SEMINAR 2. PENDULUMS. V = mgl cos θ. (2) L = T V = 1 2 ml2 θ2 + mgl cos θ, (3) d dt ml2 θ2 + mgl sin θ = 0, (4) θ + g l

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1 Probem 7. Simpe Penduum SEMINAR. PENDULUMS A simpe penduum means a mass m suspended by a string weightess rigid rod of ength so that it can swing in a pane. The y-axis is directed down, x-axis is directed hizontay,i.e. x = sin θ, y = cos θ. The kinetic energy is then T = 1 mv = 1 m(ẍ + ÿ ) = 1 m( θ). (1) If we put the potentia energy to be zero when the string is hizonta, then at ange θ it is So the Lagrangian is which yieds Lagrange s equation of motion V = mg cos θ. () L = T V = 1 m θ + mg cos θ, (3) d dt m θ + mg sin θ =, (4) θ + g sin θ =. (5) This equation ooks simpe but, in genera, it is not easy to sove. However, if we assume that the osciations are sma (say, θ << π/), then sin θ can be approximated by θ, and Eq. (5) takes the fm of the usua inear equation f a simpe harmonic motion, namey θ + g θ =, (6) θ + ω θ =, (7) where ω = g. (8) The soution of this equation is we-known: θ = C cos(ωt + δ), (9) where ω and δ are the anguar frequency and the phase of the osciations, whie C is arbitrary constant which determine the ampitude of the osciations. The period of the osciations is then T = π ω = π g. (1)

2 Note that the period of osciations is independent of the ampitude, provided the ampitude is sma enough so that Eq. (7) is a good approximation. Now we return back to the consideration of the penduum equation in its genera fm (5). Mutipying both sides of it by θ and integrating, we obtain θ θ = g sin θ θ, (11) θd θ = g sin θ, (1) and 1 θ = g cos θ + const. (13) F a moment, et use this equation f finding the period in a particuar case of argeampitude swinging when the penduum is going back and fce between the turning points 9 and +9. By definition, in these points θ =, (14) so the constant in (13) must be zero, and we have 1 θ = g cos θ, = g dt cos θ, cos θ = g dt. (15) By observation that in our particuar case the change in θ from θ = to θ = 9 cresponds just one-quarter of a period, it foows π/ cos θ = that is the period of the 18 swings is g T/4 dt = g T 4. (16) T = 4 g π/ cos θ. (17) You might be famiiar with the function invoved in the r.h.s. of this expression: it is nothing but a particuar case of the Beta B-function. Finay, et consider swings of any ampitude, say α. Then we may use the turningpoint-condition (14) with θ = α which eads to the reation θ = g (cos θ cos α). (18)

3 Hence Eq. (17) must be changed to T α = 4 g α cos θ cos α. (19) Here T α is the period f swings from α to +α and back. The integra invoved in this expression can be transfmed to a tabe integra as foows: cos θ = 1 sin θ cos α = 1 sin α I = α cos θ cos α = (sin α sin θ) cos θ cos α = α (sin = 1 α α sin θ ) 1 sin θ sin α sin α () Introduce new variabe: In terms of this variabe I = 1 x = sin θ sin α dx = 1 cos θ sin α dx (1 x )(1 x sin α ) (1) ( α ) K sin, () where we used the notation K f an eiptic integra. Thus the period (19) takes the fm ( α ) T α = 4 K sin = 4 g g K( sin α ). (3) This expression f the period can be used to exactify the vaue of the period as compared with its sma-osciation approximation given by Eq. (1) due to the existence of the foowing expansion f an eiptic integra: K ( sin α ( ) π = 1 + ( 1) sin α + ( 1 3) sin 4 α ) (4) F α sma enough so that sin α/ can be approximated by α/, it foows K ( sin α ( ) ) π = 1 + α , (5) and hence the period can be approximated as ( ) T α = π 1 + α g (6) We see that this fmua differs from our previous one f simpe harmonic motion, T = π /g, by the presence of the second- (and higher-)der terms on α. Naturay,

4 f very sma α this difference is negigibe. However, f somewhat arge α, say, α =.5 radian (about 3 ), we get T α = π ( ) g (7) It means, f exampe, that a penduum started at 3 woud get exacty out of phase with a penduum started at very sma ange in about 3 periods. Physicay, the motion of a penduum at different ampitudes can be easiy understood if we consider the sum of the kinetic and potentia energy, T + V = 1 m θ mg cos θ = E, (8) where E is the initia energy eve of the system. The potentia energy V (θ) is mg cos θ. We see that f mg < E < mg, (9) the motion is osciating one because of the existence of the turning point where the tota energy is equa the potentia energy. On the other hand, f E > mg, (3) there is no turning point, and the motion is nonosciaty: θ is steadiy increases steadiy decreases, whie θ osciates between a maximum and minimum vaue, as cabn be shown in the phase diagram θ = f(θ). In this case a penduum has enough energy to swing around in a compete circe. Note that this motion is not osciaty but sti periodic, a penduum making one compete revoution each time θ increases decreases by π. Finay, f E = mg, (31) there exist the positions θ = ±(n + 1)π, (n =, 1,...) (3) which are caed the bifurcation points of the soution of the equation of motion f a simpe penduum. Probem 8. Doube Penduum Consider the motion of a doube penduum that consist of two simpe pendua, each of mass m and enght, as shown in Figure from the Set of Probems. The first one is attached to a fixed suppt, and the second one is attached to the mass of the first. (a) Assuming that the penduum executes sma osciations confined to a singe pane, find

5 the modes of osciations. (b) Find numericay the genera soution of the equations of motion. The configuration of the system is specified by the two anges θ and ϕ, as shown in the Figure. The Cartesian codinates of the two masses reate to these generaized codinates accding to: x 1 = sin θ y 1 = cos θ (33) x = x 1 + sin ϕ = (sin θ + sin ϕ) y = y 1 + cos ϕ = (cos θ + cos ϕ). The cresponding veocities are ẋ 1 = cos θ θ ẏ 1 = sin θ θ ẋ = (cos θ θ (34) + cos ϕ ϕ) ẏ = (sin θ θ + sin ϕ ϕ), so that the kinetic and potentia energies are cacuated as T = 1 m(ẋ 1 + ẏ1 + ẋ + ẏ) = 1 m [ θ + ϕ + cos(θ ϕ) θ ϕ] (35) and V = mgy 1 mgy = mg( cos θ + cos ϕ). (36) The Lagrangian L = T V = 1 m [ θ + ϕ + cos(θ ϕ) θ ϕ] + mg( cos θ + cos ϕ) (37) creates Lagrange s equations of motion { θ + ϕ cos(θ ϕ) + ϕ sin(θ ϕ) + g sin θ = ; ϕ + θ cos(θ ϕ) θ sin(θ ϕ) + g sin ϕ =. (38) In genera, this system of equations is rather compicated and it must be soved numericay. The anaytica soution can be obtained, as usua, in the case of sma osciations when we can use the sma ange approximation f a sine and cosine functions invoved, that is cos(θ ϕ) 1, sin(θ ϕ) (θ ϕ), sin θ θ sin ϕ ϕ. (39) Substituting these expressions into the equations of motion and negecting the higherder terms, we obtain the simpified version of these equations, { θ + ϕ + ω θ = ϕ + θ + ω ϕ =, where (4) ω = g. (41)

6 By substituting we transfm Eqs. ampitudes A and B, { θ = Ae iωt iωt, (4) ϕ = Be (4) to the set of the homogeneous agebraic equations f the { ( ω + ω )A ω B = ω A + ( ω + ω )B = This set of equations has the nontrivia (i.e. nonzero) soution ony if its determinant is equa zero, (43) ( ( ω det + ω) ω ) ω ( ω + ω) =, (44) ( ω + ω ) ω 4 =, (45) which yieds the equation whose roots are ω 4 4ω ω + ω 4 =, (46) ω 1, = ω. (47) These roots determines the two possibe modes of sma osciations of a doube penduum. To understand the physica sense of these modes we substitute the frequencies ω 1 and ω back into the set of equations (43). F ω = ω 1, we obtain { ( )A + ( )B = ( )A + (1 )B =, which yieds (48) B = A. (49) F ω = ω, we have { ( + )A + ( + )B = ( + )A + (1 + )B =, which yieds (5) B = A. (51) Hence the modes ω 1 and ω can be naturay caed the symmetric and antisymmetric modes, respectivey. It is interesting to notice that the ratio of these two mode frequencies is independent of a the parameters m, and g and is equa to [ ] 1/ ω ( + ) = ω 1 ( =.414, (5) ) that is the osciation in the faster, antisymmetric, mode has a frequency about two and one-haf times that of the sower, symmetric, mode.