Problem 2.261
Referring to the problem of a robot arm catching an egg
(Prob. 2.260), the strategy is that the arm and the egg must have
the same velocity and the same position at the same time for
the arm to gently catch the egg. In addition, what should be
true about the accelerations of the arm and the egg for the catch
to be successful after they rendezvous with the same velocity
at the same position and time? Describe what happens if the
accelerations of the arm and egg do not match.
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 341
Problem 2.262
Although point
P
is moving on a sphere, its motion is being studied
with the cylindrical coordinate system shown. Discuss in detail whether
or not there are incorrect elements in the sketch of the cylindrical com-
ponent system at P.
Solution
The unit vector
OuR
points in the direction of
Er
. This is incorrect. For a cylindrical coordinate system, the unit
vector OuRmust be parallel to the ✓Rplane and point in the Rdirection.
342 Solutions Manual
Problem 2.263
Although point
P
is moving on a sphere, its motion is being studied
with the cylindrical coordinate system shown. Discuss in detail whether
or not there are incorrect elements in the sketch of the cylindrical com-
ponent system at P.
Solution
The unit vector
Ou✓
points in the direction of decreasing
✓
. This is incorrect, as
Ou✓
must point in the direction
Dynamics 2e 343
Problem 2.264
Discuss in detail whether or not (a) there are incorrect elements in the
sketch of the spherical component system at
P
and (b) the formulas for
the velocity and acceleration components derived in the section can be
used with the coordinate system shown.
Solution
(a)
The unit vector
Ou
is pointing in the direction of decreasing
. This is incorrect. It must point in the
direction of increasing .
344 Solutions Manual
Problem 2.265
Discuss in detail whether or not (a) there are incorrect elements in the
sketch of the spherical component system at
P
and (b) the formulas for
the velocity and acceleration components derived in the section can be
used with the coordinate system shown.
Solution
(a) The orientations of the unit vectors in relation to the positive directions of r,, and ✓are correct.
Dynamics 2e 345
Problem 2.266
A glider is descending with a constant speed
v0D30 m=s
and a
constant descent rate of
1m=s
along a helical path with a constant
radius
RD400
m. Determine the time the glider takes to complete
a full 360ıturn about the axis of the helix (the ´axis).
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.
346 Solutions Manual
Problem 2.267
An airplane is flying horizontally at a constant speed
v0D320 mph
while
its propellers rotate at a constant angular speed
!D1500 rpm
. If the
propellers have a diameter
dD14 ft
, determine the magnitude of the
acceleration of a point on the periphery of the propeller blades.
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 347
Problem 2.268
A top-slewing crane is lifting an object
C
at a constant rate of
P´D5:3 ft=s
while rotating at a constant rate
!D0:12 rad=s
about the vertical axis. If the distance between the object and
the axis of rotation of the crane’s boom is
rD46 ft
and it is
being reduced at a constant rate of
6:5 ft=s
, find the velocity and
acceleration of
C
, assuming that the swinging motion of
C
can be
neglected.
Solution
348 Solutions Manual
Problem 2.269
The system depicted in the figure is called a spherical pendulum. The
fixed end of the pendulum is at
O
. Point
O
behaves as a spherical
joint; i.e., the location of
O
is fixed while the pendulum’s cord can
swing in any direction in the three-dimensional space. Assume that
the pendulum’s cord has a constant length
L
, and use the coordi-
nate system depicted in the figure to derive the expression for the
acceleration of the pendulum.
Solution
Keeping in mind that the length of the pendulum is constant, we have that the position vector of the pendulum
bob is described as ErDLOur, where the radial coordinate ris such that