280 Solutions Manual
Problem 2.211
The collar is mounted on the horizontal arm shown, which is originally
rotating with the angular velocity
!0
. Assume that after the cord is
cut, the collar slides along the arm in such a way that the collar’s total
acceleration is equal to zero. Determine an expression of the radial
component of the collar’s velocity as a function of
r
, the distance from
the spin axis. Hint: Using polar coordinates, observe that
d.r2P
✓/=dt D
ra✓.
Solution
We are told that the collar’s acceleration is zero, which implies that each component of its acceleration must
be zero. Looking first at the transverse component of acceleration, we must have
Dynamics 2e 281
Problem 2.212
Particle
A
slides over the semicylinder while pushed by the arm
pinned at
C
. The motion of the arm is controlled such that it starts
from rest at
✓D0
,
!
increases uniformly as a function of
✓
, and
!D0:5 rad=s
for
✓D45ı
. Letting
RD4in:
, determine the
speed and the magnitude of the acceleration of Awhen D32ı.
Solution
We observe that
A
moves along a circle of radius
R
and center
O
.
For this reason we describe the motion of
A
using a polar coordinate
282 Solutions Manual
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Dynamics 2e 283
Problem 2.213
The mechanism shown is called a swinging block
slider crank. First used in various steam locomotive
engines in the 1800s, this mechanism is often found
in door-closing systems. If the disk is rotating with
a constant angular velocity
P
✓D60 rpm
,
HD4ft
,
RD1:5 ft
, and
r
denotes the distance between
B
and O, compute Pr,P
,Rr, and R
when ✓D90ı.
Solution
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284 Solutions Manual
Problem 2.214
A satellite is moving along the elliptical orbit shown. Using the polar
coordinate system in the figure, the satellite’s orbit is described by
the equation
r.✓/D2b2aCpa2b2cos ✓
a2Cb2.a2b2/cos.2✓/;
which implies the following identity
rr00 2.r0/2r2
r3Da
b2;
where the prime indicates differentiation with respect to
✓
. Using
this identity and knowing that the satellite moves so that
KDr2P
✓
with
K
constant (i.e., according to Kepler’s laws), show that the
radial component of acceleration is proportional to 1=r2, which is
in agreement with Newton’s universal law of gravitation.
Solution
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Dynamics 2e 285
Problem 2.215
At a given instant, an airplane flying at an altitude
h0D10;000 ft
begins its descent in preparation for landing when it is
r.0/ D20 mi
from the radar station at the destination’s airport. At that instant,
the aiplane’s speed is
v0D300 mph
, the climb rate is constant
and equal to
5ft=s
, and the horizontal component of velocity is
decreasing steadily at a rate of
15 ft=s2
. Determine the
Pr
,
P
✓
,
Rr
, and
R
✓that would be observed by the radar station.
Solution
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permission of McGraw-Hill, is prohibited.
286 Solutions Manual
Problem 2.216
Considering the system analyzed in Example 2.21, let
hD15 ft
,
v0D55 mph
, and
D25ı
. Plot the
trajectory of the projectile in two different ways: (1) by solving the projectile motion problem using
Cartesian coordinates and plotting
y
versus
x
and (2) by using a computer to solve Eqs. (3), (4), (9), and
(10) in Example 2.21. You should, of course, get the same trajectory regardless of the coordinate system
used.
Solution
Using the Cartesian coordinate system shown, the acceleration of the
projectile is
Dynamics 2e 287
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288 Solutions Manual
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permission of McGraw-Hill, is prohibited.
Dynamics 2e 289
Problem 2.217
Reference frame
A
is translating relative to reference frame
B
. Both frames track the motion of a particle
C
. If at one instant the velocity of particle
C
is the same in the two frames, what can you infer about the
motion of frames Aand Bat that instant?
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