A.8 Newton’s Laws of Motion
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Section A.9 Conservation of Linear Momentum
275
Second Law: The time rate of change of the linear momentum of a body is
equal to the force F applied to it.
Except at very high velocities, where relativistic eﬀects must be considered, the second law can be expressed mathematically in terms of the
mass m and acceleration a of the object as2
F
(A.10)
ma
This is one of the most commonly used equations in mechanics. It
shows that if the applied force and the mass of the object are known, the
acceleration can be calculated. When the acceleration is known, the
velocity of the object and the distance traveled can be computed from
the previously given equations.
The Earth’s gravitational force, like all other forces, causes an acceleration. By observing the motion of freely falling bodies, this acceleration has been measured. Near the surface of the Earth, it is approximately 9.8 m/sec2 . Because gravitational acceleration is frequently used
in computations, it has been given a special symbol g. Therefore, the
gravitational force on an object with mass m is
Fgravity
(A.11)
mg
This is, of course, also the weight of the object.
Third Law: For every action, there is an equal and opposite reaction. This
law implies that when two bodies A and B interact so that A exerts a
force on B, a force of the same magnitude but opposite in direction is
exerted by B on A. A number of illustrations of the third law are given
in the text.
A.9
Conservation of Linear Momentum
It follows from Newton’s laws that the total linear momentum of a system of
objects remains unchanged unless acted on by an outside force.
2
The second law can be expressed mathematically in terms of the time derivative of
momentum: that is,
Force
t→0
mv(t +
t) − mv(t)
t
d
(mv)
dt
m
dv
dt
ma
276
Appendix A Basic Concepts in Mechanics
FIGURE A.1
A.10
The radian.
Radian
In the analysis of rotational motion, it is convenient to measure angles in a
unit called a radian. With reference to Fig. A.1, the angle in radian units is
deﬁned as
θ
s
r
(A.12)
where s is the length of the circular arc and r is the radius of rotation. In a full
circle, the arc length is the circumference 2πr. Therefore in radian units the
angle in a full circle is
2πr
r
θ
2π rad
Hence,
1 rad
A.11
360◦
2π
57.3◦
Angular Velocity
The angular velocity ω is the angular displacement per unit time; that is, if a
body rotates through an angle θ (in radians) in a time t, the angular velocity is
ω
θ
(rad/sec)
t
(A.13)
Section A.14 Equations for Angular Momentum
A.12
277
Angular Acceleration
Angular acceleration α is the time rate of change of angular velocity. If the
initial angular velocity is ω0 and the ﬁnal angular velocity after a time t is ωf ,
the angular acceleration is3
ωf − ω0
t
α
A.13
(A.14)
Relations between Angular and Linear Motion
As an object rotates about an axis, each point in the object travels along the
circumference of a circle; therefore, each point is also in linear motion. The
linear distance s traversed in angular motion is
s
rθ
The linear velocity v of a point that is rotating at an angular velocity ω a
distance r from the center of rotation is
v
(A.15)
rω
The direction of the vector v is at all points tangential to the path s. The linear
acceleration along the path s is
a
A.14
(A.16)
rα
Equations for Angular Momentum
The equations for angular motion are analogous to the equations for translational motion. For a body moving with a constant angular acceleration α and
initial angular velocity ω0 , the relationships are shown in Table A.1.
3
Both angular velocity and angular acceleration may vary along the path. In general, the
instantaneous angular velocity and acceleration are deﬁned as
ω
dθ
;
dt
α
dω
dt
d 2θ
dt 2
278
Appendix A Basic Concepts in Mechanics
TABLE A.1
Equations for
Rotational Motion (angular
constant)
acceleration, α
A.15
ω
ω0 + αt
θ
ω0 t + 21 αt 2
ω2
ω20 + 2αθ
ωav
(ω0 + ω)
2
Centripetal Acceleration
As an object rotates uniformly around an axis, the magnitude of the linear
velocity remains constant, but the direction of the linear velocity is continuously changing. The change in velocity always points toward the center
of rotation. Therefore, a rotating body is accelerated toward the center of
rotation. This acceleration is called centripetal (center-seeking) acceleration.
The magnitude of the centripetal acceleration is given by
ac
v2
r
ω2 r
(A.17)
where r is the radius of rotation and v is the speed tangential to the path of
rotation. Because the body is accelerated toward its center of rotation, we
conclude from Newton’s second law that a force pointing toward the center of
rotation must act on the body. This force, called the centripetal force Fc , is
given by
Fc
mac
mv2
r
mω2 r
(A.18)
where m is the mass of the rotating body.
For a body to move along a curved path, a centripetal force must be applied
to it. In the absence of such a force, the body moves in a straight line, as
required by Newton’s ﬁrst law. Consider, for example, an object twirled at
the end of a rope. The centripetal force is applied by the rope on the object.
From Newton’s third law, an equal but opposite reaction force is applied on
the rope by the object. The reaction to the centripetal force is called the centrifugal force. This force is in the direction away from the center of rotation.
The centripetal force, which is required to keep the body in rotation, always
acts perpendicular to the direction of motion and, therefore, does no work
Section A.17 Torque
TABLE A.2
279
Moments of Inertia of Some Simple Bodies
Body
Location of axis
Moment of inertia
A thin rod of length l
A thin rod of length l
Sphere of radius r
Cylinder of radius r
Through the center
Through one end
Along a diameter
Along axis of symmetry
ml2 /12
ml2 /3
2mr2 /5
mr2 /2
(see Eq. A.28). In the absence of friction, energy is not required to keep a
body rotating at a constant angular velocity.
A.16
Moment of Inertia
The moment of inertia in angular motion is analogous to mass in translational
motion. The moment of inertia I of an element of mass m located a distance r
from the center of rotation is
I
mr 2
(A.19)
In general, when an object is in angular motion, the mass elements in the
body are located at diﬀerent distances from the center of rotation. The total
moment of inertia is the sum of the moments of inertia of the mass elements
in the body.
Unlike mass, which is a constant for a given body, the moment of inertia
depends on the location of the center of rotation. In general, the moment of
inertia is calculated by using integral calculus. The moments of inertia for a
few objects useful for our calculations are shown in Table A.2.
A.17
Torque
Torque is deﬁned as the tendency of a force to produce rotation about an axis.
Torque, which is usually designated by the letter L, is given by the product of
the perpendicular force and the distance d from the point of application to the
axis of rotation; that is (see Fig. A.2),
L
F cos θ × d
The distance d is called the lever arm or moment arm.
(A.20)
280
Appendix A Basic Concepts in Mechanics
FIGURE A.2
A.18
Torque produced by a force.
Newton’s Laws of Angular Motion
The laws governing angular motion are analogous to the laws of translational
motion. Torque is analogous to force, and the moment of inertia is analogous
to mass.
First Law: A body in rotation will continue its rotation with a constant angular velocity unless acted upon by an external torque.
Second Law: The mathematical expression of the second law in angular
motion is analogous to Eq. A.10. It states that the torque is equal to the
product of the moment of inertia and the angular acceleration; that is,
L
(A.21)
Iα
Third Law: For every torque, there is an equal and opposite reaction torque.
A.19
Angular Momentum
Angular momentum is deﬁned as
Angular momentum
Iω
(A.22)
From Newton’s laws, it can be shown that angular momentum of a body is
conserved if there is no unbalanced external torque acting on the body.
Section A.20 Addition of Forces and Torques
FIGURE A.3
A.20
281
The resolution of a force into its vertical and horizontal components.
Addition of Forces and Torques
Any number of forces and torques can be applied simultaneously to a given
object. Because forces and torques are vectors, characterized by both a magnitude and a direction, their net eﬀect on a body is obtained by vectorial addition.
When it is required to obtain the total force acting on a body, it is often convenient to break up each force into mutually perpendicular components. This
is illustrated for the two-dimensional case in Fig. A.3. Here we have chosen
the horizontal x- and the vertical y-directions as the mutually perpendicular
axes. In a more general three-dimensional case, a third axis is required for the
analysis.
The two perpendicular components of the force F are
Fx
Fy
F cos θ
F sin θ
(A.23)
The magnitude of the force F is given by
F
Fx2 + Fy2
(A.24)
When adding a number of forces (F1 , F2 , F3 , . . .) the mutually perpendicular components of the total force FT are obtained by adding the corresponding