Basic Theoretical Physics: A Concise Overview P3 pdf

Finally, the problem of planetary motion dating back to the time of New-ton where one must in principle distinguish between the inertial mass mt entering 2.2 and a gravitational mass ms,

8 2 Force and Mass

mentioned, a different viewpoint is also possible, and it is better to keep an

open mind on these matters than to fix our ideas unnecessarily.

Finally, the problem of planetary motion dating back to the time of

New-ton where one must in principle distinguish between the inertial mass mt entering (2.2) and a gravitational mass ms, which is numerically identical to

mt(apart from a universal constant, which is usually replaced by unity), is

far from being trivial; msis defined by the gravitational law:

F (r) = −γ Ms· ms

|r − R|2 · r − R

|r − R| ,

wherer and msrefer to the planet, andR and Msto the central star (“sun”),

while γ is the gravitational constant Here the [gravitational] masses play

the role of gravitational charges, similar to the case of Coulomb’s law in electromagnetism In particular, as in Coulomb’s law, the proportionality

of the gravitational force to Ms and ms can be considered as representing

an active and a passive aspect of gravitation.4 The fact that inertial and gravitational mass are indeed equal was first proved experimentally by E¨otv¨os

(Budapest, 1911 [6]); thus we may write ms = mt ≡ m.

2.4 Newton’s Third Axiom (“Action and Reaction ”)

Newton’s third axiom states that action and reaction are equal in magni-tude and opposite in direction.5This implies inter alia that the “active” and

“passive” gravitational masses are equal (see the end of the preceding

sec-tion), i.e., on the one hand, a body with an (active) gravitational charge Ms

generates a gravitational field

G(r) = −γ Ms

|r − R|2 · |r − R| r − R ,

in which, on the other hand, a different body with a (passive) gravitational

charge ms is acted upon by a force, i.e., F = ms· G(r) The relations are

analogous to the electrical case (Coulomb’s law) The equality of active and passive gravitational charge is again not self-evident, but in the considered

context it is implied that no torque arises (see also Sect 5.2) Newton also

recognized the general importance of his third axiom, e.g., with regard to the application of tensile stresses or compression forces between two bodies Three additional consequences of this and the preceding sections will now

be discussed

4

If one only considers the relative motion, active and passive aspects cannot be distinguished

5

In some countries this is described by the abbreviation in Latin “actio=reactio”

2.4 Newton’s Third Axiom (“Action and Reaction ”) 9 a) As a consequence of equating the inertial and gravitational masses in Newton’s equationF (r) = ms· G(r) it follows that all bodies fall equally

fast (if only gravitational forces are considered), i.e.: a(t) = G(r(t)).

This corresponds to Galileo’s experiment6, or rather thought experiment,

of dropping different masses simultanously from the top of the Leaning Tower of Pisa

b) The principle of superposition applies with respect to gravitational forces:

G(r) = −γ

k

(ΔMs) k

|r − R k |2 · r − R k

|r − R k | . Here (ΔMs)k k ΔV k is the mass of a small volume element ΔV k, and

kis the mass density An analogous “superposition principle” also applies for electrostatic forces, but, e.g., not to nuclear forces For the principle

of superposition to apply, the equations of motion must be linear

c) Gravitational (and Coulomb) forces act in the direction of the line joining the point masses i and k This implies a different emphasis on the meaning

of Newton’s third axiom In its weak form, the postulate means that

F i,k =−F k,i; in an intensified or “strong” form it means that F i,k = (r i − r k)· f(r i,k ), where f (r ik ) is a scalar function of the distance r i,k :=

|r i − r k |.

As we will see below, the above intensification yields a sufficient condition that Newton’s third axiom not only implies F i,k =−F k,i, but also D i,k =

−D k,i, whereD i,k is the torque acting on a particle at r i by a particle atr k

6

In essence, the early statement of Galileo already contained the basis not only of

the later equation ms= mt, but also of the E¨otv¨os experiment, [6] (see also [4]), and of Einstein’s equivalence principle (see below)

3 Basic Mechanics of Motion

in One Dimension

3.1 Geometrical Relations for Curves in Space

In this section, motion is considered to take place on a fixed curve in

three-dimensional Euclidean space This means that it is essentially one-dimensional; motion in a straight line is a special case of this

For such trajectories we assume they are described by the radius vector

r(t), which is assumed to be continuously differentiable at least twice, for

t ∈ [t a , t b ] (where t a and t b correspond to the beginning and end of the

motion, respectively) The instantaneous velocity is

v(t) := dr

dt , and the instantaneous acceleration is

a(t) := dv

dt =

d2r

dt2 , where for convenience we differentiate all three components, x(t), y(t) and z(t) in a fixed Cartesian coordinate system,

r(t) = x(t)e x + y(t) e y + z(t) e z: dr(t)

dt = ˙x(t) e x+ ˙y(t) e y+ ˙z(t) e z For the velocity vector we can thus simply write: v(t) = v(t)τ (t), where

v(t) =

( ˙x(t))2+ ( ˙y(t))2+ ( ˙z(t))2

is the magnitude of the velocity and

τ (t) := v(t)

|v(t)|

the tangential unit vector to the curve (assuming v = 0).

v(t) and τ(t) are thus dynamical and geometrical quantities, respectively,

with an absolute meaning, i.e., independent of the coordinates used

In the following we assume thatτ(t) is not constant; as we will show, the

acceleration can then be decomposed into, (i), a tangential component, and,

12 3 Basic Mechanics of Motion in One Dimension

(ii), a normal component (typically: radially inwards), which has the direction

of a so-called osculating normal n to the curve, where the unit vector n is

proportional to dτ

dt, and the magnitude of the force (ii) corresponds to the well-known “centripetal” expression v R2 (see below); the quantity R in this formula is the (instantaneous) so-called radius of curvature (or osculating radius) and can be evaluated as follows:

1

R =| dτ

v · dt | 1 Only the tangential force, (i), is relevant at all, whereas the centripetal expression (ii) is compensated for by forces of constraint2, which keep the motion on the considered curve, and need no evaluation except in special instances

The quantity

t



t a

v(t) dt

is called the arc length s(t), with the differential ds := v(t) dt As already mentioned, the centripetal acceleration, directed towards the center of the osculating circle, is given by

acentrip..(t) = n(t) v2(t)

R(t) .

We thus have

a(t) ≡ τ(t) · dv(t)

dt +acentrip.(t)

The validity of these general statements can be illustrated simply by con-sidering the special case of circular motion at constant angular velocity, i.e.,

r(t) := R · (cos(ωt)e x + sin(ωt) e y )

The tangential vector is

τ(t) = − sin(ωt)e x + cos(ωt) e y , and the osculating normal is

n(t) = −(cos(ωt)e x + sin(ωt) e y ) , i.e., directed towards the center The radius of curvature R(t) is of course

identical with the radius of the circle The acceleration has the

above-mentioned magnitude, Rω2= v2/R, directed inwards.

1

It is strongly recommended that the reader should produce a sketch illustrating these relations

2

This is a special case of d’Alembert’s principle, which is described later.

3.2 One-dimensional Standard Problems 13

Fig 3.1 Osculating circle and radius of curvature The figure shows as a typical

example the lower part of an ellipse (described by the equation y1

b := 1±q1− x2

a2,

with a := 2 and b := 1) and a segment (the lower of the two curves!) of the osculating

circle at x = 0 (with R := a b2 ≡ 4) Usually a one-dimensional treatment suffices,

since an infinity of lines have the same osculating circle at a given point, and since

usually one does not require the radial component n · mv2

R of the force (which

is compensated by forces of constraint), as opposed to the tangential component

Ftangent:= m · ˙v(t) ·τ (t) (where t is the time, m the inertia and v the magnitude of

the considered point mass);n and τ are the osculating normal and the tangential

unit vectors, respectively A one-dimensional treatment follows

Finally, as already alluded to above, the quantiesτ(t), n(t), R(t), and s(t)

have purely geometrical meaning; i.e., they do not change due to the kine-matics of the motion, but only depend on geometrical properties of the curve

on which the particle moves The kinematics are determined by Newton’s equations (2.2), and we can specialize these equations to a one-dimensional problem, i.e., for∼ τ (t), since (as mentioned) the transverse forces, n(t) · v2

R,

are compensated for by constraining forces.

The preceding arguments are supported by Fig 3.1 above

Thus, for simplicity we shall write x(t) instead of s(t) in the following,

and we have

v(t) = dx(t)

dt and m · a(t) = m · d2x(t)

dt2 = F (t, v(t), a(t)) , where F is the tangential component of the force, i.e., F ≡ F · τ.

3.2 One-dimensional Standard Problems

In the following, for simplicity, instead of F we consider the reduced quantity

f := F If f (t, v, x) depends on only one of the three variables t, v or x, the

14 3 Basic Mechanics of Motion in One Dimension

equations of motion can be solved analytically The most simple case is where

f is a function of t By direct integration of

d2x(t)

dt2 = f (t)

one obtains:

v(t) = v0+

 t

t0

d˜tf (˜ t) and x(t) = x0+ v0 · (t − t0) +

 t

t0

d˜tv(˜ t)

(x0, v0 and t0 are the real initial values of position, velocity and time.)

The next most simple case is where f is given as an explicit function of v.

In this case a standard method is to use separation of variables ( ˆ= transition

to the inverse function), if possible: Instead of

dv

dt = f (v)

one considers

dt = dv

f (v) , or t − t0=

 v

v0

d˜v

f (˜ v) . One obtains t as a function of v, and can thus, at least implicitly, calculate v(t) and subsequently x(t).

The third case is where f ≡ f(x) In this case, for one-dimensional prob-lems, one always proceeds using the principle of conservation of energy, i.e.,

from the equation of motion,

m dv

dt = F (x) ,

by multiplication with

v = dx dt and subsequent integration, with the substitution v dt = dx, it follows that

v2

2m + V (x) ≡ E

is constant, with a potential energy

V (x) := −

x



x0

d˜xF (˜ x)

Therefore,

v(x) =

 2

m (E − V (x)) ,

3.2 One-dimensional Standard Problems 15

2

m (E − V (x))

, and finally

t − t0=

 x

x0

d˜x

 2

m (E − V (˜x)) .

This relation is very useful, and we shall return to it often later

If f depends on two or more variables, one can only make analytical

progress in certain cases, e.g., for the driven harmonic oscillator, with damp-ing proportional to the magnitude of the velocity In this important case,

which is treated below, one has a linear equation of motion, which makes the

problem solvable; i.e.,

¨

x = −ω2x −2

τ v + f (t) ,

where useful general statements can be made (see below) (The above-mentioned ordinary differential equation applies to harmonic springs with

a spring constant k and mass m, corresponding to the Hookean force

FH:=−k · x, where ω2= k/m, plus a linear frictional force F R:=−m2

τ · v, plus a driving force FA := m · f(t).)

There are cases where the frictional force depends quadratically on the velocity (so-called Newtonian friction),

F R:=−α · mv2

2 ,

i.e., with a so-called technical friction factor α, and a driving force depending mainly, i.e., explicitly, on x, and only implicitly on t, e.g., in motor racing, where the acceleration may be very high in certain places, F a = mf (x) The

equation of motion,

m ˙v = −α mv2

2 + mf (x) , can then be solved by multiplying by

dt dx



≡ 1 v

 : One thus obtains the ordinary first-order differential equation

dv

dx +

αv

2 =

f (x)

v ,

which can be solved by iteration On the r.h.s of this equation, one uses, for

example, an approximate expression for v(x) and obtains a refinement on the

l.h.s., which is then substituted into the r.h.s., etc., until one obtains con-vergence In almost all other cases one has to solve an ordinary second-order

differential equation numerically Many computer programs are available for

solving such problems, so that it is not necessary to go into details here

4 Mechanics of the Damped

and Driven Harmonic Oscillator

In this section the potential energy V (x) for the motion of a one-dimensional system is considered, where it is assumed that V (x) is smooth everywhere and differentiable an arbitrarily often number of times, and that for x = 0,

V (x) has a parabolic local minimum In the vicinity of x = 0 one then obtains the following Taylor expansion, with V  (0) > 0:

V (x) = V (0) + 1

2V

 (0)x2+ 1

3!V

 (0) x3+ ,

i.e.,

V (x) = V (0) + mω

2

2 x

2+O x3 , with ω2:= V  (0)/m, neglecting terms of third or higher order For small

os-cillation amplitudes we thus have the differential equation of a free harmonic

oscillator of angular frequency ω0:

m¨ x = − dV

dx , x =¨ −ω2x , whose general solution is: x(t) = x0 ·cos(ω0t −α), with arbitrary real quantities

x0 and α.

In close enough proximity to a parabolic local potential energy minimum, one always obtains a harmonic oscillation (whose frequency is given by ω0:=

V (0)

m )

If one now includes (i) a frictional force, F R:=−γv, which can be char-acterized by a so-called “relaxation time” τ (i.e., γ =: m · 2/τ), and (ii)

a driving force FA(t) = m · f(t), then one obtains the ordinary differential

equation

¨

x +2

τ ˙x + ω

2

0x = f (t) This is a linear ordinary differential equation of second order (n ≡ 2) with constant coefficients For f (t) ≡ 0 this differential equation is homogeneous, otherwise it is called inhomogeneous (For arbitrary n = 1, 2, the general

inhomogeneous form is:



dn

dn t+

n −1 ν=0

a ν d ν

dν t



x(t) = f (t)).

18 4 Mechanics of the Damped and Driven Harmonic Oscillator

For such differential equations, or for linear equations in general, the principle of superposition applies: The sum of two solutions of the homo-geneous equation, possibly weighted with real or complex coefficients, is also

a solution of the homogeneous equation; the sum of a “particular solution”

of the inhomogeneous equation plus an “arbitrary solution” of the homoge-neous equation yields another solution of the inhomogehomoge-neous equation for the same inhomogeneity; the sum of two particular solutions of the inhomo-geneous equation for different inhomogeneities yields a particular solution

of the inhomogeneous differential equation, i.e., for the sum of the inhomo-geneities.

The general solution of the inhomogeneous differential equation is

there-fore obtained by adding a relevant particular solution of the inhomogeneous (i.e., “driven”) equation of motion to the general solution of the homogeneous equation, i.e., the general “free oscillation”.

As a consequence, in what follows we shall firstly treat a “general free oscillation”, and afterwards the seemingly rather special, but actually quite general “periodically driven oscillation”, and also the seemingly very special, but actually equally general so-called “ballistically driven oscillation”.

The general solution of the equation for a free oscillation, i.e., the general solution of

dn

dn t +

n −1



ν=0

a ν

dν

dν t x(t) = 0 for n = 2 ,

is obtained by linear combination of solutions of the form x(t) ∝ e λ ·t After

elementary calculations we obtain:

x(t) = exp



− t τ



·

⎧

⎨

⎩x0cos



ω2− 1

τ2 · t

+



v0+x0

τ



· sin



ω2− 1

τ2· t



ω02− 1

τ2

⎫

⎪

This expression only looks daunting at first glance, until one realizes that the

bracketed expression converges for t → 0 to

x0+



v0+x0

τ



· t ,

as it must do

Equation (4.1) applies not only for real

ε =



ω2− 1

τ2

4 Mechanics of the Damped and Driven Harmonic Oscillator 19

but also for imaginary values, because in the limit t → 0 not only

sin(εt)

but also

sin iεt

iε → t ;

x0 and v0 are the initial position and the initial velocity, respectively Thus the above-mentioned formula applies

a) not only for damped oscillations, i.e., for ε > 0, or

ω2> 1

τ2 ,

i.e., sine or cosine oscillations with frequency

ω1:=



ω2− 1

τ2 and the damping factor e−λ1t , with λ1:= 1τ,

b) but also for the aperiodic case,

ω20< 1

τ2 ,

since for real

ε : sin(iε · t)

iε ≡ sinh εt , with sinh(x) : = 1

2(e

x − e −x ) , and

cos(iεt) ≡ cosh(εt) , with cosh(x) : = 1

2(e

x+ e−x) :

In the aperiodic case, therefore, an exponential behavior with two char-acteristic decay frequencies results (“relaxation frequencies”),

λ ±:= 1

τ ±

 1

τ2 − 1

ω2 .

Of these two relaxation frequencies the first is large, while the second is small

c) Exactly in the limiting case, ε ≡ 0, the second expression on the r.h.s.

of equation (4.1) is simply (v0+x0

τ)· t for all t, i.e., one finds the fastest

decay

4 Mechanics of the Damped and Driven Harmonic Oscillator 19

but also for imaginary values, because in the limit t → not only... for real

ε =



ω2− 1

τ2