370 Solutions Manual
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 371
Problem 2.285
The motion of a peg sliding within a rectilinear guide is controlled by
an actuator in such a way that the peg’s acceleration takes on the form
RxDa0.2 cos 2!t ˇsin !t/
, where
t
is time,
a0D3:5 m=s2
,
!D
0:5 rad=s
, and
ˇD1:5
. Determine the total distance traveled by the
peg during the time interval
0
s
t5
s if
Px.0/ Da0ˇ=! C0:3 m=s
.
When compared with Prob. 2.57, why does the addition of
0:3 m=s
in
the initial velocity turn this into a problem that requires a computer to
solve?
Solution
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372 Solutions Manual
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 373
Problem 2.286
The acceleration of an object in rectilinear free fall while immersed in a linear viscous
fluid is
aDgCdv=m
, where
g
is the acceleration of gravity,
Cd
is a constant
drag coefficient,
v
is the object’s velocity, and
m
is the object’s mass. Letting
vD0
and
sD0
for
tD0
, where
s
is position and
t
is time, determine the position as a
function of time.
Solution
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374 Solutions Manual
Problem 2.287
Heavy rains cause a stretch of road to have a coefficient of friction
that changes as a function of location. Under these conditions, the
acceleration of a car skidding while trying to stop can be approxi-
mated by
RsD.kcs/g
, where
k
is the friction coefficient un-
der dry conditions,
g
is the acceleration of gravity, and
c
, with units
of
m1
, describes the rate of friction decrement. Let
kD0:5
,
cD0:015 m1
, and
v0D45 km=h
, where
v0
is the initial velocity
of the car. Determine the time it will take the car to stop and the
percent increase in stopping time with respect to dry conditions,
i.e., when cD0.Hint:
Z1
p1Cx2dx Dlog⇣xCp1Cx2⌘:
Solution
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Dynamics 2e 375
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permission of McGraw-Hill, is prohibited.
376 Solutions Manual
Problem 2.288
The acceleration of a particle of mass
m
suspended by a linear spring with spring constant
k
and unstretched length
L0
(when the spring length is equal to
L0
, the spring exerts
no force on the particle) is given by
RxDg.k=m/.x L0/
. Assuming that at
tD0
the particle is at rest and its position is
xD0
m, derive the expression of the particle’s
position xas a function of time. Hint: A good table of integrals will come in handy.
Solution
We have acceleration as a function of position and we integrate it as follows:
dx
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Dynamics 2e 377
Problem 2.289
In a movie scene involving a car chase, a car goes over the top of a ramp
at
A
and lands at
B
below. Letting
˛D18ı
and
ˇD25ı
, determine the
speed of the car at
A
if the car is to be airborne for a full
3
s. Furthermore,
determine the distance
d
covered by the car during the stunt, as well as
the impact speed and angle at
B
. Neglect aerodynamic effects. Express
your answer using the U.S. Customary system of units.
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378 Solutions Manual
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 379
Problem 2.290
Consider the problem of launching a projectile a distance
R
from
O
to
D
with a known launch speed
v0
. It is probably clear to you
that you also need to know the launch angle
✓
if you want the
projectile to land exactly at
R
. But it turns out that the condition
determining whether or not
v0
is large enough to get to
R
does
not depend on
✓
. Determine this condition on
v0
.Hint: Find
v0
as a function of
R
and
✓
, and then remember that the sine function
is bounded by 1.
Solution
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