Mechanical Engineering Chapter 12 Homework The Peg Constrained Move The Slots The

181

SOLUTION

Velocity. Using the chain rule, the first and second time derivatives of r can be

determined.

The radial and transverse components of the velocity are

Since

u

#

is in the opposite sense to that of positive

u

, u

#

=–

6 rad

>

s. Thus, at u

=150°,

Thus, the magnitude of the velocity is

These components are shown in Fig. a

*12–172.

The rod OA rotates clockwise with a constant angular velocity

of 6 rad

>

s. Two pin-connected slider blocks, located at B, move

freely on OA and the curved rod whose shape is a limaçon

described by the equation r = 200(2 − cos

u

) mm. Determine

the speed of the slider blocks at the instant

u=150°

.

O

400 mm

200 mm

600 mm

r

6 rad/s

B

A

u

Ans:

182

SOLUTION

Acceleration. Using the chain rule, the first and second time derivatives of r can be

determined

Here, since

u

#

is constant,

u

$

=0.

Since

u

#

is in the opposite sense to that of positive

u

,

The radial and transverse components of the acceleration are

Thus, the magnitude of the acceleration is

12–173.

Determine the magnitude of the acceleration of the slider

blocks in Prob. 12–172 when

u

= 150°.

O

400 mm

200 mm

600 mm

r

6 rad/s

B

A

u

Ans:

183

SOLUTION

r2=4 cos 2u

u

12–174.

A double collar C is pin connected together such that one

collar slides over a fixed rod and the other slides over a

rotating rod. If the geometry of the fixed rod for a short

distance can be defined by a lemniscate, r2 = (4 cos 2

u

) ft2,

determine the collar’s radial and transverse components of

velocity and acceleration at the instant

u

= 0° as shown. Rod

OA is rotating at a constant rate of

u

#

= 6 rad

>

s.

r2 fi 4 cos 2 u

u fi 6 rad/s

O

rC

A

·

Ans:

184

12–175.

SOLUTION

r=4t|t=1=4r

#

=4

A block moves outward along the slot in the platform with

a speed of where tis in seconds.The platform

rotates at a constant rate of 6 rad/s. If the block starts from

rest at the center, determine the magnitudes of its velocity

and acceleration when t=1s.

r

#

=14t2m>s,

θ

·=6rad/s

r

$

Ans:

185

SOLUTION

v

=

20

m>s

Thus

r=282.84

*12–176.

The car travels around the circular track with a constant

speed of 20 m

>

s. Determine the car’s radial and transverse

components of velocity and acceleration at the instant

u

=

p

>4 rad

.

r  (400 cos u) m

r

u

Ans:

vr=–14.1 m>s

186

SOLUTION

u

=0.006

t

2

0

t=4=0.096 rad =5.50°

=0.012

At

u=0.096 rad

12–177.

The car travels around the circular track such that its

transverse component is

u

= (0.006t2) rad, where t is in

seconds. Determine the car’s radial and transverse

components of velocity and acceleration at the instant t

=

4 s.

r  (400 cos u) m

r

u

Ans:

187

12–178.

SOLUTION

r=

200

u`u=p>3 rad

=

600

p ft

The car travels along a road which for a short distance

is deﬁ ned by r=(200>u) ft, where u is in radians. If it

maintains a constant speed of v=35 ft>s, determine the

radial and transverse components of its velocity when

u=p>3 rad. r

θ

Ans:

188

SOLUTION

r=8

u=0.6 t

u

At

t=4 s

12–179.

A horse on the merry-go-round moves according to the

equations r

=

8 ft,

u

=

(0.6t) rad, and z

=

(1.5 sin

u

) ft,

where t is in seconds. Determine the cylindrical components

of the velocity and acceleration of the horse when t

=

4 s.

z

r

u

Ans:

189

SOLUTION

r=8

r

#

=0

u

#

=2

u

=0

*12–180.

A horse on the merry-go-round moves according to the

equations r

=

8 ft,

u

#

=

2 rad

>

s and z

=

(1.5 sin

u

) ft, where

t is in seconds. Determine the maximum and minimum

magnitudes of the velocity and acceleration of the horse

during the motion.

z

r

u

Ans:

vmax =16.3

ft>s

vmin =16

190

12–181.

If the slotted arm rotates counterclockwise with a

constant angular velocity of , determine the

magnitudes of the velocity and acceleration of peg at

.The peg is constrained to move in the slots of the

fixed bar and rotating bar .ABCD

u=30°

P

u

#

=2 rad>s

AB

SOLUTION

Time Derivatives:

Velocity:

Acceleration:

r=4 sec u

A

D

P

C

r (4 sec ) ft

u

4 ft

Ans:

191

12–182.

The peg is constrained to move in the slots of the fixed bar

and rotating bar .When , the angular

velocity and angular acceleration of arm are

and ,respectively.Determine

the magnitudes of the velocity and acceleration of the peg

at this instant.P

u

$

=3 rad>s2

u

#

=2 rad>s

AB

u=30°ABCD

SOLUTION

Time Derivatives:

When ,

Velocity:

Acceleration:

u=30°

r=4 sec u

A

D

B

P

C

r

fi

(4 sec

)

ft

u

4 ft

Ans:

192

12–183.

SOLUTION

Since

v=ru

#

r

#

=0

r=60

A truck is traveling along the horizontal circular curve of

radius with a constant speed

Determine the angular rate of rotation of the radial line r

and the magnitude of the truck’s acceleration.

u

#v=20 m>s.

r=60 m

rfi60 m

·

u

Ans:

193

*12–184.

A truck is traveling along the horizontal circular curve of

r

adius with a speed of which is increasing

at

Determine the truck’s radial and transverse

components of acceleration.

3m>s2.

20 m>sr=60 m

SOLUTION

r=60

rfi60 m

·

u

Ans:

194

SOLUTION

#

u=5

12–185.

The rod OA rotates counterclockwise with a constant

angular velocity of

u

#

= 5 rad

>

s. Two pin-connected slider

blocks, located at B, move freely on OA and the curved rod

whose shape is a limaçon described by the equation

r = 100(2 − cos

u

) mm. Determine the speed of the slider

blocks at the instant

u

= 120°.

A

u

O

B

r

y

x

·

u fi 5 rad/s

Ans:

195

SOLUTION

u

#

=5

u

#

=0

12–186.

Determine the magnitude of the acceleration of the slider

blocks in Prob. 12–185 when

u

= 120°.

A

u

O

B

r

y

x

·

u fi 5 rad/s

196

12–187.

The searchlight on the boat anchored 2000 ft from shore is

turned on the automobile, which is traveling along the

straight road at a constant speed of Determine the

angular rate of rotation of the light when the automobile is

from the boat.r=3000 ft

80 ft>s.

80 ft/s

ru

u

SOLUTION

r=2000 csc u

Ans:

*12–188.

$

required angular acceleration of the light at this instant.u

a velocity of at the instant determine ther=3000 ft,80 ft>s

If the car

>

is accelerating at and has 15 ft s2

80 ft/s

ru

u

SOLUTION

r

#

=-2000 csc uctn uu

#

r=2000 csc u

in Prob. 12–187

Ans:

198

12–189.

A particle moves along an Archimedean spiral ,

where is given in radians. If (constant),

determine the radial and transverse components of the

particle’s velocity and acceleration at the instant

. Sketch the curve and show the components on

the curve.

u=p>2 rad

u

#

=4 rad>su

r

=(8

u

)ft

SOLUTION

Time Derivatives: Since is constant, .

Velocity:Applying Eq. 12–25, we have

Acceleration: Applying Eq. 12–29, we have

u

$

=0u

#

y

x

u

r(8 u)ft

r

199

12–190.

SOLUTION

Time Derivatives:Here,

Velocity: Applying Eq. 12–25, we have

Acceleration: Applying Eq. 12–29, we have

Solve Prob. 12–189 i f the particle has an angular

acceleration when at rad.u=p2u

#

=4 rad>su

$

=5 rad>s2

y

x

u

r(8 u)f

t

r

>

Ans:

v

r=

32.0

ft>s

200

12–191.

SOLUTION

At ,

z=-0.8382

t=3s

u=0.5 tr=3z=3 sin 2t

r

z

A

u

The arm of the robot moves so that r = 3 ft is constant, and

4

1

its grip

A moves along the path z = 3 sin u2ft, where u is in

radians. If ut= 10.5 2rad, where t is in seconds, determine the

magnitudes

of the grip’s

velocity and acceleration when t = 3 s.

Ans: