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Problem 17.1 In Active Example 17.1, suppose that
at a given instant the hook His moving downward at
2 m/s. What is the angular velocity of gear Aat that
instant?
A
50 mm
B
200 mm
100
mm
Problem 17.2 The angle θ(in radians) is given as a
function of time by θ=0.2πt2.Att=4 s, determine
the magnitudes of (a) the velocity of point Aand (b) the
tangential and normal components of acceleration of
point A.
A
2 ft
θ
Solution: We have
Problem 17.3 The mass Astarts from rest at t=0 and
falls with a constant acceleration of 8 m/s2. When the
mass has fallen one meter, determine the magnitudes of
(a) the angular velocity of the pulley and (b) the tangen-
tial and normal components of acceleration of a point at
the outer edge of the pulley.
100 mm
Solution: We have
Problem 17.4 At the instant shown, the left disk has
an angular velocity of 3 rad/s counterclockwise and an
angular acceleration of 1 rad/s2clockwise.
(a) What are the angular velocity and angular accel-
eration of the right disk? (Assume that there is
no relative motion between the disks at their point
of contact.)
(b) What are the magnitudes of the velocity and accel-
eration of point A?
2.5 m
1 m
A
2 m
3 rad/s
1 rad/s2
Solution:
rLωL=rRωR⇒ωR=rR
rL
ωL=1m
2.5 m (3 rad/s)=1.2 rad/s
Problem 17.5 The angular velocity of the left disk is
given as a function of time by ωA=4+0.2trad/s.
(a) What are the angular velocities ωBand ωCat
t=5s?
(b) Through what angle does the right disk turn from
t=0tot=5s?
100 mm
100 mm
200 mm
200 mm
ω
A
ω
B
ω
C
334
Problem 17.6 (a) If the bicycle’s 120-mm sprocket
wheel rotates through one revolution, through how many
revolutions does the 45-mm gear turn? (b) If the angu-
lar velocity of the sprocket wheel is 1 rad/s, what is the
angular velocity of the gear? 45
mm
120
mm
Solution: The key is that the tangential accelerations and tangen-
tial velocities along the chain are of constant magnitude
vB=(0.045)ωBvA=(0.120)(1)
ωB=120
45 rad/s =2.67 rad/s
rB
dθB
dt =rA
dθA
dt
Integrating, we get
rBθB=rAθArA=0.120 m
rB=0.045 m
θB=120
45 (1)rev θA=1 rev.
rA
vA
rB
Problem 17.7 The rear wheel of the bicycle in Prob-
lem 17.6 has a 330-mm radius and is rigidly attached to
the 45-mm gear. It the rider turns the pedals, which are
rigidly attached to the 120-mm sprocket wheel, at one
revolution per second, what is the bicycle’s velocity?
Problem 17.8 The disk is rotating about the origin
with a constant clockwise angular velocity of 100 rpm.
Determine the xand ycomponents of velocity of points
Aand B(in in/s).
A
y
Solution:
Problem 17.9 The disk is rotating about the origin
with a constant clockwise angular velocity of 100 rpm.
Determine the xand ycomponents of acceleration of
points Aand B(in in/s2).
B
A
x
y
8 in
Solution:
336
Problem 17.10 The radius of the Corvette’s tires is
14 in. It is traveling at 80 mi/h when the driver applies
the brakes, subjecting the car to a deceleration of 25 ft/s2.
Assume that the tires continue to roll, not skid, on the
road surface. At that instant, what are the magnitudes
of the tangential and normal components of acceleration
(in ft/s2) of a point at the outer edge of a tire relative
to a nonrotating coordinate system with its origin at the
center of the tire?
y
x
an=rω2=14
12 ft(101 rad/s)2=11,800 ft/s2.
Problem 17.11 If the bar has a counterclockwise
angular velocity of 8 rad/s and a clockwise angular
acceleration of 40 rad/s2, what are the magnitudes of
the accelerations of points Aand B?
0.4 m
0.2 m
A
Solution:
aA=α×rA/O −ω2rA/O
aB=α×rB/O −ω2rB/O
Problem 17.12 Consider the bar shown in Problem
17.11. If |vA|=3 m/s and |aA|=28 m/s2, what are |vB|
and |aB|?
Problem 17.13 A disk of radius R=0.5 m rolls on
a horizontal surface. The relationship between the
horizontal distance xthe center of the disk moves and
the angle βthrough which the disk rotates is x=Rβ.
Suppose that the center of the disk is moving to the right
with a constant velocity of 2 m/s.
(a) What is the disk’s angular velocity?
(b) Relative to a nonrotating reference frame with
its origin at the center of the disk, what are the
magnitudes of the velocity and acceleration of a
point on the edge of the disk?
x
R
β
R
Problem 17.14 The turbine rotates relative to the coor-
dinate system at 30 rad/s about a fixed axis coincident
with the xaxis. What is its angular velocity vector?
y
338
Problem 17.15 The rectangular plate swings in the x–y
plane from arms of equal length. What is the angular
velocity of (a) the rectangular plate and (b) the bar AB?
y
x
A
B
10 rad/s
Solution: Denote the upper corners of the plate by Band B′, and
denote the distance between these points (the length of the plate) by
since L=L′, the figure AA′B′Bis a parallelogram. By definition, the
opposite sides of a parallelogram remain parallel, and since the fixed
side AA′does not rotate, then BB′cannot rotate, so that the plate does
not rotate and
ωBB′=0 .
Similarly, by inspection the angular velocity of the bar AB is
ωAB =10k(rad/s),
where by the right hand rule the direction is along the positive zaxis
(out of the paper).
A
L′
A′
Problem 17.16 Bar OQ is rotating in the clockwise
direction at 4 rad/s. What are the angular velocity vectors
of the bars OQ and PQ?
Strategy: Notice that if you know the angular velocity
of bar OQ, you also know the angular velocity of bar
PQ.
1.2 m
O
Q
1.2 m
y
x
4 rad / s
Problem 17.17 A disk of radius R=0.5 m rolls on
a horizontal surface. The relationship between the hor-
izontal distance xthe center of the disk moves and the
angle βthrough which the disk rotates is x=Rβ. Sup-
pose that the center of the disk is moving to the right
with a constant velocity of 2 m/s.
(a) What is the disk’s angular velocity?
(b) What is the disk’s angular velocity vector?
x
R
β
R
x
y
Problem 17.18 The rigid body rotates with angular
velocity ω=12 rad/s. The distance rA/B =0.4m.
(a) Determine the xand ycomponents of the velocity
of Arelative to Bby representing the velocity as
shown in Fig. 17.10b.
(b) What is the angular velocity vector of the
rigid body?
(c) Use Eq. (17.5) to determine the velocity of Arel-
ative to B.
x
y
BA
ω
rA/B
Solution:
y
340
Problem 17.19 The bar is rotating in the counterclock-
wise direction with angular velocity ω. The magnitude of
the velocity of point Ais 6 m/s. Determine the velocity
of point B.
0.4 m
A
y
Solution:
Problem 17.20 The bar is rotating in the counterclock-
wise direction with angular velocity ω. The magnitude
of the velocity of point Arelative to point Bis 6 m/s.
Determine the velocity of point B.
0.4 m
A
y
Solution:
rA/B =(0.8m)2+(0.6m)2=1m
Problem 17.21 The bracket is rotating about point O
with counterclockwise angular velocity ω. The magni-
tude of the velocity of point Arelative to point Bis
4 m/s. Determine ω.
A
B
120 mm
y
x
O
Solution:
Problem 17.22 Determine the xand ycomponents of
the velocity of point A.y
x
A
5 rad/s
O
2 m
30°
Solution: The velocity of point Ais given by:
Problem 17.23 If the angular velocity of the bar in
Problem 17.22 is constant, what are the xand ycom-
ponents of the velocity of Point A0.1 s after the instant
shown?
342
Problem 17.24 The disk is rotating about the zaxis at
50 rad/s in the clockwise direction. Determine the xand
ycomponents of the velocities of points A,B, and C.
x
y
100 mm
A
Solution: The velocity of Ais given by vA=v0+ω×rA/O ,or
vA=0+(−50k)×(0.1j)=5i(m/s).
=−3.54i−3.54j(m/s),
For C, we have
vc=v0+ω×rC/O =0+
ijk
00−50
−0.1 cos 45◦−0.1 sin 45◦0
=−3.54i+3.54j(m/s).
x
BC
45°45°
100 mm
Problem 17.25 Consider the rotating disk shown in
Problem 17.24. If the magnitude of the velocity of
point Arelative to point Bis 4 m/s, what is the
magnitude of the disk’s angular velocity?
Problem 17.26 The radius of the Corvette’s tires is
14 in. It is traveling at 80 mi/h. Assume that the tires
roll, not skid, on the road surface.
(a) What is the angular velocity of its wheels?
(b) In terms of the earth-fixed coordinate system shown,
determine the velocity (in ft/s) of the point of the
tire with coordinates (−14 in, 0,0).
y
x
Problem 17.27 Point Aof the rolling disk is moving
toward the right. The magnitude of the velocity of point
Cis 5 m/s. determine the velocities of points Band D.
y
45⬚
C
D
0.6 m
Solution: Point Bis the center of rotation (zero velocity).
rC/B =√2(0.6m)=0.849 m,
Problem 17.28 The helicopter is in planar motion in
the x–yplane. At the instant shown, the position
of its center of mass, G,isx=2m, y=2.5m,
and its velocity is vG=12i+4j(m/s). The position
of point T, where the tail rotor is mounted, is x=
−3.5m,y=4.5 m. The helicopter’s angular velocity is
0.2 (rad/s) clockwise. What is the velocity of point T?
y
x
T
G
344
Problem 17.29 The bar is moving in the x–yplane
and is rotating in the counterclockwise direction. The
velocity of point Arelative to the reference frame is
vA=12i−2j(m/s). The magnitude of the velocity of
point Arelative to point Bis 8 m/s. what is the velocity
of point Brelative to the reference frame?
y
B
2 m
30⬚
Solution:
ω=v
Problem 17.30 Points Aand Bof the 2-m bar slide
on the plane surfaces. Point Bis moving to the right at
3 m/s. What is the velocity of the midpoint Gof the bar?
Strategy: First apply Eq. (17.6) to points Aand B
to determine the bar’s angular velocity. Then apply
Eq. (17.6) to points Band G.
y
A
Solution: Take advantage of the constraints (B stays on the floor,
Problem 17.31 Bar AB rotates at 6 rad/s in the clock-
wise direction. Determine the velocity (in in/s) of the
slider C.
Solution:
vB=vA+ωAB ×rB/A
Problem 17.32 If θ=45◦and the sleeve Pis moving
to the right at 2 m/s, what are the angular velocities of
the bars OQ and PQ?
1.2 m
O
Q
P
θ
1.2 m
Solution: From the figure, v0=0, vP=vPi=2i(m/s)
Problem 17.33 In Active Example 17.2, consider the
instant when bar AB is vertical and rotating in the clock-
wise direction at 10 rad/s. Draw a sketch showing the
positions of the two bars at that instant. Determine the
angular velocity of bar BC and the velocity of point C.
0.4 m
10 rad/s
A
B
C
0.4 m 0.8 m
346
Problem 17.34 Bar AB rotates in the counterclockwise
direction at 6 rad/s. Determine the angular velocity of
bar BD and the velocity of point D.
y
0.32 m
D
Problem 17.35 At the instant shown, the piston’s velo-
city is vC=−14i(m/s). What is the angular velocity of
the crank AB?
y
Solution:
vB=vA+ωAB ×rB/A
Problem 17.36 In Example 17.3, determine the angu-
lar velocity fo the bar AB that would be necessary so
that the downward velocity of the rack VR=10 ft/s at
the instant shown.
Dv
R
6 in
B
C
A
12 in
10 rad/s
16 in6 in 6 in
10 in
Solution: We have
348
Problem 17.37 Bar AB rotates at 12 rad/s in the
clockwise direction. Determine the angular velocities of
bars BC and CD.
A
C
B
D
350
mm
200
mm
300 mm 350 mm
12 rad/s
Solution: The strategy is analogous to that used in Problem 17.36.
Problem 17.38 Bar AB is rotating at 10 rad/s in the
counterclockwise direction. The disk rolls on the circular
surface. Determine the angular velocities of bar BC and
the disk at the instant shown.
2 m
A
B
C
3 m
3 m
1 m
Solution: The point “D” at the bottom of the wheel has zero veloc-
ity.
Problem 17.39 Bar AB rotates at 2 rad/s in the coun-
terclockwise direction. Determine the velocity of the
midpoint Gof bar BC.
C
B
10 in
12 in
G
y
350
Problem 17.40 Bar AB rotates at 10 rad/s in the
counterclockwise direction. Determine the velocity of
point E.
x
y
A
B
C
10 rad/s E
D
400 mm
700 mm 700 mm400
mm
Problem 17.41 Bar AB rotates at 4 rad/s in the
counterclockwise direction. Determine the velocity of
point C.
A
B
C
D
Ex
y
600 mm
600 mm
400 mm
500 mm
300
mm
300
mm
200
mm
352
