Computation.
Substituting the force laws, the kinematic equations, and the expression for the mass moment
of inertia into the Newton-Euler equations, we obtain the following system of three equations in the three
unknowns N,aQx , and ˛c
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1537
Problem 7.63
A shop sign, with a uniformly distributed mass
mD30 kg
,
hD1:5
m,
wD
2m, and dD0:6 m, is at rest when cord AB suddenly breaks.
Modeling
AB
and
CD
as inextensible and with negligible mass, deter-
mine the tension in cord
CD
and the acceleration of the sign’s center of mass
immediately after AB breaks.
Solution
The FBD of the sign immediately after the cord
AB
breaks is shown at the
right.
Balance Principles. Based on this FBD, the Newton-Euler equations are
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.
Using these solutions, the acceleration of the center of mass of the sign is
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1539
Problem 7.64
A shop sign, with a uniformly distributed mass
mD30 kg
,
hD1:5
m,
wD
2m, and dD0:6 m, is at rest when cord AB suddenly breaks.
Modeling
AB
and
CD
as elastic cords with negligible mass and stiffness
kD8000 N=m
, determine the tension in cord
CD
and the acceleration of the
sign’s center of mass immediately after AB breaks.
Solution
The FBD of the sign immediately after the cord
AB
breaks is shown at the
right.
Balance Principles. Based on this FBD, the Newton-Euler equations are
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 7.65
Solve Example 7.3 on p. 532 by assuming that the crate just
tips. In doing so, show that this motion is not possible for the
given conditions since part of your solution will not be physically
admissible.
2Nh
2FPdh
2DIG˛c;
where ˛cis the angular acceleration of the crate and where, treating the crate as a uniform body,
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1541
Computation.
Substituting the kinematic equations and the expression for the mass moment of inertia into
the Newton-Euler equations, we obtain the following system of three equations in the three unknowns
N
,
F
,
and ˛c
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permission of McGraw-Hill, is prohibited.
Problem 7.66
The cord, which is wrapped around the inner radius of the spool of
mass
m
, is pulled vertically at
A
by a constant force
P
, causing the
spool to roll over the horizontal bar
BD
. Assuming that the cord
is inextensible and of negligible mass, that the spool rolls without
slip, and that its radius of gyration is
kG
, determine the angular
acceleration of the spool and the total force between the spool and
the bar.
R2Ck2
G!P; and ˛sD RP
m.R2Ck2
G/:
Therefore,
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Dynamics 2e 1543
Problem 7.67
Refer to the systems in Example 3.9 on p. 200 (particle separating
from semicylinder) and Example 7.10 on p. 558 (sphere separating
from semicylinder).
(a)
Determine the speed of the particle and that of the sphere
when each separates from the semicylinder.
(b)
Compare their speeds of separation and explain the sources
of any difference.
(c)
Determine the value of
such that the sphere and the particle
separate at the same speed.
Solution
Part (a)
The speed
vp
of the particle for any value of
is readily found using Eq. (10) in Example 3.9,
The speed of the center of the sphere for any angle
is found using Eq. (15) in Example 7.10, which
gives P
sas a function of for the sphere as
Since the angle of separation for the sphere was found to be
cos ss D10
17
, then the speed of the sphere at
Part (b)
Comparing
vsp
to
vss
, the two factors that determine which is larger are the factor of
2
3
versus the
factor of
10
17
and the size of
compared with that of
R
. Since
2
3>10
17
,
vss
will be smaller than
vsp
unless
is sufficiently large. We explore this in the solution to Part (c).
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permission of McGraw-Hill, is prohibited.
Part (c)
To get the
at which
vsp Dvss
, we simply set them equal to one another and solve for the
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 1545
Problem 7.68
Referring to the systems in Example 3.9 on p. 200 (particle sep-
arating from semicylinder) and Example 7.10 on p. 558 (sphere
separating from semicylinder), show that the sphere dynamically
behaves just as a particle if the interface between the sphere and
the semicylinder is frictionless. In this case, that will mean that the
sphere separates from the semicylinder at the same location as the
particle.
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