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A.8 Newton’s Laws of Motion

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◦





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







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)







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)







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





;

dt



α





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)







Third Law: For every torque, there is an equal and opposite reaction torque.



A.19



Angular Momentum



Angular momentum is defined as

Angular momentum







(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



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A.8 Newton’s Laws of Motion

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