ENGR221 Lecture 25

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Curvilinear Motion &

Equations of MotionENGR 221

April 16, 2003

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Lecture Goals

Normal and Tangential Coordinates

Rectilinear Equations of Motion

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Equations of motion

From Newtons Second Law of motion

where the force and acceleration are vectorquantities.

F ma

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Equations of motion

The vectors can be broken into components similar to

equilibrium

x y z x y zx x y y z z& &

F i F j F k m a i a j a k

F ma F ma F ma

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Equations of motion

From equilibrium equations

x

y

z

0

0

0

F

F

F

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Equations of motion

The difference between the equilibrium equations and

the equations of motion is that the right hand side is

no longer zero.

x x

y y

z z

F ma

F maF ma

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Procedure for solving the

Equations of Motion

1. Draw the Free-Body Diagram.

2. Write the equations of motion

3. Solve for the velocities and accelerations need

or obtain the equations for the components.

4. Solve for the forces

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Example ProblemEquation of

MotionA 80-kg skydiver falls at 85 m/s

when she opens her parachute. If

her speed is reduced to 5 m/sduring the next 60 m of fall,

determine the average force exerted

on her body by the parachute

during this interval.

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MotionDraw the free-body diagram

of the skydiver.

y yF ma

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MotionRewrite the equation

The acceleration is going to be a

constant

y y

2

p y80 kg 9.81 m/s 80 kg

F ma

F a

p2

y9.81 m/s

80 kg

Fa

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MotionThe acceleration is defined as

60 m 5 m/s

0 85 m/s

dv dv dxadt dx dt

adx vdv

adx vdv

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MotionPlug in the acceleration equation

5 m/s60 m 2p2

0 85 m/s

60 m

p2 2 2

0

p

9.81 m/s80 kg 2

9.81 m/s 3600 m /s80 kg

5584.8 N

F vdx

Fx

F

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Class ProblemRectilinear

MotionThe 2000-lb elevator cage is

brought to rest from an initial

speed of 25 ft/s in a distanceof 50 ft. Determine the

uniform deceleration and the

tension in the elevator cable

while the cage is coming to

rest.

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Example ProblemRectilinear

Motion

a) Determine the acceleration of body

A and tension in the cable

connecting the bodies.

b) Determine the acceleration of body

B if body A is replaced with a

constant force of 245 N.

Two bodies A and B with masses of 25 kg

and 30 kg, respectively. During motion of

the bodies,

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MotionDraw the free body diagram of the boxes A and B. The

normal weight, NB and R is resistance due to friction and

the tension T.

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xampeProblem

Equation ofMotion

Find the equations of motion

for the block.

2

Ay A Ay Ay

Bx B Bx Bx

2

By B By B

B

25 kg 9.81 m/s 25 kg

30 kg

30 kg 9.81 m/s 0

294.3 N

F m a T a

F m a T R a

F m a N

N

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xampeProblem

Equation ofMotion

If there is no friction

resistance thenaAy = aBx

2

Ay A Ay Ay

Bx B Bx Bx2

Bx Bx

2 2

Bx

25 kg 25 kg 9.81 m/s

30 kg

30 kg 25 kg 25 kg 9.81 m/s

4.46 m/s 30 kg 4.46 m/s 133.8 N

F m a T a

F m a T a

a a

a T

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Example Problem

Equation of Motion

The force in the cable is T =245 N

Bx2

Bx

245 N 30 kg

8.167 m/s

a

a

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Class Problem

Rectilinear Motion

a) Determine the acceleration of body A and tension

in the cable connecting the bodies if the fiction is

neglected.

Two bodies A and B with masses of 100 kg and

300 kg, respectively. During motion of the

bodies,

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motion

a) The acceleration of A

b) The tension in the cable

c) The velocity of body B

after 5 sec.

Twp bodies A (40 kg) and B (30 kg) are connected by a flexible

cable. The kinetic coefficient of friction between body A and the

inclined surface is mk= 0.25 and the system is released from rest.

During the motion of the body determine.

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motionDraw the free body diagrams of the bodies

xampe

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xampeProblem

Rectilinearmotion

The equations are

2

Ax A Ax A Ax

2

Ay A Ay A

A

2

By B By By

32 40 kg 9.81 m/s 40 kg

5

440 kg 9.81 m/s 05

313.92 N

30 kg 9.81 m/s 30 kg

F m a T F a

F m a N

N

F m a T a

xampe

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xampeProblem

Rectilinearmotion

Use the constraints of the two length of the cable

The direction so the block are moving the upward

A B

A B

A B

2 constant

2 0

2

L s s

s s

s s

xampe

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xampeProblem

Rectilinearmotion

The frictional force is

Rearrange the equations in terms of T

A A A 0.25 313.92 N

78.48 N

F Nm

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xampe

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xampeProblem

Rectilinearmotion

The equations are

Ax Ax

Ax

2

Ax

2 30 kg 2 294.3 N 313.92 N 40 kg

274.68 N 160 kg

1.717 m/s 191.3 N

a a

a

a T

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Equations of motion

The difference between the linear and curvilinear

equations of motion is the defined system.

n n

t t

F ma

F ma

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Example ProblemCurvilinear

motionThe bob of a 2 m pendulum describes an arc of circle in

a vertical plane. If the tension in the cord is 2.5 times

the weight of the bob for the position show, find thevelocity and acceleration of the bob at the position. .

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motionDraw the free body diagram of the mass .

t t

o

t

n n

o

n

sin 30

2 cos 30

F ma

mg ma

F ma

mg mg ma

+

+

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motionSolve for the acceleration .

o 2 o

t t

2

t

o 2 o

n n

2

t

sin 30 9.81 m/s sin 304.90 m/s

2 cos 30 9.81 m/s 2 cos 30

16.30 m/s

mg ma aa

mg mg ma a

a

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motionCompute the tangential velocity .

2t

n t n

2

t 16.30 m/s 2 m

5.71 m/s

va v a

v

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motionDetermine the rated speed of a highway curve of radius

= 400 ft. banked through an angle, q = 18o. The rated

speed of the banked curve road is the speed at which acar should travel if no lateral friction force is to be

exerted on its wheel.

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motionDetermine the equation of motion

for the car with a radius = 400 ft.

banked through an angle, q = 18o

.

y

o

n n

o

n

0

cos 18 0

sin 18

F

R WF ma

R ma

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motionSolve for R and find an.

o

o o

n o

o 2 o

n

2

cos 18

sin 18 sin 18

cos 18

tan 18 32.2 ft/s tan 18

10.46 ft/s

mg

R

mgma R

a g

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motionCompute the tangential velocity .

2

t

n t n

2

t 10.46 ft/s 400 ft

64.7 ft/s

60 mph64.7 ft/s 44.1 mph

88 ft/s

v

a v a

v

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Class ProblemCurvilinear

motionAn airplane is descending at anangle, q = 20o with respect to

the horizontal when it drops a

bomb. If the altitude at thetime of release is 5000 m and

speed of the plane 750 km/hr,

determine the range

(horizontal distance traveled)

of the bomb and the elapsed

time before it strikes the

ground. Neglect air friction.

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Homework (Due 4/23/03)

Problems:

15-10, 15-21,15-23, 15-57, 15-60, 15-61,

15-62, 15-63, 15-64

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Bonus ProblemCurvilinear

motionA block B of mass m may slidefreely on a frictionless arm OA,

which rotates in a horizontal

plane at a constant rate ,Knowing that B is released at a

distance r0 from O and express as

a function of r (a) the

components vrof the velocity of

B along AO, (b) the magnitude

of the horizontal force F exerted

on B by the arm OA.

q

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motionDraw the free body diagram ofthe block B.

r r

20

2

F ma

m r r

F ma

F m r r

q q

q

q q

+

+

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motionDetermine the velocity of the r component

Substitute into the equation for r double dot

r r

r rr r

v r r v

dv dvdrr v v

dr dt dr

2rr0

dvm v r

drq

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motionIntegrate

r

0

f

2 2rr r r

r

0.5

2 2r 0

v

o

dvv r v dv r dr dr

v r r

q q

q

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motionInsert the to find the

force.and 0r q q

0.52 2

0

0.5

2 2 20

0 2

2

F m r r r

m r r

q q

q

ENGR221 Lecture 25

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