MA4248 Weeks 1-3.

Topics Coordinate Systems, Kinematics, Newton’sLaws, Inertial Mass, Force, Momentum, Energy,Harmonic Oscillations (Springs and Pendulums)

Mechanics a drama authored by physical lawwhose stage is four dimensional space-time

Space-time has an affine structure, and additionalstructure for either classical or relativistic mechanics

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VECTOR SPACE

Definition a set, whose elements are called vectors,together with two operations, called vector additionand scalar multiplication (by elements of R = reals)

- the two operations must be related

- each operation must satisfy certain properties

Examples d-tuples of real numbers, real valuedfunctions on a specified set S, set of functions on Rhaving the form x a cos (x+b)

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AFFINE SPACEDefinition a set A, whose elements are called points,together with a vector space V and an operation

(called translation), that associates to every p in A and u in V an element in A (denoted by p+v), and that satisfies the following two properties:

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AVA

exists A therein points of)q,p(pair every for -

Examples lines, planes, space without the Euclidean

structure (dot product and derived angle and length)

qupsuch that Vin u element unique a holds following theVwu, A, p allfor -

w)(upwu)(p

EUCLIDEAN VECTOR SPACE

Definition: A vector space together with an operationV x V V, that associates to a pair (a,b) of vectorsa, b in V a real number (called their dot product and denoted by ), that satisfies the following

Definition: Length and Angle between nonzero vectors are then defined by

ba

4

0a ifonly and if0aa

2ba21ba1)2b21b1(a abba

aa|a|

|b||a|bacos

Euclidean Affine Space or Euclidean Space is an Affine Space whose Associated Vector Spacehas a Euclidean Vector Space structure

VECTOR ALGEBRA

Vectors can be represented by their coordinateswith respect to a choice of basis

1b

2b

2211 bubuu

)vu,vu(~vu 2211

and then so canvector operations

)cu,cu(~uc 21

2211 vuvuvu

5

)u,u(~u 21

If the basis orthonormal, thenijji bb

KINEMATICS

A trajectory in an affine space is a function

,],[: Adcf

Smooth trajectories in affine space define trajectoriesin the associated vector space, called velocities

,],[: Vdcf

t

)t(f)tt(f

0tlim)t(f

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KINEMATICSExample 1 Choose p, q, r be points in A and construct

]d,c[t),pq(tr)t(f

qp)qp(

0t

limt

)qp(t

0t

lim

t

))qp(tr()qp)(tt(r(

0t

lim

s

)s(f)ss(f

0s

lim)s(f

7

KINEMATICS

Smooth trajectories in affine space define trajectoriesin the associated vector space, called accelerations

t

)t(f)tt(f

0tlim)t(f

8

Remark Here, in contrast to the situation for velocity,the numerator is the difference between two vectors

KINEMATICS

Example 2 Harmonic Oscillation of a smallbody, “particle”, along a line is described by

)pq)(btcos(ar)t(f

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)pq)(btsin(a)t(f

)pq)(btcos(a)t(f 2

KINEMATICS

Example 3 Circular Motion of a small body isdescribed with by orthogonal unit vectors u, v

vtautartf )sin()cos()(

10

))2/((

)cos()sin()(

rtf

vtautatf

)(

))sin()(cos()(2

2

tf

vtutatf

NEWTON’s FIRST LAW

The velocity of an isolated body is constant

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Criticize the following versions of this law given by Halliday, Resnick and Walker:

page 73 If no force acts on a body, then the body’svelocity cannot change; that is, the body cannotacceleratepage 74 If no net force acts on a body, then the body’s velocity cannot change; that is, the body cannot accelerate

NEWTON’s FIRST LAW

Inertial Frames are preferred coordinate systems forspace-time for which Newtons laws hold

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Given an inertial frame, we can obtain others bytranslating the original in space and time, by rotatingthe original through some angle about an axis, and

by00 v)t(u)t(origf)t(newf

0v)t(origf)t(newf

)t(origf)t(newf

NEWTON’s THIRD LAW

13

Criticize the following version of this law given by Halliday, Resnick and Walker:

page 84 When two bodies interact, the forces on the bodies from each other are always equal inmagnitude and opposite in direction

Logically formulate this law by using Newton’s second law on page 84

The net force on a body is equal to the product of the bodies mass and the acceleration of the body

NEWTON’s SECOND and THIRD LAWS

14

Deal with the effects of interactions between bodies on their motion that cause them to accelerate

When bodies i and j interact (only) with each other, their acceleration magnitudes satisfy

}j,i{ja/

}j,i{ia is independent of the interaction

and for bodies i, j, k (interacting pairwise), these ratios satisfy the equation (this is not an algebraic identity)

1}i,k{ia/

}i,k{ka

}k,j{ka/

}k,j{ja

}j,i{ja/

}j,i{ia

DEFINITION OF INERTIAL MASS

15

Choose a standard body and assign it a mass, forexample the SI standard of 1 kilogram mass is that ofthe paltinum-iridium cylinder kept at the InternationalBureau of Weights and Measures near Paris

Define the mass of any body i to be

then the three body equation implies

kilograms)a/a(m }cyl,i{i

}cyl,i{cyli

ij}cyl,i{

cyl/}cyl,i{i

}cyl,j{j/}cyl,j{

cyl m/maaaa

}j,i{

j}j,i{

i aa /

DEFINITION OF FORCE

16

Define the force on a body to have magnitude ma and direction given by direction of its acceleration

Then Newton’s second law can be expressed as

1on22on1 FF

Remark: this is a consequence of Newton’s laws, as discussed in Calkin’s book, together with the definitions of mass and force

NEWTON’s SECOND LAW

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Deals with the interaction of three or more bodies

or, equivalently

Law is an empirical observation that says the netacceleration of any body is the sum of the acclerationsthat it experience from its interaction with each of theother bodies individually

j

jaa

j

jnet FFam

BOUND COMBINATIONS

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Suppose that particles 1, 2 and 3 interact. Then

1on31on211 FFam

If particles 2 and 3 interact so that they are bound together as a single particle, then

2on12on322 FFam

3on23on133 FFam

23on13on12on12323 FFFam

where 32233223 aaa,mmm

MOMENTUM

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Suppose that particles 1 and 2 interact over time

))t(vm)tt(vm(

t)t(amtFtF

t)t(am)t(vm)tt(vm

2222

222on11on2

111111

Therefore the momentum of the system, defined by

)t(vm)t(vm)t(p 2211

is constant or invariant. This is the case for anysystem of particles.

ENERGY

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Suppose that a particle accelerates under a force

)t),t(x(F)t(am)t(vm

Further assume that the force is conservative, which by definition means that there is a real valuedpotential energy function U(x) on space such that

UgradF Then energy )t(v)t(mv

2

1))t(x(U)t(E

is constant since

0)t(v)t(mv)t(v))t(x(Ugrad)t(E

HARMONIC OSCILLATIONS

Consider an object attached to aspring that moves horizontally near equilibrium Then

2

2

12

2

1 xmkxE )t(cosa)t(x

xx 0 0x

2k

2

1 xgradkxF

R,k,kE2a m 21

HARMONIC OSCILLATIONS

Consider a pendulum - an object ona swinging lever. Then for small

222

Lm θLθgE

)t(cosa)t(θ

θ Lθ

R,L

g,

LmgE2a

22

IN LINE COLLISIONS

Consider the collision of two objects

222

211

222

211 vmvmvmvm

1v1m

2m2v 1v

Since kinetic energy is conserved

2vSince momentum is conserved

22112211 vmvmvmvm

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INLINE COLLISIONS

221

21

21

211 v

mmm2

vmmmm

v

2211

12

2

1

21 vmvm

vv

v

v

mm

11

From the two equations we derive

221

121

21

12 v

mmmm

vmm

m2v

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STATICS

Compute the force that each string exerts on the body

mg

Hint: The direction of each strings force is along the string and away from the body 25