120 Solutions Manual
Problem 2.78
As we will see in Chapter 3, 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
RxD
g.k=m/.x L0/.
Let
kD8lb=ft
,
mD0:048 slug
, and
L0D2:5 ft
. If the particle is released from
rest at
xD0ft
, determine how long it takes for the spring to achieve its maximum
length. Hint: A good table of integrals will come in handy.
Solution
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Dynamics 2e 121
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122 Solutions Manual
Problem 2.79
A weight
A
with mass
mD18 kg
is attached to the free end of a nonlinear spring such
that the acceleration of
A
is
aDg.=m/.y L0/3
, where
g
is the acceleration
due to gravity,
is a constant, and
L0D0:5
m. Determine
such that
A
does not fall
below yD1m when released from rest at yDL0.
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Dynamics 2e 123
Problem 2.80
Two masses
mA
and
mB
are placed at a distance
r0
from one another. Because of their mutual gravitational
attraction, the acceleration of sphere Bas seen from sphere Ais given by
RrDG✓mACmB
r2◆;
where
GD6:674⇥1011 m3=.kgs2/D3:439⇥108ft3=.slugs2/
is the universal gravitational constant.
If the spheres are released from rest, determine
(a) The velocity of B(as seen by A) as a function of the distance r.
(b)
The velocity of
B
(as seen by
A
) at impact if
r0D7ft
, the weight of
A
is
2:1 lb
, the weight of
B
is
0:7 lb, and
(i) The diameters of Aand Bare dAD1:5 ft and dBD1:2 ft, respectively.
(ii) The diameters of Aand Bare infinitesimally small.
Solution
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permission of McGraw-Hill, is prohibited.
124 Solutions Manual
Problem 2.81
Two masses
mA
and
mB
are placed at a distance
r0
from one another. Because of their mutual gravitational
attraction, the acceleration of sphere Bas seen from sphere Ais given by
RrDG✓mACmB
r2◆;
where
GD6:674⇥1011 m3=.kgs2/D3:439⇥108ft3=.slugs2/
is the universal gravitational constant.
Assume that the particles are released from rest at rDr0.
(a) Determine the expression relating their relative position rand time. Hint:
Zpx=.1 x/ dx Dsin1pxpx.1 x/:
(b)
Determine the time it takes for the objects to come into contact if
r0D3
m,
A
and
B
have masses of
1.1 and 2:3 kg, respectively, and
(i) The diameters of Aand Bare dAD22 cm and dBD15 cm, respectively.
(ii) The diameters of Aand Bare infinitesimally small.
Solution
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 125
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permission of McGraw-Hill, is prohibited.
126 Solutions Manual
Problem 2.82
Suppose that the acceleration Rrof an object moving along a straight line takes on the form
RrDG✓mACmB
r2◆;
where the constants
G
,
mA
, and
mB
are known. If
Pr.0/
is given, under what conditions can you determine
Pr.t/ via the following integral?
Pr.t/ DPr.0/ Zt
0
GmACmB
r2dt:
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Dynamics 2e 127
Problem 2.83
If the truck brakes hard enough that the crate slides to the right
relative to the truck, the distance
d
between the crate and the front
of the trailer changes according to the relation
R
dD(kgCaTfor t<t
s;
kgfor t>t
s;
where
ts
is the time it takes the truck to stop,
aT
is the acceleration
of the truck,
g
is the acceleration of gravity, and
k
is the kinetic
friction coefficient between the truck and the crate. Suppose that
the truck and the crate are initially traveling to the right at
v0D
60 mph
and the brakes are applied so that
aTD10:0 ft=s2
.
Determine the minimum value of
k
so that the crate does not hit
the right end of the truck bed if the initial distance
d
is
12 ft
.Hint:
The truck stops before the crate stops.
Solution
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permission of McGraw-Hill, is prohibited.
128 Solutions Manual
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 129
Problem 2.84
Cars
A
and
B
are traveling at
vAD72 mph
and
vBD67 mph
,
respectively, when the driver of car
B
applies the brakes abruptly,
causing the car to slide to a stop. The driver of car
A
takes
1:5
s to
react to the situation and applies the brakes in turn, causing car
A
to
slide as well. If
A
and
B
slide with equal accelerations, i.e.,
RsAD
RsBDkg
, where
kD0:83
is the kinetic friction coefficient
and
g
is the acceleration of gravity, compute the minimum distance
d
between
A
and
B
at the time
B
starts sliding to avoid a collision.
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