Chapter 10 Homework Solutions

Document Sample

```					Chapter 10 Homework Solutions

G10.1. Express the following angles in radians: (a) 30°, (b) 57°, (c) 90°, (d) 360°, and (e)
420°. Give as numerical values and as fractions of

Solution:

G10.4. A centrifuge rotor is accelerated is accelerated from rest to 20,000 rpm in 5.0 min.
What is its average angular acceleration?

Solution:
20 000 min                          60 s                      s
2094    s
s
s

G10.9. (a) A grinding wheel 0.35 m in diameter rotates at 2500 rpm. Calculate its angular
velocity in rad/s. (b) What are the linear speed and acceleration of a point on the edge of
the grinding wheel?

Solution:
2
1 min            revolutions                                  revolutions revolution radians
2
1 sec          revolutions                                  revolutions revolution radians
2
s
revolution

s          m/s
There is no tangential acceleration, only centripetal:
m/s

G10.16. The angle through which a rotating wheel has turned in time is given by
, where is in radians and in seconds. Determine an expression
(a) for the instantaneous angular velocity and (b) for the instantaneous angular
acceleration . (c) Evaluate and at             s. (d) What is the average angular velocity,
and (e) the average angular acceleration between         s and          s?

Solution:

1
(a)
s                  s                 s

(b)
s                 s

(c)
s
s

(d)

s

(e)
s
s
s

G10.17. The angular acceleration of a wheel, as a function of time, is                 ,
where is in rad/s and in seconds. If the wheel starts from rest (            at     ),
determine a formula for (a) the angular velocity and (b) the angular position , both as a
function of time. (c) Evaluate and at           s

Solution:
(a)

s                    s

(b)
s                     s

(c)
s            s

2
G10.21. The biceps muscle exerts a vertical force on the lower arm as shown on Figs. 10-
51a and b. For each case, calculate the torque about the acis of rotation through the elbow
joint, assuming the muscle is attached 5.0 cm from the elbow as shown.

Solution:
(a)             sin           N         m sin     °         m N

sin           N         m sin     °         m N

G10.27. A 1.4-kg grindstone in the shape of a uniform cylinder of radius 0.20 m acquires
a rotational rate of 1800 rev/s from rest over a 6.0-s interval at constant angular
acceleration. Calculate the torque delivered by the motor.

Solution:
!           " #           m N

G10.32. A 0.84-m diameter solid sphere can be rotated about an axis through its center by
a torque of 10.8 m.N which accelerates it uniformly from rest through a total of 180
revolutions in 15.0 s. What is the mass of the sphere?

Solution:

rev      rev                   s                  s

!       " #
m.N      "            m        s                  m

G10.39. When discussing moments of inertia, especially for unusual or irregularly shaped
objects, it is sometimes convenient to work with the radius of gyration, k. This radius is
defined so that if all the mass of the object were concentrated at this distance from the
axis, the moment of inertia would be the same as that of the original object. Thus, the
moment of inertia of any object can be written in terms of its mass M and the radius of

3
gyration as I=MR . Determine the radius of gyration for each of the objects (hoop,
cylinder, sphere, etc.) shown in Fig.10-21

Solution:
Hoop:               !                #           \$           #
#       (
%       &'          !                #               " (                     \$
+
) * , -*+.                  '        !               #               \$                   #
#       #
% ** ( -*+.                     '!               #               #               \$
/ .+0            ) &1           ' !                  #               \$                   #
/ .+0                   '            !               2               \$                       2
/ .+0                   '            !               2           \$                       2
* (
#        1+. &*             '        !                   *       (                   \$

G10.42. Use the parallel axis theorem to show that the moment of inertial of a thin rod
about an axis perpendicular to the rod at one end is !   " * , given that if the axis
passes through the center, !      " *

Solution:
" 2
!
! 3     !           "       2
!   3
" 2
+"      2           " 2

G10.43. Use the perpendicular axis theorem and Fig.10-21h to determine a formula for
the moment of inertial of a thin, square plate of side 4 about an axis (a) through its center
and along a diagonal of the plate, (b) through the center and parallel to a side

!        !      !            !                       4       4

!        !     !             !                   4           4

G10.46. A ball of mass M and radius R on the end of a thin massless rod is rotated in a
horizontal circle of radius R about an axis of rotation AB, as shown in Fig.10-58. (a)
Considering the mass of the ball to be concentrated at its center of mass, calculate its
moment of inertia about AB. (b) Using the parallel axis theorem and considering the
finite radius of the ball, calculate the moment of inertia of the ball about AB. (c)
Calculate the percentage error introduced by the point mass approximation for R
cm and R       1.0 m

4
Solution:
!    " #
!    !   " #                       " #            " #
!            " # " # " #                     #
!                   " # " #                  #       #

G10.59. A 4.8-m-diameter merry-go-round is rotating freely with an angular velocity of
0.80 rad/s. Its total moment of inertial is 1950 kg.m . Four people standing on the ground,
each of 65 kg mass, suddenly step onto the edge of the merry-go-round. What is the
angular velocity of the merry-go-round now? what if the people were on it initially and
then jumped off in a radial direction (relative to the merry-go-round)?

Solution:

5 . 36*                     . 6      +4   .4
2    !                !

!                   kg.m                  kg           m

!               kg.m          kg       m

!               !       !&     &*

G10.65. Two masses,                kg and             kg, are connected by a rope that
hangs over a pulley (as in Fig.10-60). The pulley is a uniform cylinder of radius 0.30 m
and mass 4.8 kg. Initially    is on the ground and      rests 2.5 m above the ground. If the
system is released, use conservation of energy to determine the speed of      just before it
strikes the ground. Assume the pulley bearing is frictionless.

5
M, R0

m2

h
m1

7   1 * \$1 4 1 4             4&            4 * .3 4 1       +4   . 4+ . +. 1

#           . 36*        * + - 0 &6** - +4          *         1    .3 . + * * 0   &

!    " #

8       7   9            /
!
3        1         31

m
s

6

```
DOCUMENT INFO
Shared By:
Categories:
Stats:
 views: 182 posted: 5/12/2010 language: English pages: 6