Homework 1 Solutions: Kinematics of a Particleibilion/ · 2016. 2. 14. · Introduction to Dynamics...

Introduction to Dynamics (N. Zabaras)

Homework 1 Solutions: Kinematics of a Particle

Prof. Nicholas Zabaras

Warwick Centre for Predictive Modelling

University of Warwick

Coventry CV4 7AL

United Kingdom

Email: [email protected]

URL: http://www.zabaras.com/

February 14, 2016

1

mailto:[email protected]

http://www.zabaras.com/


2

A A

2

B B

18 m/s , a 2 m/s

12 m/s , a 3 m/s

B A B/AV V V

o o

B/A-12 j (-18 cos 60 i 18 sin 60 j) V

B/AV { 9 i 3.588 j} m/s

2 2

/ (9) (3.558) 9.69 m/sB A /1 1

/

( ) 3.588tan tan ( )

( ) 9

B A y

B A x

21.7o o o

Aa {2 cos 60 i 2 sin 60 j}

2 22

B n t

(12)a a i a j - i - 3 j - i 3j { 1.44i 3 j} /

100m s

B A B/Aa a a

B/Aa {-2.44 i - 4.732 j }

2 2 2

/ (2.44) (4.732) 5.32 m/sB Aa /

/

( ) 4.732tan

( ) 2.44

B A y

B A x

62.7o

At the instant shown cars A and B are traveling with

speeds of 18m/s and 12m/s, respectively. Also at this

instant A has a decrease in peed of 2m/s2 and B has an

increa. in speed of 3m/s2. Determine the velocity and

acceleration of B with respect to A.

Problem 1

2


Problem 2

Position coordinate

“cord equation”

Velocity

Acceleration

3A Bs s l

3 0A B

3 0A Ba a

3A B

3A Ba a

3* 6 18 ft/sA

Determine the speed of block A in the Fig.

if block B has an upward speed of 6 ft/s.

3


Problem 3

Position coordinate

“cord equation”

Velocity

Acceleration

1 22 ( )A C B B Cs s l and s s s l

4 0A B

4 0A Ba a

2 14 2A Bs s l l const

4 4* 6 24 ft/sA B

4A Ba a

Determine the speed of A in the Fig. if

B has an upward speed of 6ft/sec.

4


1

2( ) ( )

C B

A C B C B

s s l

s s s s s l

Position coordinate “cord equation”

2 14 2A Bs s l l const

4 0A B

1 1*2 0.5 ft/s

4 4B B

Determine the speed of block B in the

Fig. if the end of the cord at A is pulled

down with a speed of 2m/s

Problem 4

5


Problem 5

6

Establish a relation y = f(x)

Taking time derivatives gives (check it!):

2 2(15) & 15DA CDl x l y

DA CDl l l

2 230 (15) (15 )x y

2225 15 ( 10 , 20 )y x at y m x m

/ /s Ady dt and dx dt

2

1 2

2 225s

dy x dx

dt dtx

2

20

0.4 /225

S A

x m

xm s

x

Cord =30 m

A man at A is hoisting a safe A as shown by

walking to the right with a constant velocity vA

= 0.5 m/sec. Determine the velocity and

acceleration of the safe when it reaches the

elevation of 10m. The rope is 30m long and

passes over a small pulley at D.

22 2

3/2 3/2 3/22 2 2

20

225 225 2250.5 / 0.0036 /

225 225 225 20

S SS A A A

x m

d d dxa m s m s

dt dx dt x x m


Problem 6

Pulley D is attached to a collar

which is pulled down at 3 in./s. At t

= 0, collar A starts moving down

from K with constant acceleration

and zero initial velocity. Knowing

that velocity of collar A is 12 in./s as

it passes L, determine the change in

elevation, velocity, and acceleration

of block B when block A is at L.

SOLUTION:

• Define origin at upper horizontal surface

with positive displacement downward.

• Collar A has uniformly accelerated

rectilinear motion. Solve for acceleration

and time t to reach L.

• Pulley D has uniform rectilinear motion.

Calculate change of position at time t.

• Block B motion is dependent on motions

of collar A and pulley D. Write motion

relationship and solve for change of

block B position at time t.

• Differentiate motion relation twice to

develop equations for velocity and

acceleration of block B.7


Problem 6

SOLUTION:

• Define origin at upper horizontal surface with

positive displacement downward.

• Collar A has uniformly accelerated rectilinear

motion. Solve for acceleration and time t to reach

L.

2

2

020

2

s

in.9in.82

s

in.12

2

AA

AAAAA

aa

xxavv

s 333.1s

in.9

s

in.12

2

0

tt

tavv AAA

8


Problem 6

• Pulley D has uniform rectilinear motion.

Calculate change of position at time t.

in. 4s333.1s

in.30

0

DD

DDD

xx

tvxx

• Block B motion is dependent on motions of collar

A and pulley D. Write motion relationship and

solve for change of block B position at time t.

Total length of cable remains constant,

0in.42in.8

02

22

0

000

000

BB

BBDDAA

BDABDA

xx

xxxxxx

xxxxxx

in.160 BB xx

9


Problem 6

• Differentiate motion relation twice to develop

equations for velocity and acceleration of block B.

0s

in.32

s

in.12

02

constant2

B

BDA

BDA

v

vvv

xxx

s

in.18Bv

0s

in.9

02

2

B

BDA

v

aaa

2s

in.9Ba

10


Problem 7

The slotted fork is rotating about O at a

constant rate of 3 rad/s. Determine the radial

and transverse components of velocity and

acceleration of the pin A at the instant =

360o. The path is defined by the spiral

groove r = (5+/p in., where is in radians.

2

2

35 7 in in/s 0 in/sr r r

p

p

p p p

2360 2 rad 3 rad/s 0 rad/so p

30.955 in/srv r

p

7(3) 21in/sv r

2 2 20 7(3) 63 in/sra r r

232 0 2( )(3) 5.73 in/sa r r

p

11


2 o

0.5(1 cos ) ft

4 ft/s

30 ft/s at 180

find and

r

v

a

0.5(1 cos )r

0.5(sin )r

0.5(cos ) ( ) 0.5(sin )r

o

2

at 180

1 ft 0 0 5r r r - . θ

Due to the rotation of the forked rod, the ball travels

around the slotted path a portion of which is in the

shape of a cardioid r = 0.5(1 - cos) ft where in

radian s. If the ball’s velocity is v=4ft/s and its

acceleration a=30 ft /s2 at the instant = 180°,

determine the angular velocity and angular

acceleration of the fork.

Problem 8

12


0.5(1 cos )r

0.5(sin )r

0.5(cos ) ( ) 0.5(sin )r o

2

at 180

1 ft 0 0 5r r r - . θ

2 2 2 2( ) ( ) (0) (1 ) 4 4 rad/sr r

2 2 2 2 2 2 2( ) ( 2 ) [ 0.5(4) 1(4) ] [1 2(0)(4)] 30a r r r r

218 rad/s

Problem 8

13


Problem 9

2* 2*4 8 / 02

r m r m s rp

p

2 2(8) (12.56) 14.89 / m s 2 2 2(50.24) (64) 81.36 /a m s

A collar slides along the smooth vertical spiral rod, r =

(2) m, where is in radians. If its angular rate of

rotation is constant and equal 4 rad/s, at the instant =

90o. Determine

- The collar radial and transverse component of velocity

- The collar radial and transverse component of

acceleration.

- The magnitude of velocity and acceleration

2 2 2r r r

24 / 0 /2

rad rad s rad sp

8 m/srv r

(4) 12.56 m/sv r p

2 2 20 (4) 50.24 m/sra r r p

22 0 2(8)(4) 64 m/sa r r

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Homework 1 Solutions: Kinematics of a Particleibilion/ · 2016. 2. 14. · Introduction to Dynamics...

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Transcript of Homework 1 Solutions: Kinematics of a Particleibilion/ · 2016. 2. 14. · Introduction to Dynamics...