Problem 7.93
The uniform ball of radius
and mass
m
is gently placed in the
bowl
B
with inner radius
R
and is released. The angle
measures
the position of the center of the ball at
G
with respect to a vertical
line through
O
. Assume that the system lies in the vertical plane.
Hint: In working the following problems, we recommend using
the r coordinate system shown.
Assume that friction is sufficient to prevent slipping.
(a)
Derive the equation(s) of motion of the ball in terms of the
angle .
(b) Determine the friction force as a function of .
(c)
Letting
.0/ D0
and
P
.0/ D0
, with
0ı< 0< 90ı
,
integrate the equation(s) of motion to determine the normal
force as a function of .
(d)
Using the results of Parts (b) and (c), and given a value for
s
, determine the maximum value of
.0/ D0
so that the
ball does not slip.
Solution
Part (a)
Dynamics 2e 1597
Part (b)
Part (d)
Problem 7.94
An important problem in billiards or pool is the determination of the
height at which you should hit the cue ball to give it backspin, topspin,
or no spin. With this in mind, at what height
h
should the cue hit the
ball so that the ball always rolls without slip, regardless of how hard the
ball is hit and how much friction is available? Assume a uniform ball of
mass
m
and radius
r
. You can determine this position without having
to worry about the impact between the cue and the ball by studying an
arbitrary horizontal force applied to a ball at height h.
IGD2
5mr2:
Force Laws.
For the sphere to roll without slipping, the following inequality must be satisfied for any value
of s
F
7r ;and ˛bD 5hP
7mr2:
To find h, we can proceed in a couple of different ways.
7r D0)hD7
Dynamics 2e 1599
Problem 7.95
The Pioneer 3 spacecraft was a spin-stabilized spacecraft
launched on December 6, 1958, by the U.S. Army Ballistic Mis-
sile agency in conjunction with NASA. It was designed with
a despin mechanism consisting of two equal masses
A
and
B
,
each of mass
m
, that could be spooled out to the ends of two
wires of variable length
`.t/
when triggered by a hydraulic timer.
As the masses unwound, they would slow the spacecraft’s spin
from an initial angular velocity
!s.0/
to the final angular veloc-
ity
.!s/final
, and then the weights and wires would be released.
Assume that masses
A
and
B
are initially at positions
A0
and
B0
, respectively, before the wire begins to unwind, that the mass
moment of inertia of the spacecraft is
IO
(this does not include
the two masses
A
and
B
), and that gravity and the mass of each
wire are negligible. Hint: Refer to Prob. 6.160 if you need help
with the kinematics.
Derive the equation(s) of motion of the system in terms of
the dependent variables `.t/ and !s.t/.
Dynamics 2e 1603
Problem 7.96
The Pioneer 3 spacecraft was a spin-stabilized spacecraft
launched on December 6, 1958, by the U.S. Army Ballistic Mis-
sile agency in conjunction with NASA. It was designed with
a despin mechanism consisting of two equal masses
A
and
B
,
each of mass
m
, that could be spooled out to the ends of two
wires of variable length
`.t/
when triggered by a hydraulic timer.
As the masses unwound, they would slow the spacecraft’s spin
from an initial angular velocity
!s.0/
to the final angular veloc-
ity
.!s/final
, and then the weights and wires would be released.
Assume that masses
A
and
B
are initially at positions
A0
and
B0
, respectively, before the wire begins to unwind, that the mass
moment of inertia of the spacecraft is
IO
(this does not include
the two masses
A
and
B
), and that gravity and the mass of each
wire are negligible. Hint: Refer to Prob. 6.160 if you need help
with the kinematics.
Derive the equation(s) of motion of the system in terms of
the dependent variables `.t/ and !s.t/. After doing so:
(a)
Use a computer to solve the equations of motion for
0t4
s, using
RD12:5 cm
,
mD7
g,
IOD
0:0277 kgm2
, and the initial conditions
!s.0/ D400 rpm
,
`.0/ D0:01 m, and P
`.0/ D0m=s.
(b)
Determine the time at which the angular velocity of the
spacecraft becomes zero (this can be done by plotting the
solution for
!s
and then estimating the time or by using
numerical root finding to determine when !sD0).
(c)
Determine the length
`
of each of the wires at the instant
that the angular velocity of the spacecraft becomes zero.
Solution
XMOW 2TR DIO˛s;(3)
Dynamics 2e 1605
Computation.
Recalling that
˛sD P!s
, substituting Eq. (14) into the Newton-Euler equations (note that
the first two Newton-Euler equations simply give 0D0), we obtain