A Solutions Manual

7/14/2019 A Solutions Manual

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Partial Solutions Manual

Ruina and PratapIntroduction to Statics and Dynamics

This draft: January 14, 2013

Have a suggestion? Want to contribute a solution?

Contact [email protected] with Subject: Solutions Manual

Note, the numbering of hand-written solutions is most-often wrong (corre-

sponding to an old numbering scheme). The hand-written problem numbers

should be ignored.


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9.1.15 Consider a force acting ona cart over a 3 second span. In case (a),the force acts in two impulses of one sec-ond duration each as shown in fig. 9.1.15.In case (b), the force acts continuously fortwo seconds and then is zero for the last

second. Given that the mass of the cart is10 kg,

, and

N, for eachforce profile,

a) Find the speed of the cart at the endof 3 seconds, and

b) Find the distance travelled by thecart in 3 seconds.

Comment on your answers for the two

cases.

Filename:pfigure9-1-fcompare

t (sec)

t (sec)0 1 2 3

0 1 2 3

F0

F(t)

F(t)

F0

(a)

(b)

Problem 9.15

2 Introduction to Statics and Dynamics, c Andy Ruina and Rudra Pratap 1994-2012.


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Chapter 9.1. Force and motion in 1D Problem 9.1.16 3

9.1.16 A car of mass is accelerated byapplying a triangular force profile shownin fig. 9.1.16(a). Find the speed of the carat

seconds. If the same speed isto be achieved at

seconds with asinusoidal force profile,

sin

,

find the required force magnitude

. Isthe peak higher or lower? Why?

Filename:pfigure9-1-fcompare2

t

t

0

0

FT

F(t)

F(t)

Fs

TT/2

TT/2

(a)

(b)

Problem 9.16

Introduction to Statics and Dynamics, c Andy Ruina and Rudra Pratap 1994-2012.


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4 Chapter 9.1. Force and motion in 1D Problem 9.1.22

9.1.22 A grain of sugar falling throughhoney has a negative acceleration propor-tional to the difference between its veloc-ity and its terminal velocity, which is aknown constant

. Write this sentence asa differential equation, defining any con-

stants you need. Solve the equation assum-ing some given initial velocity

.



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Chapter 9.1. Force and motion in 1D Problem 9.1.26 5

9.1.26 A bullet penetrating flesh slowsapproximately as it would if penetratingwater. The drag on the bullet is about

where

is the den-sity of water,

is the instantaneous speedof the bullet,

is the cross sectional area

of the bullet, and

is a drag coefficientwhich is about

. Assume that thebullet has mass

where

isthe density of lead,

is the cross sec-tional area of the bullet and is the lengthof the bullet (approximated as cylindrical).Assume grams, entering velocity

m

s,

, and bullet

diameter

mm.

a) Plot the bullet position vs time.

b) Assume the bullet has effectivelystopped when its speed has droppedto

m

s, what is its total penetra-tion distance?

c) According to the equations impliedabove, what is the penetration dis-tance in the limit

?

d) How would you change the modelto make it more reasonable in itspredictions for long time?



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6 Chapter 9.1. Force and motion in 1D Problem 9.1.26 (continued)



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Chapter 9.1. Force and motion in 1D Problem 9.1.26 (continued) 7



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8 Chapter 9.1. Force and motion in 1D Problem 9.1.26 (continued)

0 5 100

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8Plot of position vs tim

time (s)

position(m)



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Chapter 9.2. Energy methods in 1D Problem 9.2.3 9

9.2.3 A force

sin

acts on aparticle with mass

kg which hasposition m, velocity m s at

s.

N and

s. At

s evaluate (give numbers and units):

a)

,

b) K,

c)

,

d) K,

e) the rate at which the force is doingwork.



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10 Chapter 9.2. Energy methods in 1D Problem 9.2.3 (continued)



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Chapter 9.2. Energy methods in 1D Problem 9.2.10 11

9.2.10 A kid ( lbm) stands on a

ft wall and jumps down, acceler-ating with

ft

s . Upon hitting theground with straight legs, she bends themso her body slows to a stop over a distance

ft. Neglect the mass of her legs. As-

sume constant deceleration as she brakesthe fall.

a) What is the total distance her body

falls?

b) What is the potential energy lost?

c) How much work must be absorbedby her legs?

d) What is the force of her legs on her

body? Answer in symbols, numbersand numbers of body weight (i.e.,find

).



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12 Chapter 9.2. Energy methods in 1D Problem 9.2.11

9.2.11 In traditional archery, when pullingan arrow back the force starts from 0 andincreases approximately linearly up to thepeak draw force

. The draw forcevaries from about

lbf fora bow made for a small person to about

lbf for a bow made for a bigstrong person. The distance the arrow ispulled back, the draw length

, variesfrom about

ft for a small adultto about inch for a big adult. An ar-

row has mass of about 300 grain (1 grain

milli gm, so an arrow has mass ofabout

gm

ounce). Giveall answers in symbols and numbers.

a) What is the range of speeds you canexpect an arrow to fly?

b) What is the range of heights an ar-row might go if shot straight up (itsa bad approximation, but for thisproblem neglect air friction)?



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Chapter 9.2. Energy methods in 1D Problem 9.2.11 (continued) 13



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14 Chapter 9.2. Energy methods in 1D Problem 9.2.16

9.2.16 The power available to a verystrong accelerating cyclist over short peri-ods of time (up to, say, about 1 minute) isabout

horsepower. Assume a rider startsfrom rest and uses this constant power. As-sume a mass (bike + rider) of

lbm, a

realistic drag force of

lbf

ft

s

.Neglect other drag forces.

a) What is the peak (steady state)speed of the cyclist?

b) Using analytic or numerical meth-ods make an accurate plot of speedvs. time.

c) What is the acceleration as

in this solution?

d) What is the acceleration as

in your solution?

e) How would you improve the modelto fix the problem with the answerabove?



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Chapter 9.2. Energy methods in 1D Problem 9.2.16 (continued) 15



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16 Chapter 9.2. Energy methods in 1D Problem 9.2.16 (continued)



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Chapter 9.3. Simple Harmonic Oscillator Problem 9.3.6 17

9.3.6 Mass hangs from a springwithconstant

and which has the length

when it is relaxed (i.e., when no mass isattached). It only moves vertically.

a) Draw a Free Body Diagram of themass.

b) Write the equation of linear mo-mentum balance.

c) Reduce this equation to a standarddifferential equation in

, the posi-tion of the mass.

d) Verify that one solution is that is constant at

.

e) What is the meaning of that solu-tion? (That is, describe in wordswhat is going on.)

f) Define a new variable

. Substitute

into your differential equa-tion and note that the equation issimpler in terms of the variable

.

g) Assume that the mass is releasedfrom an an initial position of

. What is the motion of the mass?

h) What is the period of oscillation ofthis oscillating mass?

i) Why might this solution not makephysical sense for a long, softspring if the initial stretch is large.In other words, what is wrong withthis solution if

?

Filename:pg141-1

k

x

l0

m

Problem 9.6



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18 Chapter 9.3. Simple Harmonic Oscillator Problem 9.3.8

9.3.8 A person jumps on a trampoline.The trampoline is modeled as having aneffective vertical undamped linear springwith stiffness

lbf

ft. The personis modeled as a rigid mass

lbm.

ft

s .

a) What is the period of motion if thepersons motion is so small that herfeet never leave the trampoline?

b) What is the maximum amplitude ofmotion (amplitude of the sine wave)for which her feet never leave thetrampoline?

c) (harder) If she repeatedly jumps sothat her feet clear the trampoline bya height

ft, what is the pe-

riod of this motion (note, the con-tact time isnotexactly half of a vi-bration period)? [Hint, a neat graphof height vs time will help.]

Filename:pfigure3-trampoline

Problem 9.8: A person jumps on a tram-poline.



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Chapter 9.3. Simple Harmonic Oscillator Problem 9.3.8 (continued) 19



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20 Chapter 9.3. Simple Harmonic Oscillator Problem 9.3.8 (continued)



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Chapter 9.4. Coupled motion in 1D Problem 9.4.14 21

The primary emphasis of this section

is setting up correct differential equations

(without sign errors) and solving these

equations on the computer.

9.4.14 Two masses are connected to fixedsupports and each other with the three

springs and dashpot shown. The force acts on mass 2. The displacements

and

are defined so that

whenthe springs are unstretched. The ground isfrictionless. The governing equations forthe system shown can be written in firstorder form if we define

and

.

a) Write the governing equations in aneat first order form. Your equa-tions should be in terms of any or allof the constants

,

,

,

,

,

, the constant force

, and

. Get-ting the signs right is important.

b) Write computer commands to findand plot

for 10 units of time.

Make up appropriate initial condi-tions.

c) For constants and initial conditionsof your choosing, plot

vs

forenough time so that decaying erraticoscillations can be observed.

Filename:p-f96-f-3

k1 k2 k3

x1

x2

F c

m2m1

Problem 9.14



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22 Chapter 9.4. Coupled motion in 1D Problem 9.4.14 (continued)



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Chapter 9.4. Coupled motion in 1D Problem 9.4.14 (continued) 23



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24 Chapter 9.5. 1D Collisions Problem 9.5.6

9.5.6 Before a collision two particles,

kg and

kg, have veloc-ities of

m

s and

m

s. Thecoefficient of restitution is

. Findthe impulse of mass A on mass B and thevelocities of the two masses after the colli-

sion.



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Chapter 9.5. 1D Collisions Problem 9.5.6 (continued) 25



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26 Chapter 9.5. 1D Collisions Problem 9.5.6 (continued)



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Problem 9.84

If you assumed v+A

= 6 m/s, than the following answers will changed) 6 kg m/sf) 14 kg m/sg) 4 kg m/s. You get this by solving v+

B = 7 m/s

h) 4 kg m/sj) 67 Jk) 0.2

1



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9.5.10 A basketball with mass

isdropped from height

onto the hard solidground on which it has coefficient of resti-tution

. Just on top of the basketball,falling with it and then bouncing againstit after the basketball hits the ground, is a

small rubber ball with mass

that has acoefficient of restitution

with the bas-ketball.

a) In terms of some or all of

,

,

,

,

and

how high does therubber ball bounce (measure heightrelative to the collision point)?

b) Assuming the coefficients of resti-tution are less than or equal toone, for given , what mass and

restitution parameters maximize theheight of the bounce of the rubberball and what is that height?



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9.5.12 According to the problem above,unless energy is created in the collision (asin an explosion),

. Show that,for given masses and given initial veloci-ties, that the loss of system kinetic energyis maximized by

.



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10.1.2 A spring with rest length

is at-tached to a mass

which slides friction-lessly on a horizontal ground as shown.At time

the mass is released fromrest with the spring stretched a distance .Measure the mass position

relative to the

wall.a) What is the acceleration of the mass

just after release?

b) Find a differential equation whichdescribes the horizontal motion ofthe mass.

c) What is the position of the mass atan arbitrary time

?

d) What is the speed of the mass whenit passes through

(the posi-tion where the spring is relaxed)?

Filename:s97f1

m

d0

Problem 10.2



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Chapter 10.1. Free Vibration of a SDOF System Problem 10.1.2 (continued) 37



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38 Chapter 10.1. Free Vibration of a SDOF System Problem 10.1.2 (continued)



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Chapter 10.1. Free Vibration of a SDOF System Problem 10.1.2 (continued) 39



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40 Chapter 10.3. Normal Modes Problem 10.3.3

10.3.3

and

are measured po-sitions on two points of a vibrating struc-ture.

is shown. Some candidatesfor

are shown. Which of the

could possibly be associated with a normalmode vibration of the structure? Answer

could or could not next to each choiceand briefly explain your answer (If a curvelooks like it is meant to be a sine/cosinecurve, it is.)

Filename:pfigure-blue-144-1

a)

b)

c)

d)

e)

X1(t)

X2?

X2?

X2?

X2?

X2?

Problem 10.3



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Chapter 10.3. Normal Modes Problem 10.3.8 41

10.3.8 For the three-mass system shown,assume

when allthe springs are fully relaxed. One of thenormal modes is described with the initialcondition

.

a) What is the angular frequency

for this mode? Answer in terms of , and . (Hint: Note thatin this mode of vibration the middlemass does not move.)

b) Make a neat plot of

versus

for one cycle of vibration with thismode.


m

x1 x2 x3

m m

kkkk

L L L L

Problem 10.8



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42 Chapter 10.3. Normal Modes Problem 10.3.8 (continued)



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Chapter 10.3. Normal Modes Problem 10.3.8 (continued) 43



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11.1.22 An object C of mass kg is pulledby three strings as shown. The accelerationof the object at the position shown is

m

s .

a) Draw a free body diagram of the

mass.b) Write the equation of linear mo-

mentum balance for the mass. Use

s as unit vectors along the strings.

c) Find the three tensions

,

, and

at the instant shown. You mayfind these tensions by using handalgebra with the scalar equations,

using a computer with the matrixequation, or by using a cross prod-uct on the vector equation.

Filename:pfigure-s94h2p9

y

x

z

C

m

T1

T2

T3

1m

4m

1.5m

2m

2m

Problem 11.22



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Chapter 11.1. Dynamics of a particle in space Problem 11.1.22 (continued) 45



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46 Chapter 11.1. Dynamics of a particle in space Problem 11.1.26

11.1.26 Bungy Jumping. In a relativelysafe bungy jumping system, people jumpup from the ground while being pulled upby a rope that runs over a pulley at O andis connected to a stretched spring anchoredat B. The ideal pulley has negligible size,

mass, and friction. For the situation shownthe spring AB has rest length

m anda stiffness of N m. The inexten-sible massless rope from A to P has length

m, the person has a mass of kg.

Take O to be the origin of an

coordinatesystem aligned with the unit vectors and

a) Assume you are given the positionof the person and thevelocity of the person

. Find her acceleration in terms

of some or all of her position, hervelocity, and the other parameters

given. Then use the numbers given,where supplied, in your final an-swer.

b) Given that bungy jumpers initialposition and velocity are

m

m

and

write com-puter commands to find her position

at

s.

c) Find the answer to part (b) with

pencil and paper (that is, find ananalytic solution to the differentialequations, a final numerical answeris desired).

Filename:s97p1-3

k

m

10 m

AO

P

B

= 10 m/s2

ground, no contactafter jump off

g

Problem 11.26: Conceptual setup for abungy jumping system.



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48 Chapter 11.1. Dynamics of a particle in space Problem 11.1.26 (continued)



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Chapter 11.1. Dynamics of a particle in space Problem 11.1.30 49

11.1.30 The equations of motion fromproblem ?? are nonlinear and cannot besolved in closed form for the position ofthe baseball. Instead, solve the equationsnumerically. Make a computer simulationof the flight of the baseball, as follows.

a) Convert the equation of motion intoa system of first order differentialequations.

b) Pick values for the gravitationalconstant

, the coefficient of resis-tance , and initial speed

, solvefor the

and

coordinates of theball and make a plots its trajectoryfor various initial angles

.

c) Use Eulers, Runge-Kutta, or othersuitable method to numerically in-tegrate the system of equations.

d) Use your simulation to find the ini-tial angle that maximizes the dis-tance of travel for ball, with and

without air resistance.

e) If the air resistance is very high,what is a qualitative description forthe curve described by the path ofthe ball? Show this with an accurateplot of the trajectory. (Make sure tointegrate long enough for the ball toget back to the ground.)



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10.30 (continued)

b). See attached codes and results

%problem 10.30(a)

functionsolution1030a

%solution to 10.30

%September 23,2008

b=1; m=1; g=10; % give values for b,m and g here

%Initial conditions and time span

tspan=[0:0.001:5]; %integrate for 50 seconds

x0=0;

y0=0; %initial position

v0=50; %magnitude of initial velocity (m/s)

theta0=20; %angle of initial velocity (in degrees)

z0=[x0,y0,v0*cos(theta0*pi/180),v0*sin(theta0*pi/180)]';

%solves the ODEs

[t,z] = ode45(@rhs,tspan,z0,[],b,m,g);

%Unpack the variables

x= z(:,1);

y =z(:,2);

v_x = z(:,3);

v_y=z(:,4);

%plot the resultsplot(x,y);

xlabel('x(m)');

ylabel('y(m)');

%set grid,xmin,xmax,ymin,ymax

axis([0,5,0,5]);

title(['Plot of Trajectory for theta= ',num2str(theta0),' degrees']);

end

%-----------------------------------------------------------------------%

functionzdot = rhs(t,z,b,m,g) %function to define ODE

x=z(1); y=z(2); v_x=z(3); v_y=z(4);

%the linear momentum balance eqns

xdot=v_x;

v_xdot=-(b/m)*v_x*(v_x^2+v_y^2)^0.5;

ydot=v_y;



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v_ydot=-g-(b/m)*v_y*(v_x^2+v_y^2)^0.5;

zdot=[xdot; ydot; v_xdot; v_ydot]; %this is what the function returns (column vector)

end

%-----------------------------------------------------------------------%



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c). Disregard this question. This question intends to ask you develop your own ode solver similar

to ode45, using Eulers method or more sophisticated method (Ruger-Kutta method).

d). To find out x distance, we use stopevent to terminate the integration at y=0. Then loop over

for theta from 0.1 to 89.1 degree with an increment of 1 degree.

%problem 10.30(d)

functionsolution1030d

%solution to 10.30

%September 23,2008

b=1; m=1; g=10; % give values for b,m and g here

%Initial conditions and time span

tspan=[0 50]; %integrate for 50 seconds

x0=0;

y0=0; %initial position

v0=50; %magnitude of initial velocity (m/s)

theta0=[0.1:1:89.1]'; %angle of initial velocity (in degrees)

distance=zeros(size(theta0)); %arrays to record x distance at y=0 for each angle

fori=1:length(theta0)



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z0=[x0,y0,v0*cos(theta0(i)*pi/180),v0*sin(theta0(i)*pi/180)]';

options=odeset('events', @stopevent);

%solves the ODEs

[t,z] = ode45(@rhs,tspan,z0,options,b,m,g);

%Unpack the variables

x= z(:,1);

distance(i)=x(end);% the last component of x is the distance we want

end

plot(theta0,distance,'*')

xlabel('theta(degrees)');

ylabel('distance(m)');%set grid,xmin,xmax,ymin,ymax

title(['plot of x distance for various theta']);

[maxd,j]=max(distance);

fprintf(1,'\nThe maximum distance is %6.4f m when theta=%2.0f degrees\n', maxd,theta0(j));

%print the results

end

%-----------------------------------------------------------------------%

functionzdot = rhs(t,z,b,m,g) %function to define ODE

x=z(1); y=z(2); v_x=z(3); v_y=z(4);

%the linear momentum balance eqnsxdot=v_x;

v_xdot=-(b/m)*v_x*(v_x^2+v_y^2)^0.5;

ydot=v_y;

v_ydot=-g-(b/m)*v_y*(v_x^2+v_y^2)^0.5;

zdot=[xdot; ydot; v_xdot; v_ydot]; %this is what the function returns (column vector)

end

%-----------------------------------------------------------------------%

function[value, isterminal, dir]= stopevent(t,z,b,m,g,v0,theta)

% terminate the integration at y=0

x=z(1);

y=z(2);

value= y;

isterminal=1;

dir=-1;

end



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Matlab out put: The maximum distance is 3.3806 m when theta=23 degrees

10.30 (Continued)

The x distance at y=0 for various theta is plotted below

e). Use the code for (a) and change bto a very large number, 100000. The trajectory looks like

,

which is approximately a triangle.



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10.30 Another solution (more detailed)

The m file attached does the following.

a) uses events and x(end) to calculate range.

b) has that embedded in a loop so that there is an angle(i) and

a range(i)

c) Makes a nice plot of range vs angle

d) uses MAX to find the maximum range and corresponding angle

e) has good numerics to show that the trajectory shape converges to

a triangle as the speed -> infinity.



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function baseball_trajectory

% Calculates the trajectory of a baseball.% Calculates maximum range for given speed,

% with and without air friction.

% Shows shape of path at high speed.

disp(['Start time: ' datestr(now)])

cla

% (a) ODEs are in the function rhs far below.

% The 'event' fn that stops the integration

% when the ball hits the ground is in 'eventfn'

% even further below.

% (b) Coefficients for a real baseball taken

% from a google search, which finds a paper

% Sawicki et al, Am. J. Phys. 71(11), Nov 2003.

% Greg Sawicki, by the way, learned some dynamics% in TAM 203 from Ruina at Cornell.

% All parameters in MKS.

m = 0.145; % mass of baseball, 5.1 oz

rho = 1.23; % density of air in kg/m^3

r = 0.0366; % baseball radius (1.44 in)

A = pi*r^2; % cross sectional area of ball

C_d = 0.35; % varies, this is typical

g = 9.81; % typical g on earth

b = C_d*rho*A/2; % net coeff of v^2 in drag force

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% (b-d) Use typical homerun hit speed and look

% at various angles of hit.

tspan=linspace(0,100,1001); % give plenty of time

n = 45; % number of simulations

angle = linspace(1,89,n); % launch from 1 to 89 degrees

r0=[0 0]'; % Launch x and y position.

% First case: No air friction.

b = 0;

subplot(3,2,1)

hold off

% Try lots of launch angles, one simulation for

% each launch angle.

for i = 1:n

inspeed = 44; % typical homerun hit (m/s), 98 mph.

theta0 = angle(i)*pi/180; % initial angle this simulation

v0=inspeed*[cos(theta0) sin(theta0)]'; %launch velocity

z0=[r0; v0]; % initial position and velocity



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options=odeset('events',@eventfn);

[t zarray]=ode45(@rhs,tspan,z0,options,g,b,m); %Solve ODE

x=zarray(:,1); y=zarray(:,2); %Unpack positions

range(i)= x(end); % x value at end, when ball hits ground

plot(x,y); title('Jane Cho: Baseball trajectories, no air friction')

xlabel('x, meters'); ylabel('y, meters'); axis('equal')

axis([0 200 0 200])

hold on % save plot for over-writing

end % end of for loop for no-friction trajectories

%Plot range vs angle, no friction case

subplot(3,2,2); hold off;

plot(angle,range);

title('Range vs hit angle, no air friction')xlabel('Launch angle, in degrees')

ylabel('Hit distance, in meters')

% Pick out best angle and distance

[bestx besti] = max(range);

disp(['No friction case:'])

best_theta_deg = angle(besti)

bestx

% Second case: WITH air friction

% Identical to code above but now b is NOT zero.

b = C_d*rho*A/2; % net coeff of v^2 in drag force

subplot(3,2,3)

hold off % clear plot overwrites

% Try lots of launch angles

for i = 1:n %

inspeed = 44; % typical homerun hit (m/s), 98 mph.

theta0 = angle(i)*pi/180; % initial angle this simulation





x=zarray(:,1); y=zarray(:,2); %Unpack positionsrange(i)= x(end); % x value at end, when ball hits ground

plot(x,y); title('Baseball trajectories, with air friction')


axis([0 120 0 120])



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end % end of for loop for with-friction trajectories

%Plot range vs angle, no friction case

subplot(3,2,4);

plot(angle,range);

title('Range vs hit angle, with air friction')

xlabel('Launch angle, in degrees')

ylabel('Hit distance, in meters')

%Find Max range and corresponding launch angle

[bestx besti] = max(range);

disp(['With Friction:'])

best_theta_deg = angle(besti)

bestx

%%%%%%%%%%%%%%%%%%%%%%%%%%

% Now look at trajectories at a variety of speeds

% Try lots of launch angles

subplot(3,2,6)

hold off

speeds = 10.^linspace(1,8,30); % speeds from 1 to 100 million m/s

for i = 1:30 %

inspeed = speeds(i); % typical homerun hit (m/s), 98 mph.

theta0 = pi/4; % initial angle is 45 degrees at all speeds





x=zarray(:,1); y=zarray(:,2); %Unpack positions

range(i)= x(end); % x value at end, when ball hits ground

plot(x,y); title('Trajectories, with air friction, various speeds ')


axis([0 2000 0 2000])


end % end of for loop for range at various speeds

disp(['End time: ' datestr(now)])

end % end of Baseball_trajectory.m

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% Governing Ord Diff Eqs.



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function zdot=rhs(t,z,g,b,m)

% Unpack the variablesx=z(1); y=z(2);

vx=z(3); vy=z(4);

%The ODEs

xdot=vx; ydot=vy; v = sqrt(vx^2+vy^2);

vxdot=-b*vx*v/m;

vydot=-b*vy*v/m - g;

zdot= [xdot;ydot;vxdot;vydot]; % Packed up again.

end

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

% 'Event' that ball hits the ground

function [value isterminal dir] = eventfn(t,z,g,b,m)

y=z(2);

value = y; % When this is zero, integration stops

isterminal = 1; % 1 means stop.

dir= -1; % -1 means ball is falling when it hits

end



60/140


0 100 2000

50

100

150

200Jane Cho: Baseball trajectories, no air friction

x, meters

y,meters

0 50 1000

50

100

150

200Range vs hit angle, no air friction

Launch angle, in degrees

Hit

distance,

inmeters

0 50 1000

20

40

60

80

100

120Baseball trajectories, with air friction

x, meters

y,meters

0 50 1000

50

100

150Range vs hit angle, with air friction


Hitdistance,

inmeters

0 1000 20000

500

1000

1500

2000Trajectories, with air friction, various speeds

x, meters

y,meters



61/140


0 50 100 150 2000

50

100

150

200Jane Cho: Baseball trajectories, no air friction

x, meters

y,meters

0 20 40 60 80 1000

50

100

150

200Range vs hit angle, no air friction


Hitdistance,

inmeters

0 20 40 60 80 100 1200

20

40

60

80

100

120Baseball trajectories, with air friction

x, meters

y,meters

0 20 40 60 80 1000

20

40

60

80

100

120Range vs hit angle, with air friction


Hitdistance,

inmeters

0 500 1000 1500 20000

500

1000

1500

2000Trajectories, with air friction, various speeds

x, meters

y,meter

s

A whole bunch oftrajectories. The one

launched at 45 degrees

goes the farthest.

As expected from

simple calculations,

the best angle, when

there is no friction, is

45 degrees.

With friction, the best

launch velocity is less.

At this speed, 44 m/s,

the best angle is about41 degrees.

Note that with friction the

ball doesnt go as far. Nor

as high when popped up.

Baseball. For the first 4 plots realistic ball properties are used and the launch speed

is always 44 m/s (typical home run hit). Spin is ignored.

At right are a bunch of trajectories. The

slowest launch is 10 m/s, the fastest

is 100,000,000 m/s. Such a ball would

burn up, tear apart etc... but ignore that.

Note that as the speed gets large thetrajectory gets closer and closer to,

its a strange and beautiful shape, to a

triangle. The same would happen if the

speed were fixed and the drag progressively

increased.

With no friction the range increases with the

square of the speed. With quadratic drag, at high

speeds the range goes up with the log of the launch speed. Like the penetration distance of a bullet.



62/140

62 Chapter 11.2. Momentum and energy for particle motion Problem 11.2.22

11.2.22 At a time of interest, a particlewith mass

kg has position, ve-locity, and acceleration

m

,

m

s

, and

m

s

, respec-tively. Another particle with mass

kg has position, velocity, and accelera-

tion

m

,

m

s

, and

m

s

, respectively. For thissystem of two particles, and at this time,find its

a) linear momentum

,

b) rate of change of linear momentum

c) angular momentum about the origin

O,

d) rate of change of angular momen-

tum about the origin

O,

e) kinetic energy K, and

f) rate of change of kinetic energy K.



63/140

Chapter 11.2. Momentum and energy for particle motion Problem 11.2.22 (continued) 63



64/140

64 Chapter 11.3. Central force motion Problem 11.3.5

Experts note that these problems do

not use polar coordinates or any other

fancy coordinate systems. Such descrip-

tions come later in the text. At this point

we want to lay out the basic equations and

the qualitative features that can be found

by numerical integration of the equationsusing Cartesian (

) coordinates.

11.3.5 An intercontinental misile, mod-elled as a particle, is launched on a ballis-tic trajectory from the surface of the earth.The force on the missile from the earthsgravity is

and is directedtowards the center of the earth. When it islaunched from the equator it has speed

and in the direction shown, from hor-izontal (both measured relative to a New-tonian reference frame). For the purposesof this calculation ignore the earths rota-

tion. You can think of this problem as two-dimensional in the plane shown. If youneed numbers, use the following values:

kg = missile mass

m

s at the earths surface,

m = earths radius,and

m s.

The distance of the missile from the centerof the earth is .

a) Draw a free body diagram of themissile. Write the linear momen-tum balance equation. Break thisequation into and components.

Rewrite these equations as a systemof-4 first order ODEs suitable forcomputer solution. Write appropri-ate initial conditions for the ODEs.

b) Using the computer (or any othermeans) plot the trajectory of therocket after it is launched for a timeof

seconds. [Hint: use a muchshorter time when debugging yourprogram.] On the same plot draw a(round) circle for the earth.

Filename:pfigure-s94q12p1

45o

x

y

Problem 11.5: In intercontinental ballisticmissile launch.

V

D

t PSD'

Hne problem coi-fh problem

Hie

X P A

3/C

o H i . 5m 3 / r

-kosloo ac-Vs

A\0s



85/140

Chapter 13.2. 1D motion with 2D and 3D forces Problem 13.2.25 85

13.2.25 Car braking: front brakes ver-sus rear brakes versus all four brakes.What is the peak deceleration of a car whenyou apply: the front brakes till they skid,the rear brakes till they skid, and all fourbrakes till they skid? Assume that the

coefficient of friction between rubber androad is

(about right, the coeffi-cient of friction between rubber and roadvaries between about

and

) and that

m s (2% error). Pick the dimen-

sions and mass of the car, but assume thecenter of mass height is greater than zerobut is less than half the wheel base

, thedistance between the front and rear wheel.Also assume that the

is halfway be-tween the front and back wheels (i.e.,

). The car has a stiff sus-pension so the car does not move up ordown or tip appreciably during braking.Neglect the mass of the rotating wheels

in the linear and angular momentum bal-ance equations. Treat this problem as two-dimensional problem; i.e., the car is sym-metric left to right, does not turn left orright, and that the left and right wheelscarry the same loads. To organize yourwork, here are some steps to follow.

a) Draw a FBD of the car assumingrear wheel is skidding. The FBDshould show the dimensions, thegravity force, what you knowa pri-ori about the forces on the wheelsfrom the ground (i.e., that the fric-tion force

, and that thereis no friction at the front wheels),

and the coordinate directions. Labelpoints of interest that you will use inyour momentum balance equations.(Hint: also draw a free body dia-gram of the rear wheel.)

b) Write the equation of linear mo-mentum balance.

c) Write the equation of angular mo-mentum balance relative to a pointof your choosing. Some particu-larly useful points to use are:

the point above the frontwheel and at the height of thecenter of mass;

the point at the height of thecenter of mass, behind therear wheel that makes a degree angle line down tothe rear wheel ground contactpoint; and

the point on the groundstraight under the front wheelthat is as far below ground asthe wheel base is long.

d) Solve the momentum balance equa-

tions for the wheel contact forcesand the deceleration of the car. Ifyou have used any or all of therecommendations from part (c) youwill have the pleasure of only solv-ing one equation in one unknown ata time.

e) Repeat steps (a) to (d) for front-wheel skidding. Note that the ad-vantageous points to use for angularmomentum balance are now differ-ent. Does a car stop faster or slower

or the same by skidding the frontinstead of the rear wheels? Would

your solution to (e) be different ifthe center of mass of the car were atground level(

=0)?

f) Repeat steps (a) to (d) for all-wheelskidding. There are some shortcutshere. You determine the car de-celeration without ever knowing thewheel reactions (or using angularmomentum balance) if you look atthe linear momentum balance equa-tions carefully.

g) Does the deceleration in (f) equalthe sum of the decelerations in (d)and (e)? Why or why not?

h) What peculiarity occurs in the solu-tion for front-wheel skidding if thewheel base is twice the height of theCM above ground and ?

i) What impossibility does the solu-tion predict if the wheel base isshorter than twice the CM height?What wrong assumption gives riseto this impossibility? What wouldreally happen if one tried to skid acar this way?


x

y

lr lf

hC D

Problem 13.25



86/140

86 Chapter 13.2. 1D motion with 2D and 3D forces Problem 13.2.25 (continued)



87/140

Chapter 13.2. 1D motion with 2D and 3D forces Problem 13.2.43 87

13.2.43 The uniform kg plate DBFHis held by six massless rods (AF, CB, CF,GH, ED, and EH) which are hinged at theirends. The support points A, C, G, and E areall accelerating in the

-direction with ac-celeration

m

s

. There is no grav-

ity.

a) What is

for the forces act-ing on the plate?

b) What is the tension in bar CB?Filename:pfigure-s94q3p1

y

z

1m

1m

1mA B

CD

E F

GH

x

Problem 13.43



88/140

88 Chapter 13.2. 1D motion with 2D and 3D forces Problem 13.2.47

13.2.47 A rear-wheel drive car on levelground. The two left wheels are on per-fectly slippery ice. The right wheels are ondry pavement. The negligible-mass frontright wheel at

is steered straight aheadand rolls without slip. The right rear wheel

at

also rolls without slip and drives thecar forward with velocity

and ac-celeration . Dimensions are asshown and the car has mass

. What isthe sideways force from the ground on theright front wheel at

? Answer in terms ofany or all of , , , , , , and .

Filename:pfigure3-f95p1p3

ice

cartoon to show

dimensionsb

b

w w/2

C

m

B

C

B

k

w/2w

h

h

Problem 13.47: The left wheels of this carare on ice.



89/140

Chapter 13.2. 1D motion with 2D and 3D forces Problem 13.2.47 (continued) 89



90/140

14.1.1 A particle goes on a circular pathwith radius

making the angle

measured counter clockwise from the pos-itive

axis. Assume

cm and

s .

a) Plot the path.

b) What is the angular rate in revolu-tions per second?

c) Put a dot on the path for the locationof the particle at

s.

d) What are the and coordinatesof the particle position at

?Mark them on your plot.

e) Draw the vectors

and

at

.

f) What are the

and

componentsof

and

at ?

g) What are the

and

componentsof

and

at

?

h) Draw an arrow representing boththe velocity and the acceleration at

.

i) Find the

and

components ofposition

, velocity

and accelera-tion at .

j) Find the

and

components of po-sition , velocity and acceleration

at

. Find the velocity andacceleration two ways:

1. Differentiate the positiongiven as

.

2. Differentiate the position giveas

and then convertthe results to Cartesian coor-dinates.

4Aff|9,.

^)

7 n 5Z r0I

r) =r A V

y = g E i n G )f t t { = h EX3 T c (+)\ee yr YA

\ /

,)

d)e)+)3)h)i)

br i n g 3 u * o s 7 ue a JrayA7= R U R90*i = K 7 A u =0 , r ' 6 nf i= -| 'Kb*+KiA,= -'zof6r

l r ev /5

L



91/140

Chapter 14.1. Kinematics of a particle in circular motion Problem 14.1.1 (continued) 91

S.?r+,

YIY

4X l +l .

j ) 2Y =It, t -v=

Cgn6n',Le d

t *zr) L

4/ lJr SpX

+ (-BslnP 't lz coEd);(r (r,^7') r ) 'r o);

n 2 , nI t Y Jcosgz ps ino i ) t

XGr-'5/

1 ,R c s s g f| 5 , r r%)Gd;

t . 1x L +-Kq, r ,

Y rYO ,

+4

ar =*nv =

Z

i =-91-s,rt, r j= Y(zo) ,? = y(d 6n- ;r)

+:

= f ( o -*lo r'i --l l Tu i

(or)'6r)

, , ,7 ) ( ro) ' - ) i *&; - tt f t U , lT r4tl



92/140

92 Chapter 14.1. Kinematics of a particle in circular motion Problem 14.1.15

14.1.15 A particle moves in circles so thatits acceleration

always makes a fixed an-gle

with the position vector

, with

. For example,

wouldbe constant rate circular motion. Assume

,

m and

rad

s.

How long does it take the particle to reach

a) the speed of sound (

m

s)?

b) the speed of light ( m s)?

c) ?



93/140

Chapter 14.2. Dynamics of a particle in circular motion Problem 14.2.30 93

14.2.30 Bead on a hoop with friction. Abead slides on a rigid, stationary, circularwire. The coefficient of friction betweenthe bead and the wire is

. The bead isloose on the wire (not a tight fit but notso loose that you have to worry about rat-

tling). Assume gravity is negligible.

a) Given

,

,

, &

; what is

?

b) If

, how does de-pend on

,

,

and

?Filename:pfigure-s94h10p1

O

R

m

x

y

v

Problem 14.30

r)

x r \/Voredyvil-hRong ongf truffusrln*MoFfiCn otlrS & r./*ltt fS

KirnDunpl'tyHW+-3 d*e,owqTnf{ 2f'3C.&chtrr36tl2sTH,rear5o1yi

gingon rigid, roTl0nArUtcirc;u-lwtre;qDswrle\- ' FtsD-Sf/ N',rffilffr*-ruNA0t 0NeI1 v, T),

-J.aF=md^tNe' - pNoa?C 4 ^ /L J , a , N =N =Zffio= e,ffi, vor-pi'ra'Ke,) N /g,e,,)- {rt^NR -- K n i 83.[T. I =7 p NK -Rrn'j' / y : , t t tN m

K,&r , f rndvd =Kee - R#&mKe rnK e3-lr1lR6' e = t- nnVli -- -r3r),:

ffid'* rnV)

&oLurlnNt3+5- Beadt

l= #f#)l-i = -uirlI T IIntroduction to Statics and Dynamics, c Andy Ruina and Rudra Pratap 1994-2012.


94/140

94 Chapter 14.2. Dynamics of a particle in circular motion Problem 14.2.30 (continued)

1 l

vtd lb



95/140

Chapter 14.2. Dynamics of a particle in circular motion Problem 14.2.34 95

14.2.34 A block with mass is movingto the right at speed

when it reaches acircular frictionless portion of the ramp.

a) What is the speed of the block whenit reaches point B? Solve in terms of

,

,

and

.

b) What is the force on the block fromthe ramp just after it gets onto theramp at point A? Solve in terms of

,

,

and

. Remember, forceis a vector. Filename:s93q4Sachse

R

A

B

g

v0

Problem 14.34

4 l?ffi-r blocKwfth fnfl$$m) ryeedVo,fn'chonlest mp

0t B in terms aF K,Vo rn, &9

r Eh=t.u=" V - o

a, speed FblocKFBD+,t ^ JN'4't, n4 ,,Enw .#3i *J -.-A $F=) NJ v _-7N

BN*+J_^strnv"' a+nJzmvr, Ep rngh d+ B-) S daesmfrKz nou,wrgYEtrn+6Ks Ef * consfianf

b.t\ -zF=rnd^ c^J-ffC'nL+= nKeennne"erT-13 '&,*rN-nnq=-rnKo'v - -mKW

:wetctockiqDrndmpa* fr

a'=Rei, R6, A=* at f rb nq L -mgtIntroduction to Statics and Dynamics, c Andy Ruina and Rudra Pratap 1994-2012.


96/140

96 Chapter 14.2. Dynamics of a particle in circular motion Problem 14.2.34 (continued)

l t ,n ti^,,ted.0 wh) fl+ d : K Aee- F0-a',- 4 -Z f = ^ q

4 4 / a- N r t ^ J e r = n ( B ? e,1

l r o\./ f- M * rnq : - * Ka zJ = ' N 'K

_Ko.A ),.1



97/140

15.1.8 Write a computer program toanimate the rotation of an object. Yourinput should be a set of

and

co-ordinates defining the object (such that

plot y vs x draws the object on thescreen) and the rotation angle

. The out-

put should be the rotated coordinates of theobject.

a) From the geometric informationgiven in the figure, generate coordi-nates of enough points to define thegiven object.

b) Using your program, plot the objectat

and

.

c) Assume that the object rotates

with constant angular speed

rad

s. Find and plot the positionof the object at

s

s

and

s.

Filename:pfigure13-3-objectrotation

x

y

O

= 30 cm

= 30o

Problem 15.8

/ * SoemQ* 7P

g,./(

-r d

l ) , td (o,z \

x

t r ) = V * ) = l l ? . 6

(t 1rrrf l ,o)

& r^l /s) (

cb), ) st^t,

,lutr----J,(

r o l

2 Lrot' - 1 '{ t

ttttt

In,

ist

etbjnt>\g

tbo), tD I)+

(

fl0

Ao)il

z z1 .? r ) -z( r l t

e5

IEL

tet

Iht,I' (u l 3 w , 7 7 0biorl

( X, osg,t;^i,(f *in0, ,rnr )

Introduction to Statics and Dynamics, c Andy Ruina and Rudra Pratap 1994-2012. 97


98/140

98 Chapter 15.1. Rotation of a rigid body Problem 15.1.8 (continued)

r 0 / 2 0 / 0 8 I : \ a n i m a t i o n b . m

? Johannes Feng? S o l u t i - o n t o 1 3 . 5 8 b , d u e L 0 / 2 t / 0 8f unct ion an imat ion b

? CLEAN UPc l o s e a l l ;

? USER INPUT (angle of rotation)Ehe ta = inpu t ( 'Ang le ,o f ro ta t ion ( in deg rees ) : \n ' ) ;? CONSTANTS

phi = 30; ? geometry of shape, in degrees? DEFINE COORDINATESFOR NON-ROUNDPARTx = ; 6 1 * g l n d ( p h i ) O 1 * c o s d ( p h i ) 2 * 1 * c o s d ( p h i ) 1y = [ 2 *1 *c o s d ( p h i ) l r , c o s d ( p h i ) O I * s i n d ( p h i ) 0 ]^ ^ . i n f - l ' * . r r ' l .I / v a r r u - L ^ t I J I

? DEFINE COORDINATES FOR QUARTER-CIRCLEp t s = l i n s p a c e ( 0 , p i / 2 , 1 0 0 0 ) ;xc i rc = 2* f * cosa ( p h i ) * cos (p ts ) ;y c i r c = 2 * f * c o s d ( p h i ) s i n ( p t s ) ;c i r c l e = [ x c i r c ; y c i r c ; ] ;? DEFINE ROTATION MATRIXR = [ c o s d ( t h e t a ) - s i n d ( t h e t a ) ; s i n d ( t h e t a ) c o s d ( t h e t a ) ] ;? DEF]NE COORDINATES FOR ROTATED SHAPEn a i n t - - P * n n i n i 'y v l r 5 g - . y v f l r g ,x r o t = p o i n t ( 1 , : ) ;y r o t = p o i n t ( 2 , : ) ;c i r c l e = R *c i r c l e ;x c i r c r o t = c i r c l e ( l - , : ) ;y c i r c r o t = c i r c l e ( 2 , : ) ;? DISPLAY THE RESULTSf igu re ;t i t l e ( 'Ro ta t ing ob jec t , by Johannes Feng ' ) ;ho ld on;p l o t ( x , y , ' : b ' ) ; ? o r i g i n a l s h a p ep l o t ( x c i r c , y c i r c , ' : b ' ) ;p lo t (x ro t , y ro t , ' r ' ) i ? ro ta ted shapen ] n l - / v a i r d r ^ f r r c i r c r o l - l r l ) :v & v e t , ,a x i s ( - . 8 . 8 - . 6 . 6 ] ;g r id on ;h o l C o f f ;en d



99/140

Chapter 15.1. Rotation of a rigid body Problem 15.1.8 (continued) 99

L 0 / 2 0 / o B animation

? Johannes Feng? S o l u t i o n t o 1 3 . 5 8 c ,funct ion animat ion c? CI,EAN UPc f o s e a l l ;

? USER INPUT (t imet = input ( 'T ime of

d u e L o / 2 L / 0 8

of rotat ion, g iven angular speed)r o t a t i o n ( s ) : \ n ' r ;

? CONSTANTS

phi = 30; ? geometry of shape, in degreesw = 2; ? angular sPeed, in radians? DEFINE COORDINATES FOR NON-ROUNDPARTx = 1 g 1 * s 1 n f l ( p h i ) 0 1 * c o s d ( p h i ) 2 * L * c o s d ( p h i ) Jy = [ 2 * 1 * c o s d ( p h i ) 1 * s e s f l ( p h i ) O 1 * s i n d ( p h i ) 0 ]po in t = [x ; y l ;? DEF]NE COORDINATESFOR QUARTER-CIRCLEp t s = l i n s p a c e ( 0 , P i / 2 ' 1 0 0 0 ) ;x c i r c = * f * g e s A ( p h i ) * c o s ( P t s ) ;y c i r c - 2 * 1 * c o s a ( P h i ) * s i n ( P t s ) ;c i r c l e = [ x c i r c ; y c i r c ; 7 ;? DEFINE ROTATION MATRIXR = l c o s ( t * w ) - s i n ( t * w ) ; s i n ( t * w ) c o s ( t * w ) ] ;? DEFINE COORDINATES* ^ . i n t s - D * n i n l - .I / v r r r u - 1 \ y v + . . v ,v v a r - n n i n t - / ' l . ) ; V u - y v + r . v \ - t t. . , ^ f - n a i n F / a ' ) ;J l v u - y v r . . e \ s , .c i r c l e - R * c i r c l e ;x c i r c r o t = c i r c l e ( 1 , : )y c i r c r o t = c i r c l e ( 2 , : )? DISPLAY THE RESULTS. F i a r r r a . + Y $ r v tt i t le ( RoEat ing obj ect , by Johannes Fengr ) ;hol -d on;p l o t ( x , Y , r : b ' ) ; ? o r i g i n a l s h a P ep l - o L x c i r c , Y c i r c , ' : b ' ) ;p t o t { x r o t , Y r o t , ' r ' ) ; ? r o t a t e d s h a P e^ l ^ r / v a . i r a r a l ' r r n i r c r n l - r r l ) :p f L ) L \ i u t u v 9 , J v r v r v e t t ta x i s ( t - . 8 . 8 - . 6 . 6 ) ) ig r id o n ;hol -d of f ;

FOR ROTATED SHAPE



100/140


ofrotat ion0=20

ofrotation 0 = 60o



101/140


ofrotatione = 100

eo f r o t a t i o n0=1



102/140


ofrotation 0 =270

T i m e = 1 s



103/140


Time: s

T ime=3 s



104/140

104 Chapter 15.2. Angular velocity Problem 15.2.14

15.2.14 A m long rod has manyholes along its length such that it can bepegged at any of the various locations. Itrotates counter-clockwise at a constant an-gular speed about a peg whose location isnot known. At some instant

, the velocity

of end

is

m

s

. After

s,the velocity of end

is

m

s

. Ifthe rod has not completed one revolutionduring this period,

a) find the angular velocity of the rod,and

b) find the location of the peg along thelength of the rod.

Filename:pfigure4-3-rp9

B

A

Problem 15.14

t3, 1A t t tr til-,q./ lUY"av-)

: per;o)

+^ls)

frtIJL_>i

,)

b)

Astu&srol }t^r nof "n^y r{d on2rf counler*r loek,,, fse.lara 1

2 r L q+ t - zo )- T T T T' 9 (zo ) | Eaynlav eluiQg = ?= ry=

i { rofofes

90 r^) sleyfA of rodrn)ul r l

l r l =l , r l=

Lorot i

lo "tion o)uy {lea0

lv * 3n1sT - #g 90hl,/s0,1 ,rr

t7/)on (An fry il, I h4 fro*



105/140

Chapter 15.2. Angular velocity Problem 15.2.22 105

15.2.22 2-D constant rate gear train.The angular velocity of the input shaft(driven by a motor not shown) is a con-stant,

input

. What is the angularvelocity

output

of the output shaftand the speed of a point on the outer edge

of disc

, in terms of

,

,

, and

?

Filename:ch4-3

RA RB

RC

A

A B

C

no slip

C

Problem 15.22: Gear B is welded to C andengages with A.



106/140

106 Chapter 15.4. Dynamics Problem 15.4.10

15.4.10 Motor turns a bent bar. Twouniform bars of length

and uniform mass

are welded at right angles. One end isattached to a hinge at O where a motorkeeps the structure rotating at a constantrate

(counterclockwise). What is the net

force and moment that the motor and hingecause on the structure at the instant shown.

a) neglecting gravity

b) including gravity.Filename:pg85-3

m

mO

x

y

motor

Problem 15.10: A bent bar is rotated by amotor.



107/140

Chapter 15.4. Dynamics Problem 15.4.10 (continued) 107



108/140


15.4.20 Atthe input toa gearbox a lbfforce is applied to gear A. At the output,the machinery (not shown) applies a forceof

to the output gear. Gear

rotates atconstant angular rate

rad

s, clock-wise.

a) What is the angular speed of theright gear?

b) What is the velocity of point

?

c) What is

?

d) If the gear bearings had friction,would

have to be larger orsmaller in order to achieve the sameconstant velocity?

e) If instead of applying a lbf tothe left gear it is driven by a mo-tor (not shown) at constant angularspeed

, what is the angular speedof the right gear?

Filename:pg131-3

RA

RC

RB

A B P

no slip

FA= 100 lb

FB= ?

C

Problem 15.20: Two gears with end loads.



109/140




110/140

110 Chapter 15.4. Dynamics Problem 15.4.20 (continued)



111/140




112/140


15.4.34 A pegged compound pendu-lum. A uniform bar of mass and length

hangs from a peg at point C and swingsin the vertical plane about an axis passingthrough the peg. The distance

from thecenter of mass of the rod to the peg can be

changed by putting the peg at some otherpoint along the length of the rod.

a) Find the angular momentum of therod about point C.

b) Find the rate of change of angularmomentum of the rod about C.

c) How does the period of the pendu-lum vary with

? Show the varia-tion by plotting the period against

. [Hint, you must first find theequations of motion, linearize forsmall , and then solve.]

d) Find the total energy of the rod (us-

ing point C as a datum for potentialenergy).

e) Find

when

.

f) Find the reaction force on the rod atC, as a function of

,

,

,

, and

.

g) For the given rod, what should bethe value of

(in terms of

) in or-der to have the fastest pendulum?

h) Test of Schulers pendulum. Thependulum with the value of ob-tained in (g) is called the Schulers

pendulum. It is not only the fastestpendulum but also the most accu-rate pendulum. The claim is thateven if

changes slightly over timedue to wear at the support point,the period of the pendulum does not

change much. Verify this claim bycalculating the percent error in thetime period of a pendulum of length

m under the following three

conditions: (i) initial

mand after some wear m,(ii) initial

m and aftersome wear m, and (iii)initial

m and after somewear m. Which pendu-lum shows the least error in its timeperiod? What is the connection be-tween this result and the plot ob-tained in (c)?


A

C

d

G

B

m

Problem 15.34



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114 Chapter 15.4. Dynamics Problem 15.4.34 (continued)



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16.1.1 A disk of radius R is hinged atpoint O at the edge of the disk, approxi-mately as shown. It rotates counterclock-

wise with angular velocity . A boltis fixed on the disk at point P at a distance

from the center of the disk. A frame is

fixed to the disk with its origin at the centerC of the disk. The bolt position P makes anangle

with the

-axis. At the instant ofinterest, the disk has rotated by an angle .

a) Write the position vector of point Prelative to C in the

coordinatesin terms of given quantities.

b) Write the position vector of point Prelative to O in the

coordinatesin terms of given quantities.

c) Write the expressions for the rota-tion matrix

and the angularvelocity matrix S

.

d) Find the velocity of point P relative

to C using

and the angular ve-locity matrix S

.

e) Using

cm,

cm,

, and , find C

0

,

and P

0

at the instant shown.

f) Assuming that the angular speed is

rad s at the instant shown,

find

C

0

and

P

0

takingother quantities as specified above.

Filename:pfigure14-1-doormat1

O

C

P

x

y

y

x

R

r

Problem 16.1

Introduction to Statics and Dynamics, c Andy Ruina and Rudra Pratap 1994-2012. 115


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116 Chapter 16.1. Rigid object kinematics Problem 16.1.1 (continued)



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Chapter 16.1. Rigid object kinematics Problem 16.1.1 (continued) 117



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118 Chapter 16.1. Rigid object kinematics Problem 16.1.12

16.1.12 The center of mass of a javelintravels on a more or less parabolic pathwhile the javelin rotates during its flight. Ina particular throw, the velocity of the cen-ter of mass of a javelin is measured to be

C m s when the center of mass

is at its highest point

m. As thejavelin lands on the ground, its nose hitsthe ground at G such that the javelin is al-most tangent to the path of the center ofmass at G. Neglect the air drag and lift onthe javelin.

a) Given that the javelin is at an angle

at the highest point, find

the angular velocity of the javelin.Assume the angular velocity iscon-stant during the flight and that thejavelin makes less than a full revo-lution.

Filename:pfigure14-1-javeline

vCC

G

h

Problem 16.12



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Chapter 16.2. Dynamics of a rigid object Problem 16.2.7 119

16.2.7 A uniform 1kg plate that is onemeter on a side is initially at rest in the po-

sition shown. A constant force

N

is applied at

and maintained hence-forth. If you need to calculate any quantitythat you dont know, but cant do the cal-

culation to find it, assume that the value isgiven.

a) Find the position of G as a func-tion of time (the answer should havenumbers and units).

b) Find a differential equation, and ini-tial conditions, that when solvedwould give

as a function of time. is the counterclockwise rotationof the plate from the configurationshown.

c) Write computer commands thatwould generate a drawing of theoutline of the plate at

s.

You can use hand calculations or

the computer for as many of the in-termediate commands as you like.Hand work and sketches should beprovided as needed to justify or ex-plain the computer work.

d) Run your code and show clear out-

put with labeled plots. Mark outputby hand to clarify any points.

Filename:S02p2p2flyingplate

x

y

G

EF

Problem 16.7



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120 Chapter 16.2. Dynamics of a rigid object Problem 16.2.7 (continued)



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Chapter 16.2. Dynamics of a rigid object Problem 16.2.7 (continued) 121



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Chapter 16.2. Dynamics of a rigid object Problem 16.2.9 123

16.2.9 A uniform slender bar AB of mass

is suspended from two springs (each ofspring constant

) as shown. Immediatelyafter spring 2 breaks, determine

a) the angular acceleration of the bar,

b) the acceleration of point

, andc) the acceleration of point

.


BA

1 2

L

Problem 16.9



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Chapter 16.3. Kinematics of rolling and sliding Problem 16.3.3 125

The next

several problems concern Work, power and

energy

16.3.3 Rolling at constant rate.A rounddisk rolls on the ground at constant rate. It

rolls

revolutions over the time of inter-

est.

a) Particle paths. Accurately plot thepaths of three points: the center ofthe disk C, a point on the outer edgethat is initially on the ground, anda point that is initially half way be-tween the former two points. [Hint:Write a parametric equation for theposition of the points. First find arelation between and

. Thennote that the position of a point isthe position of the center plus theposition of the point relative to thecenter.] Draw the paths on the com-

puter, make sure

and

scales arethe same.

b) Velocity of points. Find the veloc-ity of the points at a few instants in

the motion: after

,

,

, and revolution. Draw the velocity vec-tor (by hand) on your plot. Draw

the direction accurately and drawthe lengths of the vectors in propor-tion to their magnitude. You can

find the velocity by differentiatingthe position vector or by using rela-tive motion formulas appropriately.Draw the disk at its position afterone quarter revolution. Note thatthe velocity of the points is perpen-dicular to the line connecting thepoints to the ground contact.

c) Acceleration of points. Do thesame as above but for acceleration.Note that the acceleration of thepoints is parallel to the line connect-ing the points to the center of thedisk.


C

P

Problem 16.3



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126 Chapter 16.3. Kinematics of rolling and sliding Problem 16.3.3 (continued)



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Chapter 16.3. Kinematics of rolling and sliding Problem 16.3.3 (continued) 127



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Chapter 16.3. Kinematics of rolling and sliding Problem 16.3.3 (continued) 129



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Chapter 16.4. Mechanics of contact Problem 16.4.6 131

16.4.6 Spool Rolling without Slip andPulled by a Cord. The light-weight spoolis nearly empty but a lead ball with mass

has been placed at its center. A force

is applied in the horizontal direction to thecord wound around the wheel. Dimensions

are as marked. Coordinate directions are asmarked.

a) What is the acceleration of the cen-ter of the spool?

b) What is the horizontal force of theground on the spool?


Ro

RiCroll without

slip F

Problem 16.6



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132 Chapter 16.4. Mechanics of contact Problem 16.4.9

16.4.9 A napkin ring lies on a thick velvettablecloth. The thin ring (of mass

, radius

) rolls without slip as a mischievous childpulls the tablecloth (mass

) out with ac-celeration

. The ring starts at the rightend (

). You can make a reason-

able physical model of this situation withan empty soda can and a piece of paper ona flat table.

a) What is the rings acceleration asthe tablecloth is being withdrawn?

b) How far has the tablecloth movedto the right from its starting point

when the ring rolls off itsleft-hand end?

c) Clearly describe the subsequentmotion of the ring. Which way doesit end up rolling at what speed?

d) Would your answer to the previ-ous question be different if the ringslipped on the cloth as the cloth was

being pulled out?


dA

x

napkin ring

tableclothr

m

Problem 16.9



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Chapter 16.4. Mechanics of contact Problem 16.4.9 (continued) 133



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134 Chapter 16.4. Mechanics of contact Problem 16.4.9 (continued)



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Chapter 16.4. Mechanics of contact Problem 16.4.23 135

16.4.23 A disk rolls in a cylinder. For allof the problems below, the disk rolls with-out slip and rocks back and forth due togravity.

a) Sketch. Draw a neat sketch ofthe disk in the cylinder. The sketch

should show all variables, coordi-nates and dimension used in theproblem.

b) FBD.Draw a free body diagram ofthe disk.

c) Momentum balance. Write theequations of linear and angular mo-mentum balance for the disk. Usethe point on the cylinder whichtouches the disk for the angular mo-mentum balance equation. Leaveas unknown in these equations vari-ables which you do not know.

d) Kinematics. The disk rolling

in the cylinder is a one-degree-of-freedom system. That is, the val-ues of only one coordinate and itsderivatives are enough to determinethe positions, velocities and accel-erations of all points. The anglethat the line from the center of thecylinder to the center of the diskmakes from the vertical can be usedas such a variable. Find all of the

velocities and accelerations neededin the momentum balance equationin terms of this variable and itsderivative. [Hint: youll need tothink about the rolling contact in or-der to do this part.]

e) Equation of motion. Write the an-gular momentum balance equationas a single second order differentialequation.

f) Simple pendulum? Does thisequation reduce to the equation fora pendulum with a point mass andlength equal to the radius of thecylinder, when the disk radius getsarbitrarily small? Why, or why not?

Filename:h12-3

RC

RD

Problem 16.23: A disk rolls without slipinside a bigger cylinder.



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140 Chapter 16.5. Collisions Problem 16.5.8

16.5.8 An acrobat modeled as a rigidbodywith uniform rigid mass of length

. She falls without rotation in the positionshown from height

where she was sta-tionary. She then grabs a bar with a firmbut slippery grip. What is

so that after

the subsequent motion the acrobat ends upin a stationary handstand? [ Hint: Whatquantities are preserved in what parts of themotion?]


h

bar

BEFORE

DURING

AFTER

Problem 16.8

A Solutions Manual

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