��⃗ 𝑖𝑖). Substituting Eq. (6) into

this, we get

𝐾𝐾𝑖𝑖 =

12𝑚𝑚𝑖𝑖�𝒗𝒗��⃗ cm + 𝒗𝒗��⃗ 𝑖𝑖

′� ⋅ �𝒗𝒗��⃗ cm + 𝒗𝒗��⃗ 𝑖𝑖′�

=12𝑚𝑚𝑖𝑖�𝑣𝑣cm2 + 2𝒗𝒗��⃗ cm ⋅ 𝒗𝒗��⃗ 𝑖𝑖

′ + 𝑣𝑣𝑖𝑖′2�

(7)

The total kinetic energy 𝐾𝐾 is the sum ∑𝐾𝐾𝑖𝑖 for all particles

making up the body.

𝐾𝐾 = �𝐾𝐾𝑖𝑖

=12 ��𝑚𝑚𝑖𝑖� 𝑣𝑣cm2 + 𝒗𝒗��⃗ cm ⋅ ��𝑚𝑚𝑖𝑖𝒗𝒗��⃗ 𝑖𝑖

′� + ��12𝑚𝑚𝑖𝑖𝑣𝑣𝑖𝑖′

2� (8)

In the first term, ∑𝑚𝑚𝑖𝑖 is the total mass 𝑀𝑀. The second term

is zero because ∑𝑚𝑚𝑖𝑖𝒗𝒗��⃗ 𝑖𝑖′ is 𝑀𝑀 times the velocity of the cen-

ter of mass relative to the center of mass, and this is zero by

definition. The last term is (1 2⁄ )𝐼𝐼cm𝜔𝜔2 from the steps (2) to

(4), where 𝐼𝐼cm is the moment of inertia with respect to the

axis through the center of mass. So Eq. (8) becomes

𝐾𝐾 =12𝑀𝑀𝑣𝑣cm

2 +12 𝐼𝐼cm𝜔𝜔

2 (9)

Fig. 3 A rigid body with both translation and rotation


Lab Manual




The motion of rolling wheel is a case of combined translation

and rotation (Fig. 4). When the wheel rolls without slipping,

the velocity 𝒗𝒗��⃗ 1′ of the point contact relative to the center of

mass has the same magnitude but opposite direction as the

center-of-mass velocity 𝒗𝒗��⃗ cm. If the radius of the wheel is 𝑅𝑅

and its angular speed about the center of mass is 𝜔𝜔, then

𝑣𝑣cm = 𝑅𝑅𝜔𝜔 (10)

The velocity of a rigid body which rolls without slipping down

a ramp can be expressed using conservation of energy. The

total mechanical energy of a body with height ℎ is

𝐸𝐸 = 𝐾𝐾 + 𝑈𝑈 =12𝑀𝑀𝑣𝑣cm


2 + 𝑀𝑀𝘨𝘨ℎ (11)

Figure 5 shows that a solid ball of mass 𝑀𝑀 and radius 𝑅𝑅

starts from rest and rolls without slipping down the track. Us-

ing equations (12) to (15), the speed 𝑣𝑣cm,2 at 𝑃𝑃2 after de-

scending a vertical distance ℎ is given by Eq. (16).

𝐾𝐾1 + 𝑈𝑈1 = 𝐾𝐾2 + 𝑈𝑈2 (12)

0 + 0 + 𝑀𝑀𝘨𝘨ℎ =12𝑀𝑀𝑣𝑣cm,2

2 +12 𝐼𝐼cm𝜔𝜔2

2 + 0 (13)

𝐼𝐼cm =25𝑀𝑀𝑅𝑅

2 (Solid sphere) (14)

𝜔𝜔2 =𝑣𝑣cm,2

𝑅𝑅 (15)

𝑣𝑣cm,2 = �107 𝘨𝘨ℎ (16)

Fig. 4 The motion of a rolling wheel is the sum of the trans-

lational motion of the center of mass plus the rota-tional motion of the wheel around the center of mass

When the ball is at the top (𝑃𝑃3) of the circular part (radius 𝑟𝑟)

of the track, the speed 𝑣𝑣cm,3 of the ball becomes

𝑣𝑣cm,3 = �107 𝘨𝘨(ℎ − 2𝑟𝑟) (17)

The ball makes a complete loop-to-loop on the circular part

of the track if

𝑀𝑀𝑣𝑣cm,32

𝑟𝑟 = 𝑀𝑀𝘨𝘨 or 𝑣𝑣cm,32 = 𝘨𝘨𝑟𝑟 (18)

Substituting Eq. (17) into (18) gives the minimum height for

the ball not to fall off the track:

ℎ =2710 𝑟𝑟

(19)

We can also express the moments of inertia of the bodies in

Fig. 2 as 𝐼𝐼cm = 𝑐𝑐𝑀𝑀𝑅𝑅2, where 𝑐𝑐 is a number that depends on

the shape of the body (solid sphere: 𝑐𝑐 = 2 5⁄ , solid cylinder:

𝑐𝑐 = 1 2⁄ , hollow cylinder(𝑅𝑅2:𝑅𝑅1 = 𝑛𝑛 ∶ 1) : 𝑐𝑐 = (𝑛𝑛2 + 1) 2𝑛𝑛2⁄ ).

Then, from Eqs. (12), (13), (15), and 𝐼𝐼cm = 𝑐𝑐𝑀𝑀𝑅𝑅2, the speed

𝑣𝑣cm,2 after descending a vertical distance ℎ is given by

𝑣𝑣cm,2 = � 2𝘨𝘨ℎ1 + 𝑐𝑐

(20)

Fig. 5 The motion of a solid sphere


Lab Manual




1. List

Item(s) Qty. Description

PC / Software Video Analysis: SG PRO

1 Records, displays and analyzes videos.

Camera

1 Feeds or streams its image in real time to a computer.

Tripod

1 Supports a camera.

Screen

1 PVC foam board, white, 900 × 1200mm

Loop-the-Loop Track

1 Sphere balls run on this track.

Straight Track

1 Cylinders run on this track.

Solid Sphere Ball

1 Solid sphere rigid-body

Cylinder Set

1 set Cylinder rigid-bodies

A-shaped Base Support Rod (600mm) Support Rod (300mm) Multiclamp

1 1 1 2

Provide stable support for experiment set-ups.

Equipment


Lab Manual




Item(s) Qty. Description

Clothespin

1 Marks the starting position of the ball on the track.

Vernier Caliper

1 Measures external diameter, internal diameter, or depth of an object with a precision to 0.05mm.

Ruler

1 Measures distance.

2. Details

(1) Video Camera

The video camera feeds or streams its image in real time to

a computer. It offers 640 × 480 pixels resolution with 30

frames per second.

(2) SG PRO: Video Analysis Software

The SG PRO software records, displays and analyzes vide-

os. It can manually or automatically track the position of ob-

jects on each frame in the video clip.

(3) Vernier Caliper

The Vernier caliper measures external, internal diameter or

depth of an object with a precision to 0.05mm.

① 22 mm is to the immediate left of the zero on the vernier

scale. Hence, the main scale reading is 22 mm.

② Look closely for and alignment of the scale lines of the

main scale and vernier scale. In the figure, the aligned (13th)

line corresponds to 0.65 mm (= 0.05 × 13).

③ The final measurement is given by the sum of the two

readings. This gives 22.65 mm (= 22 + 0.65).


Lab Manual




Setup 1. Equipment setup

Connect the camera to the USB port of your PC.

① Motion of Balls: Use the Loop-the-Loop Track.

② Motion of Cylinders: Use the Straight Track.

Setup


Lab Manual




Setup 2. Software Setup

(1) Run SG PRO software.

(2) Adjust the camera position.

To begin monitoring, click [Start Monitor] icon in the [화면조

정] (Start Monitor) palette of [실험영상] (Video) tab. Adjust the

camera position. Click [Stop Monitor] icon to finish monitoring.

(3) Start / Stop recording videos.

In [영상녹화] (Video Record) palette, set [프레임] (Frame

Rates) of the video to [30 fps] and click [Start Recording] or

[Stop Recording] icon.

NOTE

In this experiment, you will use the auto-tracker of the

video analysis software to automatically track objects.

This eliminates the need to mark every frame manually,

thus speeding up the tracking process.

If it looks complicated to use auto-tracker, you can also

manually track objects. See [Appendix] for manual track-

ing prcedure.

NOTE

How to adjust the camera position

① Distance between Camera and Track

- As above

② Camera Height/Position

- Top of loop-the-loop → on the middle horizontal line

- 3rd (from left) white point → on the middle vertical line

③ Camera Tilt

- Make sure the support rod is vertical.

- Please refer to step (8) for complete adjustment.

① Distance between Camera and Track

- As above

② Camera Height/Position

- 2nd (from left) white point → on the middle

③ Camera Tilt

- Make sure the support rod is vertical.

- Please refer to step (8) for complete adjustment.


Lab Manual




(4) Export the recorded video as a movie file.

We will extract a video clip (a subset of frames) of interest in

the recorded video in order to reduce the file size.

① Import the video temporarily stored in memory.

② All frames are initially selected (highlighted in yellow).

③ Drag the slider to scan the video. Find the first frame of

the video clip of interest.

④ Click |◀.

⑤ The previous frames become dehighlighted.

⑥ In the same way, find the last frame and click ▶|.

⑦ The next frames become dehighlighted.

⑧ Save the highlighted frames as a movie file.

(5) Analyze the video.

Click [분석] (Analysis) tab.

(6) Open the movie file.

Enter the path and file name of the movie in the [영상파일]

(Files), or click [Folder] icon and find the movie file.

Click [Open] icon in the [분석] (Analysis) palette.

(7) Set the position of the origin.

Display the coordinate by clicking [원점설정] (Origin) button

in the [설정] palette. Click a desired location in the main video

view. Finish setting by clicking [원점설정] (Origin) again.


Lab Manual




(8) Calibrate the video scale.

Enter [450mm] in the [설정] (Properties) palette. Click [스케

일설정] (Calibration), and then successively click two calibra-

tion points in the main video view.

Measure distances between calibration points to check the

camera is tilted. Click [길이측정] (Length) and then succes-

sively click calibration points. If not all of them are equal, it

means the video is distorted, i.e. the camera is at a tilt. (You

must readjust the camera position and record the video

again.)

Check [좌표보기] (Coordinate) in the [보기] (View) palette,

and make sure the coordinate system is modified in order to

maintain the assigned real coordinate.

(9) Adjust a search area.

Auto-tracker works by selecting one template color of a fea-

ture of interest and then searching each frame for the best

match to that template color. If the similarly colored objects

are in view, auto-tracker will mark on the wrong position. For

this reason, auto-tracker limits its search for a match in each

frame to a user-defined rectangular search area.

Click [분석영역설정] (Search Area) icon in the [분석] (Analy-

sis) palette, and then adjust the rectangular search area. Fin-

ish the adjustment by clicking [분석영역설정] icon again.

NOTE

Auto-tracking: Follow steps (9) to (12).

Manual-tracking: Skip steps (9) to (12), & see [Appendix].


Lab Manual




(10) Select a target color.

Before using auto-tracker, scan through the video and verify

that the feature of interest is visible and reasonably con-

sistent in all frames. If not, re-record a video or use manual

track process.

① Set a target object.

Check the checkbox of the 1st row in the [대상물체] (Object)

palette and then click [▼] for configuration of color.

[컬러설정] (Color Properties) window will appear.

② Select a target color.

Click the video feature of interest on the video.

③ Make sure only the pixels of interest are selected.

The best match pixels become highlighted at the target posi-

tion on the lower black screen.

NOTE

If the following message appears, select any other frame

by dragging the slider.


Lab Manual




④ Adjust the color level, if required.

If the highlighted screen includes wrong pixels, you need to

adjust the color level. (Those wrong pixels interfere auto-

tracking. Adjust [Hue], [Saturation], and [Lightness] so the

highlighted screen includes only the pixels of the object.

⑤ Check all frame.

By dragging the slider of the frame bar, verify that only the

pixels of interest are highlighted in all frames. If not, repeat

adjusting the color level.

⑥ Click [저장] (Save) to save the configuration.

(11) Begin auto-tracking.

Click [분석시작] (Search) icon. Auto-tracker will search and

mark each frame as it goes.

(12) Correct any track errors.

Manually correct track errors.

① Find the frame with a track error.

You can find the frame that the target marker is linked to by

clicking the marker on the video view or using the slider.


Lab Manual




② Delete the marker.

Click [제거] (Delete) icon. Then a pop-up window [00 프레

임]의 대상물체를 삭제 하시겠습니까?] (Delete the marker of

the frame 00?) will appear. Click [예] (Yes).

③ Manually add a new marker.

In the same frame, click [추가] (Add) icon, and then mark the

new position using the mouse on the video view.

(13) Plot the tracks.

① Click [결과] (Result) tab to display tables and graphs of

the track data.

② Whenever you modify track positions in step (12), you

have to click [분석데이터 다시 읽기] (Data Update) to update

track data.

③ You can export track data to EXCEL by clicking [내보내

기] (Export). The EXCEL may show numeric values in the

form of currency (software bug). Change [표시형식] (Format

Cell) from [통화] (Currency) to [일반] (General) in EXCEL.

If you have any problem using the SG PRO software, please

close and run the software again. (It has several bugs.)


Lab Manual




Experiment 1. Motion of a Solid Sphere, Part 1

(1) Set up equipment.

Use the Loop-the-Loop Track to observe the rolling motion

of a sphere ball. Remove the Straight Track and place a white

screen behind the Loop-the-Loop Track.

(2) Measure mass 𝑀𝑀 and radius 𝑅𝑅 of the solid sphere ball.

𝑀𝑀 = _________

𝑅𝑅 = _________

(3) Record a video.

Release the ball from rest at any appropriate height.

(Repeat more than three times.)

(4) Analyze the video.

See [Setup 2. Software Setup]. Using coordinates, calculate

ℎ, 𝑟𝑟, 𝑣𝑣cm,2, and 𝑣𝑣cm,3.

Use the followings to calculate 𝑣𝑣cm on the loop.

Verify the following equations.

𝑣𝑣cm,2 = �107 𝘨𝘨ℎ (16)

𝑣𝑣cm,3 = �107 𝘨𝘨(ℎ − 2𝑟𝑟) (17)

If you cannot calculate 𝑣𝑣cm,3 at the top (𝑃𝑃3) of the loop,

choose any point such as 𝑃𝑃4. If the height of 𝑃𝑃4 is ℎ′, then

𝑣𝑣cm,4 = �107𝘨𝘨(ℎ − ℎ′)

Find and compare the total mechanical energy of the ball at

every point, and verify the conservation of energy

𝐸𝐸 = 𝐾𝐾 + 𝑈𝑈 =12𝑀𝑀𝑣𝑣cm


2 + 𝑀𝑀𝘨𝘨ℎ (11)

Procedure


Lab Manual




Experiment 2. Motion of a Solid Sphere, Part 2

We will find the minimum height for which the ball makes a

complete loop-the-loop on the circular part of the track.

(1) Follow the setup of experiment 1.

(2) Mark a starting position. (Use a clothespin.)

(3) Find the minimum height.

Vary the starting position and find the minimum height for

which the ball makes a complete loop-the-loop on the circular

part of the track. (Use your senses of sight and hearing.)

(4) Record a video.

Release the ball at this height.

(5) Analyze the result.

Verify the following equations.

𝑀𝑀𝘨𝘨 = 𝑀𝑀𝑣𝑣cm,32

𝑟𝑟 or 𝑣𝑣cm,32 = 𝘨𝘨𝑟𝑟 (18)

ℎ =2710 𝑟𝑟

(19)

Experiment 3. Motion of Solid or Hollow Cylinders

Use the Straight Track to observe the rolling motion of cylin-

ders.

(1) Measure inner radius 𝑅𝑅1, outer radius 𝑅𝑅2, & mass 𝑀𝑀.

(2) Mount the Straight Track and place the white screen.

Do NOT remove the Loop-the-Loop Track. It could be warped.

(3) Start measurement and analyze your results.

Verify equation (20) for all cylinders.

𝑣𝑣cm,2 = � 2𝘨𝘨ℎ1 + 𝑐𝑐

(20)

Solid cylinder: 𝑐𝑐 = 1 2⁄

Hollow cylinder (𝑅𝑅2:𝑅𝑅1 = 𝑛𝑛 ∶ 1): 𝑐𝑐 = (𝑛𝑛2 + 1) 2𝑛𝑛2⁄

Find and compare the total mechanical energy of the ball at

every point, and verify the conservation of energy.


Lab Manual




How to do manual tracking

① Check the checkboxes of the 1st row in the [대상물체]

(Object) palette. (We will track one object.)

② Select any frame of interest by moving the slider.

③ Click the [추가] (Add) icon of the 1st row.

④ Mark the position of the object using the mouse.

⑤ Repeat steps ② to ④ for all frames.

Markers will appear as below.

When any marker is on the wrong position, follow the steps

below to delete the wrong marker.

① Find the target marker you want to delete.

② Find the frame that the target marker is linked to.

③ Click [제거] (Delete) icon of the 1st row.

④ Click [예] (Yes) button in the popup window.

When you finish tracking objects for all frames, you can see

the markers as below.

Appendix

NOTE

When markers do not appear (due to software bugs),

① Click [Auto-tracker] icon and wait a second until auto-

tracking process ends. (Ignore the result.)

② Open the movie file AGAIN.

③ Repeat the manual tracking process again.


Lab Manual




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.

□ Keep the White Screens together at the front of the laboratory.

□ Place the Camera and Tripod assembly on any safe place.

□ Leave the equipment assembled.

Result & Discussion

End of LAB Checklist

objective theory - yonsei universityphylab.yonsei.ac.kr/exp_ref/105_rigidbody_eng.pdfgeneral physics...

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