REPORT Rotation

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SIMULATIONS RELATED TO MOTION OFPARTICLES IN ROTATING SYSTEMS

Birla Institute of Technology and Science, Pilani

K.K. Bir la Goa Campus

18 Apr il 2011


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A

Repor t onSIMULATIONS RELATED TO MOTION OF

PARTICLES IN ROTATING SYSTEMS

Submitted in partial fulfillment for the completion of the course

PHY C231 Physics Project Lab

Prepar ed by: Under the supervision of:

Akshat Agha 2009A4PS380G Dr. Toby Joseph

Sahil Gupta 2009 A4PS393 G


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Acknowledgement

We would like to take this opportunity to thank our instructor Dr. Toby Joseph

for giving us an opportunity to undertake this study. We are thankful to him for

constantly supporting and guiding us throughout this project. Also, we are

thankful to our friends who helped us in making the code and rectifying it.

Thank You!


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Table of Contents

Acknowledgements 3

Abstract 4

1. Introduction 62.

Rotating Disc 7

3. Rotating Cone 104. The MATLAB Code- Rotating disc 125. Observations and Findings 14

5.1.CASE 15.1.1. Projection Point at R=1m 15

5.1.2. Trajectory traced by the body at different speeds 16

5.1.3. Variation of angle of projection with escape velocity 18

5.2. CASE 2

5.2.1. Projection Point at R=5m 19

5.2.2. Variation of angle of projection with escape velocity 20

5.3. CASE 3

5.3.1. Projection Point outside at R=10.6m 215.3.2. Trajectory traced by the body at different speeds 22

6. Results and Conclusions 24

6.1. Inferences 25

7. References and Bibliography 27


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Rotating Disc

The first system we studied is a bullet entering a disc just sliding along itssurface with initial velocity V, angular velocity of disc being and frictioncoefficient .

3D View:

Bullet at radius

Top View:


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The system involves friction between particle and disc. As, the disc rotates,

centripetal force is provided to particle by friction force. As a result particle

tends to follow a curved path. The forces acting on the particle are friction and

centrifugal force.

The quantities we require for solving the problem are:

= Friction coefficient of disc

= Angular velocity of the diskVni = Velocity w.r.t non-inertial frame

ni = Angle of bullet w.r.t non-inertial frame

= Position of particle

The other symbols used are:

Fi = force in inertial frame

Vi = velocity w.r.t inertial frame

i = angle of bullet w.r.t inertial frame

Now, the velocity of particle in inertial frame is:

= [() + + 2 cos2 ] = [ cos() ] = cos() = sin() = =


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The friction force acts on the particle in the direction opposite to the

instantaneous relative velocity of the particle w.r.t disc i.e. (

r

-

) So, the

magnitude of acceleration is given by:

= (( ) + )

= + = ( ) =


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Rotating Cone

The second system we studied is a particle projected in an arbitrary direction ina cone rotating at an angular velocity and friction coefficient .

Equations of motion in spherical coordinate system:

=

+

+

0(Since particle moving along the cone, = 0 ) = + +


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The MATLAB Code-Rotating Disc

%[output] = tablerot(v0,th0,mu,omg)%DIALOGUE BOX GENERATION

prompt={'Coefficient of friction:','Angular speed of disc in rps:','Initial Speed of

bullet in m/s:','Initial angle of bullet(th)in m/s:','Position of gun in metres:'};

name='Simulation on rotating disc';

%SETTING THE DEFAULT VALUES

numlines=1;defaultanswer={'1','1','10.73','0','1'};

options.Resize='on';

options.WindowStyle='normal';

options.Interpreter='tex';

%INPUTTING THE VALUES

answer=inputdlg(prompt,name,numlines,defaultanswer,options);

%CHANGING THE INPUTTED STRING INTO NUMERICAL VALUES

mu=str2num(answer{1});

omg=str2num(answer{2});

vrot=str2num(answer{3});

throt=str2num(answer{4});

r0=str2num(answer{5});

%CONVERTING THE SPEED W.R.T. DISC TO SPEED AS PER INERTIAL FRAME

v0 = sqrt((omg*r0)^2 + vrot^2 + 2*(omg*r0)*vrot*cos(pi/2-throt));

th0 = acos(vrot*cos(throt)/v0);

ifthrot>pi

ifthrot


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x(1) = r0;

y(1) = 0;

mug = mu*g;%CALCULATING POSITION AFTER TIME dT, GETTING THE NEW ACCELERATION

THEN NEW VELOCITY TO CALCULATE THE NEW POSITION RECURSIVELY

for i = 1:N

omgt = omg*i*dt;

xr(i) = x(i)*cos(omgt) + y(i)*sin(omgt);

yr(i) = -1*x(i)*sin(omgt) + y(i)*cos(omgt);

x(i+1) = x(i) + vx*dt;

y(i+1) = y(i) + vy*dt;

vtx = -1*omg*y(i+1);

vty = omg*x(i+1);

Fint = mug/(sqrt((vtx - vx)^2 + (vty - vy)^2));

Fx = Fint*(vtx - vx);

Fy = Fint*(vty - vy);

vx = vx + Fx*dt;

vy = vy + Fy*dt;

end

% PLOTTING THE TRAJECTORY ON A GRAPH

figure(1)

hold off

plot(x,y,'r');lm = max([max(abs(x)) max(abs(y))]);

lm = lm+2;

axis([-lm lm -lm lm]);

axis square

hold on

plot(xr,yr,'b')


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OBSERVATIONSAND

FINDINGS


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CASE 1: Projection Point at R=1m

For the values,

1. Coefficient of friction = 1,2. Gravitational acceleration= 10m/3. = 1 rad/s4. Starting position = 15. Angle ofprojection w.r.t. disc= 0

Graph of Radius of circular loop (Y-axis) versus Speed of projection (X-axis):

0

2

4

6

8

10

12

0 2 4 6 8 10 12

Speed of projection w.r.t.

disc m/s

Radius of Circular loops

Meters

0 1.000

1 1.050

2 1.222

3 1.500

4 1.890

5 2.410

6 3.050

7 3.860

8 4.840

9 6.060

10 7.650

10.4 8.520

10.5 8.790

10.6 9.110

10.68 9.450

10.69 9.500

10.70 9.560

10.71 9.630

10.72 9.710

10.73 9.828

10.735 9.929

10.736 9.985

>10.736 Flies off to infinity


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Tr ajector ies Tr aver sed By the Body at Different Speeds:

*Colour codes in graphs:

Red colour- As observed from inertial frame of reference (outside the disc).

Blue colour- As observed by non-inertial frame of reference (rotating along with the centre of the

disc)

V=0.8 m/s V=2 m/s

V=4 m/s V=6 m/s

V=7 m/s V=9 m/s


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V=10.0 m/sV=10.20 m/s

V=10.60 m/s V=10.736 m/s

V=10.74 m/s V=10.75 m/s


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Var iat ion of Angle of Pr ojection with Escape Velocity

For the values,

1. Coefficient of friction = 1,2. Gravitational acceleration= 10 m/3. = 1 rad/s4. Starting position = 1

Angle of projection w.r.t disc Escape Velocity w.r.t disc

m/s

0 10.736

/8 10.52 /6 10.48

/4 10.46

/3 10.51

/2 10.80

2 /3 11.28

3 /4 11.59

5 /6 11.89

7 /8 12.04

12.45

9 /8 12.73

7 /6 12.79

5 /4 12.83

4 /3 12.79

3 /2 12.44

5 /3 11.85

7 /4 11.53

11 /6 11.22

15 /8 11.08

2 10.73

Graph of Escape velocity w.r.t. disc (Y-axis) versus Angle of Projection (X-axis):

0

2

4

6

8

10

12

14

0 100 200 300 400


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CASE 2: Projection Point at R=5m

For the values,

1. Coefficient of friction = 1,2. Gravitational acceleration= 10 m/3. = 1 rad/s4. Starting position = 55. Angle of projection w.r.t. disc= 0


0

2

4

6

8

10

12

0 1 2 3 4 5 6 7

Speed of projection w.r.t.

disc m/s

Radius of Circular loops

Meters

0 5.0000.2 5.008

0.4 5.018

0.6 5.014

0.8 5.065

1 5.101

2 5.403

3 5.908

4 6.639

4.5 7.106

5 7.66

5.5 8.3616 9.456

6.07 9.83

6.075 9.903

>6.075 Flies off to infinity


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Var iat ion of Angle of Pr ojection with Escape Velocity

For the values,

1. Coefficient of friction = 1,2. Gravitational acceleration= 10 m/3. = 1 rad/s4. Starting position = 5

Graph of Escape velocity w.r.t. disc (Y-axis) versus Angle of Projection (X-axis):

0

5

10

15

20

0 100 200 300 400

Angle of projection w.r.t disc Escape Velocity w.r.t disc

m/s

0 6.075

/8 5.730

/6 5.690 /4 5.756

/3 5.960

/2 6.870

2 /3 8.570

3 /4 9.790

5 /6 11.21

7 /8 11.97

14.39

9 /8 16.44

7 /6 16.94

5 /4 17.514 /3 17.39

3 /2 14.81

5 /3 10.37

7 /4 8.60

11 /6 6.90

15 /8 6.14

2 6.075


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CASE 3: Projection Point outside at R=10.6mFor the values,

1. Coefficient of friction = 12. Gravitational acceleration= 10 m/3. = 1 rad/s4. Starting position = 10.6 m5. Angle ofprojection w.r.t. disc=


9.2

9.3

9.4

9.5

9.6

9.7

9.8

9.9

10

10.1

0 2 4 6 8 10 12

Speed of projection w.r.t. disc m/s Radius of Circular loops

Meters11.037 Flies off to infinity


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Tr ajector ies Tr aver sed by the Body at Dif ferent Speeds:

*Colour codes in graphs:

Red colour- As observed from inertial frame of reference (outside the disc).

Blue colour- As observed by non-inertial frame of reference (rotating along with the centre of the

disc)

V=6.0 m/s V=6.08 m/s

V=7 m/s V=8 m/s

V=9 m/s V=10 m/s


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V=10.5 m/s V=10.8 m/s

V=10.9 m/sV=11.0 m/s

V=11.037 m/s V=11.04 m/s


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RESULTSAND

CONCLUSION


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Inferences

1.

The particle behaves in a very unique way on varying the velocity ofprojection. Under a certain limiting value of velocity, the particle

appears to enter a stable circular orbit of radius . At velocitieshigher than the critical value or the escape velocity, the particle leaves

the orbit and flies off to infinity.

This phenomenon can be explained as the friction force (providing

centripetal force) and centrifugal force balance each other. The particle

moves and gets into the circular orbit till the friction force is able to

balance the centrifugal force. = The value of depends on the value of angular velocity of disc,coefficient of friction between the disc and particle and the value of

acceleration due to gravity.

In our simulations, we took the values of=1rad/s, =1 and g=10m/ .Theoretically, the value of should be 10m. And it is verified by thesimulations also. The particle appears to rotate in circular orbit of radiusapproaching 10m just below the escape velocity.

2. When the particle is projected from outside at angle , it entersthe circular orbit for a certain range of velocities. The particle is

observed to go into infinity below and above those critical values.

This behaviour can be explained as the particle will not be able to enter

the region of if velocity of projection is too small, and it will not gointo circular orbit if velocity of projection is too high. In both theviolating cases, particle will go off to infinity, as observed in simulations.

3. The angle of projection of the particle is seen to have an unexpectedeffect on the escape velocity of particle. On increasing the angle , the

escape velocity first decreases, as expected but it increases suddenly at

=/2, instead of decreasing (as expected by common observation). The


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References and Bibliography

Books

Kleppner .D & Kolenkow .R (1999), An Introduction to Mechanics. New Delhi:

Tata McGraw Hill Publications.

Tutorials

MIT open courseware, MATLAB tutorial

http://ocw.mit.edu/index.htm

http://www.mathworks.com/help/techdoc/learn_matlab/bqr_2pl.html

Webpages

Spherical coordinate system, Wikipedia.

Documents

REPORT Rotation