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Dynamics 2e 1567
Problem 7.78
An inextensible cord of negligible mass is wound around a homogeneous
circular object. Assume that the cord is pulled to the right while remain-
ing horizontal, and determine the value of the object’s mass moment of
inertia
IG
, such that the object rolls without slip no matter how large the
tension in the cord. What is the shape of such an object?
Solution
The FBD of the circular object as it is being pulled by a force
P
is shown on the
right.
Balance Principles.
Based on the FBD shown, the Newton-Euler equations for
the object are
XFxW FCPDmaGx ;(1)
XFyWNmg DmaGy ;(2)
XMGW PR FR DIG˛o;(3)
where ˛ois the angular acceleration of the object and its unknown mass moment of inertia is IG.
Force Laws. For the object to roll without slip we must have
jFj sjNj:(4)
Kinematic Equations. Since the object rolls without slip, we have
aGx D R˛oand aGy D0: (5)
Computation.
Substituting the kinematic equations into the Newton-Euler equations and treating the force
P
and the mass moment of inertia
IG
as a given parameters, we obtain the following system of three equations
in the three unknowns N,F, and ˛o
FCPD mR˛o; N mg D0; and PR FR DIG˛o
the solution of which is
NDmg; F DIGmR2
IGCmR2P; and ˛oD 2PR
IGCmR2:(6)
This expression for the force
F
shows that, unless
F
were completely independent of
P
, for arbitrarily large
values of
P
the force
F
would take on corresponding large values that would cause the no slip condition
jFj sjNj
to be violated. In order for the force
F
to be independent of
P
, the numerator of the expression
for Fmust be equal to zero, that is,
IGDmR2:
The only homogeneous circular symmetric object with mass moment of inertia equal to
mR2
is a thin circular
ring of radius R.
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Problem 7.79
The spool of mass
m
, radius of gyration
kG
, inner radius
ri
, and
outer radius
ro
is placed on a horizontal conveyer belt. The cable that
is wrapped around the spool and attached to the wall is initially taut.
Both the spool and the conveyer belt are initially at rest when the
conveyer belt starts moving with acceleration
ac
. If the coefficient of
static friction between the conveyer belt and spool is
s
, determine
(a)
The maximum acceleration of the conveyer belt so that the
spool rolls without slipping on the belt
(b)
The initial tension in the cable that attaches the spool to the
wall
(c) The angular acceleration of the spool
Evaluate your answers for
mD500 kg
,
kGD1:3
m,
sD0:5
,
riD0:8 m, and roD1:6 m.
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Dynamics 2e 1569
Enforcing the rolling without slip condition at B, we have that aBx D acand so
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Problem 7.80
The thin uniform bar
AB
of mass
m
and length
L
hangs from a wheel
at
A
, which rolls freely on the horizontal bar
DE
. In the following
problems, neglect the mass of the wheel and assume that the wheel
never separates from the horizontal bar.
If the bar is released from rest at the angle
, determine, immediately
after release, the angular acceleration of the bar, the force on the bar at
A, and the acceleration of end A.
Solution
The FBD of the bar immediately after it is released is shown on the right.
Balance Principles.
Based on this FBD, the Newton-Euler equations for
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Dynamics 2e 1571
Problem 7.81
The thin uniform bar
AB
of mass
m
and length
L
hangs from a wheel
at
A
, which rolls freely on the horizontal bar
DE
. In the following
problems, neglect the mass of the wheel and assume that the wheel
never separates from the horizontal bar.
Find the equation(s) of motion of the bar, using the coordinates
x
and shown on the figure as the dependent variables.
Solution
The FBD of the bar is shown on the right.
Balance Principles.
Based on this FBD, the Newton-Euler equations for
the bar 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.
and FA
L
2sin D1
12mL2R
:
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Dynamics 2e 1573
Problem 7.82
The thin uniform bar
AB
of mass
m
and length
L
hangs from a wheel
at
A
, which rolls freely on the horizontal bar
DE
. In the following
problems, neglect the mass of the wheel and assume that the wheel
never separates from the horizontal bar.
Find the equation(s) of motion of the bar using the coordinates
x
and
shown on the figure as the dependent variables and then simulate the
system’s behavior by numerically solving the equations of motion for
5
s,
using
mD2kg
,
LD0:6
m,
x.0/ D0
m,
Px.0/ D0m=s
,
.0/ D60ı
,
and P
.0/ D0rad=s. Plot xand for 0t5s.
2sin DIG˛b;
where
˛b
is the angular acceleration of the bar and its mass moment of inertia
Force Laws. All forces have been accounted for on the FBD.
Kinematic Equations. Relating the acceleration of Gto that of A, we obtain
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Dynamics 2e 1575
Problem 7.83
A uniform thin rod is slightly nudged at
B
from the
D0
position so that it falls to the
right. The coefficient of static friction between the rod and the floor is s.
(a)
Determine as a function of
the normal force (
N
) and the frictional force (
F
)
exerted by the ground on the rod as the rod falls over.
(b)
Knowing that the rod will slip when
jF=N j
exceeds
s
, determine whether the rod
will slip as it falls.
Solution
Part (a)
The FBD of the rod as it falls is shown on the right.
Balance Principles. Based on this FBD, the Newton-Euler equations for the rod are
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