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Problem 6.35
A bicycle has wheels 700 mm in diameter and a gear set with the dimensions given in the table below.
Crank
Sprocket C1 C2 C3
No. of Cogs 26 36 48
Radius (mm) 52.6 72.8 97.0
Cassette (9 speeds)
Sprocket S1 S2 S3 S4 S5 S6 S7 S8 S9
No. of Cogs 11 12 14 16 18 21 24 28 34
Radius (mm) 22.2 24.3 28.3 32.3 36.4 42.4 48.5 56.6 68.7
If a cyclist has a cadence of
1Hz
, determine the angular speed of the rear wheel in rpm when using the
combination of C3 and S2. In addition, knowing that the speed of the cyclist is equal to the radius of the
wheel times its angular speed, determine the cyclist’s speed in m=s.
Solution
Using C3 and S2, the radii of the crank, sprocket, and wheel are
RcD97:0 mm
,
RsD24:3 mm
, and
RwD0:7000
m, respectively. Let
!c
,
!s
, and
!w
be the angular speeds of the crank, sprocket, and wheel,
respectively. The sprocket is fixed to the wheel, so
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Dynamics 2e 1201
Problem 6.36
A bicycle has wheels 700 mm in diameter and a gear set with the dimensions given in the table below.
Crank
Sprocket C1 C2 C3
No. of Cogs 26 36 48
Radius (mm) 52.6 72.8 97.0
Cassette (9 speeds)
Sprocket S1 S2 S3 S4 S5 S6 S7 S8 S9
No. of Cogs 11 12 14 16 18 21 24 28 34
Radius (mm) 22.2 24.3 28.3 32.3 36.4 42.4 48.5 56.6 68.7
If a cyclist has a cadence of
68 rpm
, determine which combination of chain ring (a sprocket mounted
on the crank) and (rear) sprocket would most closely make the rear wheel rotate with an angular speed
of
127 rpm
. Having found a chain ring/sprocket combination, determine the wheel’s exact angular speed
corresponding to the chosen chain ring/sprocket combination and the given cadence.
Solution
Let
!c
,
!w
, and
!s
be the angular speeds of the crank, wheel, and sprocket, respectively. Let
Rc
,
Rw
, and
Rs
be the radii of the crank, wheel, and sprocket, respectively. The chain does not slip with respect to the
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
For this choice of gears, the corresponding angular speed of the wheel is
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 1203
Problem 6.37
A bicycle has wheels 700 mm in diameter and a gear set with the dimensions given in the table below.
Crank
Sprocket C1 C2 C3
No. of Cogs 26 36 48
Radius (mm) 52.6 72.8 97.0
Cassette (9 speeds)
Sprocket S1 S2 S3 S4 S5 S6 S7 S8 S9
No. of Cogs 11 12 14 16 18 21 24 28 34
Radius (mm) 22.2 24.3 28.3 32.3 36.4 42.4 48.5 56.6 68.7
If a cyclist is pedaling so that the rear wheel rotates with an angular speed of
16 rad=s
, determine all
possible (rear) sprocket/chain ring (crank-mounted sprocket) combinations that would allow him or her to
pedal with a frequency within the range 1:00–1:25 Hz.
Solution
Let
!c
,
!w
, and
!s
be the angular speeds of the crank, wheel, and sprocket, respectively. Let
Rc
,
Rw
, and
Rs
be the radii of the crank, wheel, and sprocket, respectively. The sprocket and wheel rotate together so that
!sD!w. In addition, since the chain does not slip with respect to the crank or sprocket, we can write
Rs
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permission of McGraw-Hill, is prohibited.
Problem 6.38
At the instant shown, the angle
D30ı
,
jEvAj D 292 ft=s
, and the
turbine is rotating clockwise. Letting
OA DR
,
OB DR=2
,
RD
182 ft
, and treating the blades as being equally spaced, determine
the velocity of point
B
at the given instant and express it using the
component system shown.
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 1205
Problem 6.39
At the instant shown, the angle
D30ı
, the turbine is rotating
clockwise, and
EaBD.70:8 O{12:8 O| / m=s2
. Letting
OA DR
,
OB DR=2
,
RD55:5
m, and assuming the blades are equally
spaced, determine the angular velocity and angular acceleration
of the turbine blades, as well as the acceleration of point
A
at the
given instant.
Solution
Since Bis in fixed axis rotation about point O, we can write the acceleration of Bas
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
where, using the given component system:
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 1207
Problem 6.40
A carrier is maneuvering so that, at the instant shown,
jEvAj D
25 knots
(1 kn is exactly equal to
1:852 km=h
) and
D33ı
.
Letting the distance between
A
and
B
be
220
m and
D22ı
,
determine
EvB
at the given instant if the ship’s turning rate at this
instant is P
D2ı=s clockwise.
Solution
Since Aand Bare on the same rigid body, we can write
EvBD EvAC E!carrier ErB=A;(1)
where, using the given component system,
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Problem 6.41
At the instant shown, the pinion is rotating between two racks with an
angular velocity
!PD55 rad=s
. If the nominal radius of the pinion
is
RD4cm
and if the lower rack is moving to the right with a speed
vLD1:2 m=s, determine the velocity of the upper rack.
Solution
Defining
A
and
B
to be the contact points between the pinion and the top
and bottom racks, respectively, because
B
must have the same velocity as the
lower rack, we can write
of McGraw-Hill, and must be surrendered upon request of McGraw-Hill. Any duplication or distribution, either in print or electronic form, without the
permission of McGraw-Hill, is prohibited.
Dynamics 2e 1209
Problem 6.42
At the instant shown, the lower rack is moving to the right with a speed
of
vLD4ft=s
, while the upper rack is fixed. If the nominal radius of
the pinion is
RD2:5 in:
, determine
!P
, the angular velocity of the
pinion, as well as the velocity of point O, i.e., the center of the pinion.
Solution
Let
A
and
B
be the contact points between the pinion and the top and bottom
racks, respectively. The velocity of point
A
must be zero since the top rack
is fixed, and the velocity of
B
must be the same as that of the bottom rack,
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