AD01_Basic Concepts in Dynamics

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Basic Concepts in Dynamics

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1 Introductory Concepts

1.1. Vector and Tensor

① Scalars, Vectors and Tensors

Scalar quantity: a quantity that can be expressed by a single real number. (ex: mass,

energy, temperature, time, etc.)

Vector quantity: a quantity that has direction as well as magnitude. vectors must have

certain transformation properties. (ex: force, moment, velocity, acceleration, etc.)

Tensor quantity: a general term to categorize physical quantities. Scalar is a tensor of rank

zero and vector is a tensor of rank one.

All physical quantities satisfy the coordinate transformation rules. For example, a scalar

quantity does not change with change of coordinate system, but a vector quantity can

change its components from one coordinate system to another. However, if the same units

are adopted, the magnitude of a vector does not change with the coordinate system.

② Types, Unit, & Equality, Addition of Vectors

Vectors have their own magnitude and direction but not location.

From the physical point of view, vector quantities can be classified into three types,

namely, free vectors, sliding vectors, and bound vectors.

Free vector: a vector quantity having the characteristics of magnitude and direction, but

no specified location or point of application

Sliding vector: a vector quantity that has the characteristics of magnitude, direction, and a

fixed line of action, but no precise location.

Bound vector: a vector quantity having a specific point of application as well as the

magnitude and direction.

Two vectors A and B are equal if A and B have the same magnitude and direction.




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A given vector can be represented as a product of a scalar magnitude and a vector of unit

length which designates its direction, i.e., A = Ae A.

Two vectors, A and B, can be added to yield another vector C . The vector C is the one

which is obtained by the polygon rule of vector addition.

Addition of vectors has the following properties: commutative, associate, i.e.

C = A + B = B + A, ( A + B) + D = A + ( B + D) = A + B + D.

③ Components of a Vector

If a vector in the three dimensional space, A, is represented as A = A1e1 + A2e2 + A3e3

where e1, e2, e3 are linearly independent unit vectors, A1, A2, A3 are called the components

of the vector A.

The unit vectors, e1, e2, e3, may not be orthogonal. They may be skewed unit vectors.

When they are orthogonal, the coordinate system that is generated using them is called an

orthogonal coordinated system.

If A = A1e1 + A2e2 + A3e3 and B = B1e1 + B2e2 + B3e3, then, A + B = ( A1 + B1)e1 + ( A2 +

B2)e2 + ( A3 + B3)e3 .

④ Scalar Product, Vector Product, Scalar & Vector Triple Product

The scalar product of two vectors A and B is defined as A B = AB·cosθ where θ is the

angle between the direction of the vectors. Note that A B = B A .

The scalar product has the distributive property: ( A + B) C = A C + B C .

If A = A1e1 + A2e2 + A3e3 and B = B1e1 + B2e2 + B3e3, A B = A1 B1 + A1 B1 + A1 B1 + ( A1 B2 +

A2 B1)e1 e2 + ( A2 B3 + A3 B2)e2 e3 + ( A3 B1 + A1 B3)e3 e1 .

If the coordinate system is orthogonal, A B = A1 B1 + A1 B1 + A1 B1 since e1 e2= e2 e3=e3 e1.

The vector product or cross product is defined as A× B = AB·sinθ e AB where e AB is the

unit vector orthogonal to A and B and pointing to the direction of the right-hand screw




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rule. If the right-handed orthogonal coordinate system is used,

A× B = ( A2 B3 – A3 B2)e1 + ( A3 B1 – A1 B3)e2 + ( A1 B2 – A2 B1)e3 .

Note that e1×e1 = e2×e2 = e3×e3 = 0, e1×e2 = – e2×e1 = e3 , e2×e3 = – e3×e2 = e1 ,

e3×e1 = – e1×e3 = e2 .

Scalar Triple Product: A ( B×C ) = B (C × A) = C ( A× B).

Vector Triple Product: A×( B×C ) = ( A C ) B – ( A B)C .

⑤ Derivative of a Vector

( )d d d

dt dt dt

+= +

A B A B, ( )d g dg d

g dt dt dt

= +A A

A ,

( )d d d

dt dt dt

⋅= ⋅ + ⋅

A B A BB A ,

( )d d d

dt dt dt

×= × + ×

A B A BB A .

( ) 3 31 2 1 21 1 2 2 3 3 1 2 3 1 2 3

ˆˆ ˆˆ ˆ ˆ ˆ ˆ ˆ

dA d dA dA d d d d A A A A A A

dt dt dt dt dt dt dt dt = + + = + + + + +

ee eAe e e e e e

If the coordinate system is orthogonal, 31 21 2 3

ˆ ˆ ˆdAdA dAd

dt dt dt dt = + +

Ae e e




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1.2. Newton’s Laws of Motion

① Every body continues in its state of rest, or of uniform motion in a

straight line, unless compelled to change that state by forces acting

upon it.

② The time rate of change of linear momentum of a body is proportional

to the force acting upon it and occurs in the direction in which the force

acts.

③ To every action there is an equal and opposite reaction; that is, the

mutual forces of two bodies acting upon each other are equal in

magnitude and opposite in direction.

④ The laws of motion for a particle can be summarized as:

Let v and a be the velocity and acceleration of a particle. Then, the first two laws of

Newton’s laws of motion yield:

( )d

k m kmdt

= =F v a

where F and m are the applied force and the mass of particle, respectively, and k is a

positive constant whose value depends upon the choice of units. The product mv is

known as the linear momentum p, that is, p = mv .

We choose the units such that k = 1, hence, we have F = ma.

The second basic law is the law of action and reaction: When two particles exert forces on

each other, these interaction forces are equal in magnitude, opposite in sense, and

directed along the straight line joining the particles.

The third basic law is the law of addition of forces: Two forces P and Q acting

simultaneously on a particle are together equivalent to a single force F = P +Q.




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⑤ Dimensions, Systems of Units, & Conversion of Units

Dimensions: qualitative aspects of a given unit (ex: length [L], time [T], speed [LT-1],

acceleration [LT-2], etc.)

A characteristic of the equations of physics is that they must exhibit dimensional

homogeneity. By this, we mean that any terms which are added or subtracted must have

the same dimensions and it also implies, of course, that the expressions on each side of an

equality must have the same dimensions.

The Newton’s laws of motion, F = ma, conclude that the fundamental dimensions

corresponding to mass and force cannot be chosen independently. We have two

possibilities for choices of the fundamental dimensions:

(1) the absolute system in which mass, length, and time are the fundamental dimensions:

(2) the gravitational system in which force, length, and time are the fundamental

dimensions.

The SI (mks) system is adopted. In this system, as the unit of force newton (N) is used.

One newton is that force which will give a mass of one kilogram an acceleration of

1 m/sec2 . In terms of the fundamental units, 1 newton (N) = 1 kg ·m/sec2.

When making checks of dimensional homogeneity, and even when performing numerical

computations, it is advisable to carry along the units, treating them as algebraic quantities.

This algebraic manipulation of units often requires the conversion from one set of units to

another set having the same dimensions.

⑥ Weight and Mass

The weight of a body is the force with which that body is attracted toward the earth. Its

mass is the quantity of matter in the body, irrespective of its location in space. Hence, the

weight w can be expressed with the mass m as




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w = mg

where g is the local acceleration of gravity.

1.3. Basis of Newtonian Mechanics

① Space

Infinite: ordinary Euclidean geometry can be applied.

Homogeneous and Isotropic: the local properties of space are independent of location or

direction.

Absolute: there exists a primary inertial frame.

② Time

Time is assumed as an absolute quantity; it is the same for all observers and, furthermore,

independent of all objects of the physical world.

Newton considered that a definition of time in terms of natural phenomena such as the

rotation of the earth was, at best, an approximation to the uniform flow of true time.

③ Mass

Since the units of length and of time are selected as the first and second fundamental

units, the third fundamental unit can be chosen as a unit of mass or of force. We have

chosen the unit of mass as the third fundamental one.

Inertial Mass: That mass determining the acceleration of a body under the action of a

given force.

Gravitational Mass: That mass determining the gravitational forces between a body and

other bodies.

Galileo was the first to test the equivalence of inertial and gravitational mass in his

(perhaps apocryphal) experiment with falling weights at the Tower of Pisa.




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Newton also tested this issue by measuring the periods of pendula of equal lengths but

with bodies of different materials.

In 1890 Eotbos devised an ingenious method to test the equivalence of inertial and

gravitational masses. Using two objects made of different materials, he compared the

effects of the Earth’s gravitational force with the effect of the inertial force caused by the

Earth’s rotation.

In 1964, Dicke showed by experiments that inertial and gravitational mass are identical to

within a few parts in 1012. [P.G..Roll, R.Krotkov, and R.H.Dicke, Ann. Physics, (N.Y.) 26,

442 (1964)]

The assertion of the exact equality of inertial and gravitational mass is termed the

principle of equivalence.

④ Force

The concept of force as a fundamental quantity in the study of mechanics has been

criticized by various scientists and philosophers of science due to the following points:

- The idea of a force, and a field force in particular, was considered to be an

intellectual construction which has no real existence.

- It is merely another name for the product of mass and acceleration which occurs in

the mathematics of solving a problem.

- Furthermore, the idea of force as a cause of motion should be discarded since the

assumed cause and effect relationships cannot be proved. (ex: a stone tied to the end

of a string and whirled in a circular path)

⑤ Inertial (Newtonian) Reference Frame

A rigid set of coordinate axes such that particle motion relative to these axes is described

by Newton’s laws of motion.




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A reference frame in which Newton’s laws are indeed valid; that is,if a body subject to no

external force moves in a straight line with constant velocity (or remains at rest), then the

coordinate system establishing this fact is an inertial reference frame.

⑥ Galilean Invariance (Principle of Newtonian Relativity)

The existence of an inertial frame implies the existence of an infinite number of other

inertial frames, all having no rotation rate relative to the fixed stars but translating with

constant velocities relative to each other. Thus, even Newtonian mechanics has no single,

preferred frame of reference. This is the Newtonian principle of relativity.

If Newton’s laws are valid in one reference frame, then they are also valid in any

reference frame in uniform motion (i.e., not accelerated) with respect to the first system.

This is a result of the fact that the equation F = m(d 2r/dt

2) involves the second time

derivative of r: A change of coordinates involving a constant velocity does not influence

the equation. This result is called Galilean invariance or the principle of Newtonian

relativity.

1.4. D’Alembert’s Principle

① Principle of Virtual Work

Let Fi(a) and f i be the forces externally applied on the body i and the force of constraints

of the body i, respectively. If the particles are at rest,

( )( ) 0a

i i i

i

δ + ⋅ =∑ F f r

since Fi(a) + f i = 0 for each i.

If the net virtual work of the forces of constraint is zero, 0i i

i

δ ⋅ =∑f r , then, we have

( ) 0a

i i

i

δ ⋅ =∑F r




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This is often called the principle of virtual work.

② D’Alembert’s Principle

The equation of motion yield Fi(a) + f i = d pi/dt for each particle i. Rewrite this Fi

(a) + f i -

d pi/dt = 0. Then this form of equation states that the particles in the system will be in

equilibrium under a force equal to the actual force plus a “reversed effective force” -

d pi/dt .

We may write the result using the infinitesimal virtual displacement δ ri as follows:

( )( ) 0a

i i i i

i

δ + − ⋅ =∑ F f p r

wherei i

d dt =p p .

We again restrict ourselves to systems for which the virtual work of the forces of

constraints vanishes and therefore obtain

( )( ) 0a

i i i

i

δ − ⋅ =∑ F p r

which is often called D’Alembert’s principle.

AD01_Basic Concepts in Dynamics

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