Problem 22-1
When a load of mass m1 is suspended from a spring, the spring is stretched a distance d.
Determine the natural frequency and the period of vibration for a load of weight m2 attached to the
same spring.
m1 = 2 kg
Given:
m2 = 0.5 kg
Solution:
k
Problem 21-1
Show that the sum of the moments of inertia of a body, Ixx+Iyy+Izz , is independent of the
orientation of the x, y, z axes and thus depends only on the location of the origin.
Solution:
2
2
Ixx + Iyy + Izz = y + z dm +
m
2
2
x + z dm +
m
m
2
Problem 20-1
The ladder of the fire truck rotates around the z axis with angular velocity 1 which is
increasing at rate 1. At the same instant it is rotating upwards at the constant rate 2.
Determine the velocity and acceleration of point A located at the
Problem 19-1
The rigid body (slab) has a mass m and is rotating with an angular velocity about an axis
passing through the fixed point O. Show that the momenta of all the particles composing the
body can be represented by a single vector having a magnitud
Problem 18-1
At a given instant the body of mass m has an angular
velocity and its mass center has a velocity vG. Show
that its kinetic energy can be represented as T = 1/2
IIC2 , where IIC is the moment of inertia of the body
computed about the instantan
Problem 17-1
The right circular cone is formed by
revolving the shaded area around the x
axis. Determine the moment of inertia Ix
and express the result in terms of the total
mass m of the cone.The cone has a
constant density .
Solution:
h
2
r x dx = 1 h
Problem 16-1
A wheel has an initial clockwise angular velocity and a constant angular acceleration .
Determine the number of revolutions it must undergo to acquire a clockwise angular velocity f
What time is required?
Units Used:
rev = 2 rad
Given:
= 10
Problem 15-1
A block of weight W slides down an inclined plane of angle with initial velocity v0. Determine
the velocity of the block at time t1 if the coefficient of kinetic friction between the block and the
plane is k.
Given:
W = 20 N
t1 = 3 s
= 30 de
Problem 14-1
A woman having a mass M stands in an elevator which has a downward acceleration a
starting from rest. Determine the work done by her weight and the work of the normal
force which the floor exerts on her when the elevator descends a distance s
Problem 13-1
Determine the gravitational attraction between two spheres which are just touching each
other. Each sphere has a mass M and radius r.
Given:
r = 200 mm
M = 10 kg
12 m
3
G = 66.73 10
kg s
Solution:
F=
GM
( 2r)
2
2
F = 41.7 nN
2
nN = 1 10
9
N
Problem 12-1 A truck traveling along a straight road at speed v1, increases its speed to v2 in time t. If its acceleration is constant, determine the distance traveled. Given: v1 = 20 Solution: a= v2 v1 t 12 at 2 a = 1.852 m s
2
km hr
v2 = 120
km hr
t = 1
Problem 11-1
The thin rod of weight W rests against the smooth wall and
floor. Determine the magnitude of force P needed to hold it
in equilibrium.
Solution:
xp = L cos ( )
yw =
U = 0;
L
2
sin ( )
xp = L sin ( )
yw =
L
2
cos ( )
Pxp Wyw = 0
P ( L sin
P roblem 10-1
Determine the moment of inertia for the shaded area about the x axis.
Given:
a := 2 m
b := 4 m
b
Solution:
y
2
Ix := 2
y a 1 dy
b
0
Ix = 39.0m
4.0
P roblem 10-2
Determine the moment of inertia for the shaded area about the y axis.
Given:
a :
Problem 9-1
Locate the center of mass of the
homogeneous rod bent in the form of a
parabola.
Given:
a := 1m
b := 2m
Solution:
y= b
dy
dx
=
x
a
2 b
2
2
x
a
a
yc :=
2
x
b
a
2
2 b x dx
1+
2
a
0
yc = 0.912 m
a
0
2
1+
2 b x dx
2
a
P roblem 9-2
Locate th
Problem 8-1
The horizontal force is P , Determine the normal and frictional forces acting on the crate of weight
W. The friction coefficients are k and s
Given:
W := 300N
P := 80N
s := 0.3
k := 0.2
:= 20deg
Solution:
Assume no slipping :
Fx = 0;
P cos
Problem 7-1
The column is fixed to the floor and is subjected to the loads shown. Determine the internal normal
force, shear force, and moment at points A and B.
Units Used:
kN := 103N
Given:
F1 := 6kN
F2 := 6kN
F3 := 8kN
a := 150mm
b := 150mm
c := 150mm
P roblem 6-1
Determine the force in each member of the truss and state if the members are in tension or
compression.
Units Used:
3
kN := 10 N
Given:
P 1 := 7kN
P 2 := 7kN
Solution:
:= 45deg
Initial Guesses:
F AB := 1kN
F DC := 1kN
F AD := 1kN
F DB := 1kN
P roblem 5-1
Draw the free-body diagram of the sphere of weight W resting between the smooth inclined
planes. Explain the significance of each force on the diagram.
Given:
W := 10 N
1 := 105deg
2 := 45deg
Solution:
NA, NB force of plane on sphere.
W for
P roblem 4-1
If A, B, and D are given vectors, prove the distributive law for the vector cross product, i.e.,
A ( B + D) = ( A B) + ( A D).
Solution:
Consider the three vectors; with A vertical.
Note obd is perpendicular to A.
od = A ( B + D) = A
(
B + D
P roblem 2-1
Determine the magnitude of the resultant force FR = F1 + F 2 and its direction, measured
counterclockwise from the positive x axis.
Given:
F 1 := 600N
F 2 := 800N
F 3 := 450N
:= 45deg
:= 60deg
:= 75deg
Solution:
:= 90deg +
F R :=
2
2
F 1
CHAPTER 12 CONTROL SYSTEM ANALYSIS USING
STATE VARIABLE METHODS
12.1
if
u
Kg +
1
sLf + Rf
qM
ia
1
s (La + Lg ) + Ra +Rg
KT
1
sJe + Be
x3
x4
qM
1
s
x2
n
x1
Kb
Je = n2J = 0.4 ; Be = n2B = 0.01
&
&
x1 = x2 ; 0.4 x 2 + 0.01x2 = 1.2x3
&
&
0.1 x 3 + 19x3 = 100
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CHAPTER 25 Mathematical Statistics
Changes
Regression and a short introduction to correlation have been combined in the same section (Sec. 25.9). SECTION 25.1. Introduction. Random Sampling, page 1044 Purpose. To explain
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Part G. PROBABILITY, STATISTICS
CHAPTER 24 Data Analysis. Probability Theory
SECTION 24.1. Data Representation. Average. Spread, page 993 Purpose. To discuss standard graphical representations of data in statistics. To in
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Page 362
CHAPTER 23 Graphs. Combinatorial Optimization
SECTION 23.1. Graphs and Digraphs, page 954 Purpose. To explain the concepts of a graph and a digraph (directed graph) and related concepts, as well as their computer represen
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Part F. Optimization. Graphs
CHAPTER 22 Unconstrained Optimization. Linear Programming
SECTION 22.1. Basic Concepts. Unconstrained Optimization, page 936 Purpose. To explain the concepts needed throughout this chapter. To
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CHAPTER 21 Numerics for ODEs and PDEs
Major Changes
These include automatic variable step size selection in modern codes, the discussion of the RungeKuttaFehlberg method, backward Eulers method and its application to stif
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CHAPTER 20 Numeric Linear Algebra
SECTION 20.1. Linear Systems: Gauss Elimination, page 833 Purpose. To explain the Gauss elimination, which is a solution method for linear systems of equations by systematic elimination (
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PART E. Numeric Analysis
The subdivision into three chapters has been retained. All three chapters have been updated in the light of computer requirements and developments. A list of suppliers of software (with addresses
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CHAPTER 18 Complex Analysis and Potential Theory
This is perhaps the most important justification for teaching complex analysis to engineers, and it also provides for nice applications of conformal mapping. SECTION 18.1.
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CHAPTER 17 Conformal Mapping
This is a new chapter. It collects and extends the material on conformal mapping contained in Chap. 12 of the previous edition. SECTION 17.1. Geometry of Analytic Functions: Conformal Mapping,