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Problem 5.179
A Pelton impulse wheel, as shown in Fig. P5.136(a), is typically found in hydroelectric power plants and
consists of a wheel with a series of buckets attached at the periphery. As shown in Fig. P5.136(b), water
jets impinge on the buckets and cause the wheel to spin about its axis (labeled
O
). Let
vw
and
.Pm
f/nz
be
the speed and the mass flow rate of the water jets at the nozzles (the nozzles are stationary), respectively.
As the wheel spins, a given water jet will impinge on a given bucket only for a very small portion of the
bucket’s trajectory. This fact allows us to model the motion of a bucket relative to a given jet (during the
time the bucket interacts with that jet) as essentially rectilinear and with constant relative speed, as was
done in Example 5.17. Although each bucket moves away from the jet, the fact that they are arranged in
a wheel results in an effective mass flow rate experienced by the vanes is
.Pm
f/nz
instead of the reduced
mass flow rate computed in Eq. (6) of Example 5.17. With this in mind, consider a bucket, as shown in Fig.
P5.136(c), that is moving with a speed
v0
horizontally away from a fixed nozzle, but subject to a mass
flow rate
.Pm
f/nz
. The inside of the bucket is shaped so as to redirect the water jet laterally out (away from
the plane of the wheel). The angle
✓
describes the orientation of the velocity of the fluid relative to the
(moving) bucket at
B
, the point at which the water leaves the bucket. Determine
✓
and
v0
such that the
power transmitted by the water to the wheel is maximum. Express v0in terms of vw.
Photo credit: Courtesy of Andritz Hydro, Austria
Solution
Due to the symmetry of the shape of a bucket, we can study the flow over half of a bucket. Using the
arguments presented in Example 5.17, under the assumption that the bucket is moving at constant velocity,
we can choose a control volume moving with the bucket. It is sufficient to study the motion only in the
horizontal direction. As explained in Example 5.17, the velocity of the water flow over the vanes must be
understood as relative velocity of the water with respect to the vanes.
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Dynamics 2e 1121
Note that we have ignored the forces in the
y
direction. This is due to the fact that the forces in the
y
direction
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Problem 5.180
In Major League Baseball, a pitched ball has been known to hit the head of the batter (sometimes
unintentionally and sometimes not). Let the pitcher be, for example, Nolan Ryan who can throw a
51
8oz
baseball that crosses the plate at
100 mph
. Studies have shown that the impact of a baseball with a person’s
head has a duration of about
1ms
. So using Eq. (5.9) on p. 315 and assuming that the rebound speed of the
ball after the collision is negligible, determine the magnitude of the average force exerted on the person’s
head during the impact.
Solution
By a straightforward application of Eq. (5.9) on p. 315, we have
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Dynamics 2e 1123
Problem 5.181
A
0:6 kg
ball that is initially at rest is dropped on the floor from a height of
1:8
m and has a rebound height of
1:25
m. If the ball spends a total of
0:01
s
in contact with the ground, determine the average force applied to the ball by
the ground during the rebound. In addition, determine the ratio between the
magnitude of the impulse provided to the ball by the ground and the magnitude
of the impulse provided to the ball by gravity during the time interval that the
ball is in contact with the ground. Neglect air resistance.
Solution
We assume that the ball is only subject to (constant) gravity and, when in contact with the ground,
to a reaction force normal to the ground. Hence, the preimpact velocity can be determined using
constant acceleration equations, as follows:
p2ghi;(1)
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impulse provided by gravity to the ball during the impact is
t.mg O|/
. Hence, referring to the expression
larger than the corresponding impulse provided by gravity.
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Dynamics 2e 1125
Problem 5.182
A person
P
is initially standing on a cart on rails, which is moving
to the right with a speed
v0D2m=s
. The cart is not being propelled
by any motor. The combined mass of person
P
, the cart, and all that
is being carried on the cart is
270 kg
. At some point a person
PA
standing on a stationary platform throws to person
P
a package
A
to
the right with a mass
mAD50 kg
. Package
A
is received by
P
with a
horizontal speed
vA=P D1:5 m=s
. After receiving the package from
A
, person
P
throws a package
B
with a mass
mBD45 kg
toward
a second person
PB
. The package intended for
PB
is thrown to the
right, i.e., in the direction of the motion of
P
, and with a horizontal
speed
vB=P D4m=s
relative to
P
. Determine the final velocity of
the person
P
. Neglect any friction or air resistance acting on
P
and
the cart.
Solution
Balance Principles.
Letting the subscripts
3
and
4
denote the time instants right before and right after the
person
P
throws the package, and applying the impulse-momentum principle as conservation of momentum
in th xdirection, we have
Force Laws. Again, all forces are accounted for on the FBD.
Kinematic Equations. We note that
Computation.
Substituting Eqs. (5) into Eq. (4), using the result in Eq. (3), and solving for
.vPx/4
, we
have
.vPx/4D.vPx/1CmA
vA=P mB
vB=P :(6)
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Dynamics 2e 1127
Problem 5.183
A Ford Excursion
A
, with a mass
mAD3900 kg
, traveling with a speed
vAD85 km=h
, collides head-on
with a Mini Cooper
B
, with a mass
mBD1200 kg
, traveling in the opposite direction with a speed
vBD40 km=h
. Determine the postimpact velocities of the two cars if the impact’s coefficient of restitution
is eD0:22. In addition, determine the percentage of kinetic energy loss.
Solution
whose solution is
vC
Ax DmBŒe.v
Bx v
Ax/Cv
BxçCmAv
Ax
mACmBD13:64 m=s;(4)
1128 Solutions Manual
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1129
Problem 5.184
The two spheres,
A
and
B
, with masses
mAD1:35 kg
and
mBD
2:72 kg
, respectively, collide with
v
AD26:2 m=s
and
v
BD
22:5 m=s
. Let
˛D45ı
. Compute the value of
ˇ
if the component
of the postimpact velocity of
B
along the LOI is equal to zero and
if the COR is eD0:63.
AD26:2 m=s
BD22:5 m=s
postimpact component of velocity along the LOI for particle Bto be equal to zero, i.e.,
vC
Bx D0: (7)
Computation.
Substituting the first of Eqs. (5), the first of Eqs. (6), and Eq. (7) into Eqs. (1) and (4) yields
two equations in the two unknowns vC
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