250 Solutions Manual
Problem 2.186
At a given instant, the merry-go-round is rotating with an angular
velocity
!D18 rpm
. When the child is
0:45
m away from the
spin axis, determine the second derivative with respect to time of
the child’s distance from the spin axis so that the child experiences
no radial acceleration.
Solution
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.
Dynamics 2e 251
Problem 2.187
A ball is dropped from rest from a height
hD5ft
. If the distance
dD3ft
, determine
the radial and transverse components of the acceleration and the velocity of the ball
when the ball has traveled a distance h=2 from its release position.
Solution
The acceleration of the ball is constant and equal to
EaDgO|: (1)
Problem 2.188
The polar coordinates of a particle are the following functions of time:
rDr0sin.t3=⌧3/and ✓D✓0cos.t=⌧/;
where
r0
and
✓0
are constants,
⌧D1
s, and where
t
is time in seconds.
Determine
r0
and
✓0
such that the velocity of the particle is completely in
the radial direction for
tD15
s and the corresponding speed is equal to
6m=s.
Solution
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.
Dynamics 2e 253
Problem 2.189
A space station is rotating in the direction shown at a constant rate
of
0:22 rad=s
. A crew member travels from the periphery to the
center of the station through one of the radial shafts at a constant
rate of
1:3 m=s
(relative to the shaft) while holding onto a handrail
in the shaft. Taking
tD0
to be the instant at which travel through
the shaft begins and knowing that the radius of the station is
200
m,
determine the velocity and acceleration of the crew member as a
function of time. Express your answer using a polar coordinate
system with origin at the center of the station.
Solution
We will use a polar coordinate system with origin at
O
, the center
of the station. The transverse coordinate
✓
is measured from a fixed
254 Solutions Manual
Problem 2.190
Solve Prob. 2.189 and express your answers as a function of posi-
tion along the shaft traveled by the astronaut.
Solution
Dynamics 2e 255
Problem 2.191
During a given time interval, a radar station tracking an airplane records
the readings
Pr.t/ DŒ449:8 cos ✓.t/ C11:78 sin ✓.t/çmph;
r.t/ P
✓.t/ DŒ11:78 cos ✓.t/ 449:8 sin ✓.t/çmph;
where
t
denotes time. Determine the speed of the plane. Furthermore,
determine whether the plane being tracked is ascending or descending
and the corresponding climbing rate (i.e., the rate of change of the
plane’s altitude) expressed in ft=s.
Solution
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.
256 Solutions Manual
Problem 2.192
The polar coordinates of a particle are the following functions of
time:
rDr0✓1Ct
⌧◆and ✓D✓0
t2
⌧2;
where r0D3ft, ✓0D1:2 rad, ⌧D20 s, and tis time in seconds.
Determine the velocity and the acceleration of the particle for
tD35
s and express the result using the polar component system
formed by the unit vectors Ourand Ou✓at tD35 s.
Solution
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.
Dynamics 2e 257
Problem 2.193
The polar coordinates of a particle are the following functions of
time:
rDr0✓1Ct
⌧◆and ✓D✓0
t2
⌧2;
where r0D3ft, ✓0D1:2 rad, ⌧D20 s, and tis time in seconds.
Determine the velocity and the acceleration of the particle for
tD35
s and express the result using the Cartesian component
system formed by the unit vectors O{and O|.
Solution
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.
258 Solutions Manual
Problem 2.194
A particle is moving such that the time rate of change of its polar
coordinates are
PrDconstant D3ft=s and P
✓Dconstant D0:25 rad=s:
Knowing that at time
tD0
, a particle has polar coordinates
r0D0:2 ft
and
✓0D15ı
, determine the position, velocity, and
acceleration of the particle for
tD10
s. Express your answers in
the polar component system formed by the unit vectors
Our
and
Ou✓
at tD10 s.
Solution
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.
Dynamics 2e 259
Problem 2.195
A particle is moving such that the time rate of change of its polar
coordinates are
PrDconstant D3ft=s and P
✓Dconstant D0:25 rad=s:
Knowing that at time
tD0
, a particle has polar coordinates
r0D0:2 ft
and
✓0D15ı
, determine the position, velocity, and
acceleration of the particle for
tD10
s. Express your answers in
the Cartesian component system formed by the unit vectors
O{
and
O|.
Solution
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.