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Figure 2.5 A swing is essentially a pendulum. The mass of the seat and of the child on it is represented by a single-point particle of mass m placed at the center of mass. The mass of the chains is neglected. and they are described by a nonstretchable string of length l whose tension T exactly balances the projection of the weight of the particle on the direction of the string. The net force on the particle is then rng sin 4>. where 4> is the angle that the rope forms with the' vertical. This net force is always tangent to the arch described by the motion of the particle.

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sin if> by if>. This is a very important special case, because then the force is proportional to the displacement and always directed against it: The basic ingredient for a harmonic oscillator. This is a case that is often found in many different branches of physics and also, fortunately, a case that leads to a simple analytical solution of the equations of motion. The equation of motion of such a frictionless swing is simply

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if> + Iif>=O.

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(2.53)

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Now let us suppose that the swing was somehow set in oscillatory motion with frequency w in the remote past and that it has been swinging with this frequency ever since; for example, (2.54) if>(t) = if>osinwt, where if>o is the amplitude of the oscillations. Then, if we substitute (2.54) in (2.53), we find that (2.55) whose only solution 16 is w = In other words, the frictionless swing can oscillate forever only at the single frequency For this reason, is said to be its natural frequency of oscillation. Any real swing, however, also has friction that will eventually stop its motion. Let us suppose that this is a sort of viscous friction proportional to the velocity, say -m'Yl(p, where "I is a constant. The effect of the person pushing the swing will be taken into account by adding a driving force: say, mIF(t), for convenience. Then the equation of motion is ., . 9 (2.56) if> + 'Yif> + Iif> = F.

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Consider now what happens if we push the swing with an arbitrary frequency w [e.g., we take F(t) = Asinwt, where A is a real number describing the strength of our pushes]. To simplify the math, we can define a new complex function 'P(t) such that 17 if> (t) = ~ {'P( t)}; then, as (2.56) is linear, 'P( t) obeys the simpler equation

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cP + 'Yrf; + g'P = Aeiwt

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(2.57)

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Equation (2.57) can be solved very quickly using an intuitive method invented by Oliver Heaviside called operational calculus [282, 283, 610]. First, we rewrite (2.57) as J2 d 9 } () A iwt (2.58)

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{ dt 2 + "I dt + I 'P t

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The expression inside the braces on the left-hand side of (2.58) is an operator that acts on 'P(t). If we multiply both sides of (2.58) from the left by the inverse of this

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that w is always a positive quantity. denotes the operation of taking the imaginary part.

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operator, we find the (particular) solution of (2.58):

<p(t) =

{ 2+ /

dt 2

(2.59)

But what is the right-hand side of (2.59)1 Fractional expressions involving differential operators are defined in terms of their power series expansions. For example, (1 dldt)-l is just 'L.':=o(dldt)n, where (dldt)n just means applying dl dtto something n times (i.e., d n Idt n ). So going back to (2.59), as each dldt applied to exp(iwt) is the same as multiplication by iw, the solution is given by

<p(t)

hw - gil'

(2.60)

/12). Then (2.60)

An interesting case is when the friction is weak (i.e., leads to

Vill

q,(t :::::: [

A [(w 2

gil) sinwt + w/coswt]

(w + Vill)

+ (/ 2)2

(w - Vill)

(2.61)

+ (/2)2

If we push the swing at its natural frequency,18 this: