Heavy pendulum

One calls heavy pendulum any solid mobile around an axis (in theory horizontal) not passing by his center of gravity and placed in a field of Pesanteur. Moved its position of balance (stable) in which the center of gravity is with the vertical of the axis, the solid puts itself at to oscillate on both sides of this position known as of balance. A beam of clock, a swing, etc, constitute heavy pendulums.

The simplest case is the pendulum made up of a small heavy object fixed on a wire (or a stem) of negligible mass in front of that of the object. Such a pendulum is called simple pendulum weighing .

The simple pendulum weighing has a historical importance owing to the fact that Galileo studied it in a detailed and scientific way.

Étude theoretical of the model of the simple pendulum weighing

One can simply establish the differential equation of the movement of oscillation starting from the mechanical conservation of energy. By neglecting the Friction S, the mechanical energy of the pendulum is constant: it is the sum of the kinetic energy and the potential energy.

In the case of the model of the simple pendulum weighing, one considers that the object brings back to a specific mass m , which moves at the distance L axis (length of the wire or stem, considered inextensible and without mass). One from of deduced:

$E\_t=\; E\_c+E\_p=\; \backslash \; frac\; \{m\; \backslash \; cdot\; l^2\; \backslash \; cdot\; (\backslash \; frac\; \{D\; \backslash \; theta\}\; \{D\; T\})\; ^2\}\; \{2\}\; -\; m\; \backslash \; cdot\; G\; \backslash \; cdot\; L\; \backslash \; cdot\; cos\; \backslash \; theta\; =\; -\; m\; \backslash \; cdot\; G\; \backslash \; cdot\; L\; \backslash \; cdot\; cos\; \backslash \; theta\_o$

where:

• G is the Accélération of the Pesanteur, and is worth on average 9,81 m·s^-2 in France.
• θ is the angle which the pendulum forms, on a date T, with the vertical
• θ₀ is the maximum angle.

The mechanical energy being constant in time, its derivative is null. By deriving the relation above compared to time one obtains after simplification:

$\backslash \; theta$ + {G \ over L} \ sin \ theta = 0

This equation is that of an oscillating not harmonic , i.e. nonsinusoidal.

The period T of the oscillations depends on the amplitude of the movement.

On the other hand, the period does not depend on the mass.

Exact expression of the period of the oscillations

By separating the variables in:

$\backslash \; frac\; \{m\; l^2\; \backslash ,\; (\backslash \; frac\; \{D\; \backslash \; theta\}\; \{D\; T\})\; ^2\}\; \{2\}\; -\; m\; G\; L\; \backslash \; cos\; (\backslash \; theta)\; =\; -\; m\; G\; L\; \backslash \; cos\; (\backslash \; theta\_0)$

$\backslash \; frac\; \{ml^2\; (\backslash \; frac\; \{D\; \backslash \; theta\}\; \{D\; T\})\; ^2\}\; \{2\}\; =\; m\; gl\; (\backslash \; cos\; (\backslash \; theta)\; -\; \backslash \; cos\; (\backslash \; theta\_0))$

$\{(\{D\; T\})\; ^2\}\; =\; \backslash \; frac\; \{L\; \backslash ,\; (D\; \backslash \; theta)\; ^2\}\; \{2g\; (\backslash \; cos\; (\backslash \; theta)\; -\; \backslash \; cos\; (\backslash \; theta\_0))\}$ and by taking the root of this last expression

$\{D\; T\}\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{L\}\; \{2g\}\}\; \backslash \; frac\; \{D\; \backslash \; theta\}\; \{\backslash \; sqrt\; \{\backslash \; cos\; (\backslash \; theta)\; -\; \backslash \; cos\; (\backslash \; theta\_0)\}\}$

one obtains the exact expression of the period of oscillations of amplitude $\backslash \; theta\_0$ which is:

$T=\; 4\; \backslash \; sqrt\; \{\backslash \; frac\; \{L\}\; \{2g\}\}\; \backslash \; int\_0^\; \{\backslash \; theta\_0\}\; \{\backslash \; frac\; \{D\; \backslash \; theta\}\; \{\backslash \; sqrt\; \{\backslash \; cos\; (\backslash \; theta)\; -\; \backslash \; cos\; (\backslash \; theta\_0)\}\}\}$

This expression results easily by noting that T= 4 times time put to go from 0 to $\backslash \; theta\_o$

$T=\; 4\; \backslash \; sqrt\; \{\backslash \; frac\; \{L\}\; \{2g\}\}\; K\; (\backslash \; sin\; \backslash \; frac\; \{\backslash \; theta\_0\}\; \{2\})\; =\; T\_0\; \{2K\; (\backslash \; sin\; \backslash \; frac\; \{\backslash \; theta\_0\}\; \{2\})\; \backslash \; over\; \backslash \; pi\}$

where K is an elliptic integral complete of first species which is worth $at\; first\; approximation\; (1+\; \backslash \; frac\; \{\backslash \; theta\_0^2\}\; \{16\})$ and where $T\_0=\; 2\; \backslash \; pi\; \backslash \; sqrt\; \{\backslash \; frac\; \{L\}\; \{G\}\}$ .

The expression $T\; =\; T\_0\; (1+\; \backslash \; frac\; \{\backslash \; theta\_0^2\}\; \{16\})$ is known under the name of formula of Bordered.

One can also express T in the form of series. If one poses $\backslash \; gamma\; =\; \backslash \; sin\; \backslash \; left\; (\{\backslash \; theta\_0\; \backslash \; over\; 2\}\; \backslash \; right)$, then:

$T\; =\; 2\; \backslash \; pi\; \backslash \; sqrt\; \{L\; \backslash \; over\; G\}\; \backslash \; sum\_\; \{n=0\}\; ^\; \backslash \; infty\; \{2n\; \backslash \; choose\; N\}\; ^2\; \{\backslash \; gamma^\; \{2n\}\; \backslash \; over\; 16^n\}$, where $\{2n\; \backslash \; choose\; N\}$ is a binomial coefficient.

If one develops γ according to θ₀, one obtains:

$T\; =\; 2\; \backslash \; pi\; \backslash \; sqrt\; \{L\; \backslash \; over\; G\}\; (1\; +\; \{\backslash \; theta\_0^2\; \backslash \; over\; 16\}\; +\; \{11\; \backslash \; theta\_0^4\; \backslash \; over\; 3072\}\; +\dots )$.

table Ci below gives the angles in degree, then in radian in the first two columns

as well as the values of $1+\; \backslash \; frac\; \{\backslash \; theta\_0^2\}\; \{16\}\; =\; T\_1/T\_0$, $1+\; \backslash \; frac\; \{\backslash \; theta\_0^2\}\; \{16\}\; +\; \{11\; \backslash \; theta\_0^4\; \backslash \; over\; 3072\}\; =T\_2/T\_0$, and finally $T/T\_0\; =\; \{2K\; (\backslash \; sin\; \backslash \; frac\; \{\backslash \; theta\_0\}\; \{2\})\; \backslash \; over\; \backslash \; pi\}$:

One can retain that with an angle of $\backslash \; theta\_o$ of 50° the period is 5% larger than that given by the simple formula: $T\_0\; =\; 2\; \backslash \; pi\; \backslash \; sqrt\; \{\backslash \; frac\; \{L\}\; \{G\}\}$ and that the correction due to the second term is perceptible only with angles higher than 70°

Approximate expression of the period of small oscillations

For low oscillations, the differential equation can be roughly written:

$\backslash \; theta$ + {G \ over L} \ theta = 0

It is thus seen that, for low amplitudes making it possible to approach the sine with its angle, the pendulum behaves like a Oscillateur harmonic. The period is then independent of the amplitude. One calls this the isochronism of the small oscillations . This period is expressed then simply by:

$T\_o\; =\; 2\; \backslash \; pi\; \backslash \; cdot\; \backslash \; sqrt\; \{\backslash \; frac\; \{L\}\; \{G\}\}$

in accordance with the limit of the expression of Bordered.

Period of the made up heavy pendulum

For an unspecified pendulum weighing, the effect of inertia on rotation cannot be brought back to a specific mass placed at the center of gravity. It is the whole of the solid which turns, and its inertia is characterized by its Moment of inertia noted J and L outdistances it center of gravity to the axis.

With the low amplitudes, the isochronism of the oscillations is also checked and the corresponding period is expressed according to J by:

$T\; =\; 2\; \backslash \; pi\; \backslash \; cdot\; \backslash \; sqrt\; \{\backslash \; frac\; \{J\}\; \{mgl\}\}$; (for the pendulum simple J= m.l ²)

Text of Galileo concerning the pendulum

(...) As for times of oscillation of mobiles suspended with wire various lengths, they have between them even proportion that the square roots of times ; so that to obtain a pendulum whose time of oscillation is double of that of another pendulum, it is advisable to give to the first a quadruple length of that of the second; same manner if a pendulum has a length nine times higher than that of another pendulum, this one will carry out three oscillations while that one achieves only one of them; from where it results that the lengths of the cords are inversely proportional to the squares of the number of the oscillations achieved in same time.

Sagredo: If I understood well, I will be able easily to thus know the length of a cord attached to an unspecified height, when well even its point of suspension would be invisible and that one would see only his lower end. So indeed I attach in this part of cord weight extremely heavy, to which I communicate back and forth pass, and if a friend then counts the number of these oscillations while myself I count the oscillations carried out by another pendulum suspended with a wire measuring one exactly bent, thanks to the number of the oscillations of these two pendulums lasting the same time, I find the length of the cord; let us suppose for example that my friend counted twenty oscillations of the large cord, in same time when I counted of them two hundred and forty for my long wire of one bent; taking the squares of the two numbers twenty and two hundred and forty, i.e. 400 and: 57600, I will say that the large cord contains: 57600 of the units whose my small pendulum contains 400, but this one measures only one bent: I will thus divide: 57600 by 400, which gives 144, and I will say that my cord has a length of 144 bent.

Salviati: You would not even be mistaken a palm especially if you take a great number of oscillations.

Sagredo: You give me with many recoveries the occasion to admire the richness and also the extreme liberality of nature, when so common things, and I would say even certain way commonplace, you make emerge from knowledge as astonishing as new, and often unforeseen for imagination. It sometimes happened to me well thousand times to pay attention to oscillations, and in particular to those of these lamps of church, suspended with long cords, and that somebody by inadvertency had put moving; but more than I knew to draw from such observations is the improbability of the opinion according to which similar movements would be maintained by the medium, i.e. by the air, which really should have a great sagacity, and at the same time few things to be made, to thus spend the hours and hours to maintain with such a regularity the swinging of a weight. As for concluding that this same mobile, suspended with a cord of one hundred bent, then isolated of its point low sometimes of four twenty ten degrees, sometimes of a degree or a half-degree only, needs the same time to cross smallest and largest of these arcs , that, I believe, would never have come to me to mind, and still maintaining seems me to hold of the impossible one.

(...) In the final analysis I took two swell, one out of lead and the other out of cork, that one at least hundred times heavier than this one, then I attached each one of them to two very fine, long wire both of four or five bent; drawing aside then from the perpendicular position, I released them at the same time and this one, according to the circumferences of the circles having equal wire for rays, exceeded the perpendicular to go up other side, consequently way; a good hundred comings and goings, achieved by the balls themselves, clearly showed me that between the period of the body weighing and that of the light body, coincidence is such as on thousand vibrations as out of hundred, the first does not acquire on the second any advance, was this tiniest, but that both have a rigorously identical rate/rhythm of movement . One also observes the action of the medium which, by obstructing the movement, well more slows down the vibrations of cork that those of lead, without however modifying their frequency; even if the arcs described by cork do not have any more that five or six degrees, against fifty or sixty for lead, they are indeed crossed in equal times.

Comment on the text :

in italics is indicated the laws of the pendulum according to Galileo. Note that it was mistaken on the second not having the precision wanted in its measurements to observe the dependence of the period according to the amplitude.

Applications

The first definition of the second was the half-period of a one meter length pendulum. (The second is not mechanically defined any more, but by a well defined number of periods of a transition in an atom from Césium).

That is to say L₀ the length of a pendulum having one period of T₀ = 2 seconds:

• a pendulum of L₀/4 will have one period of 1s
• a pendulum length L will have one period such as: $\backslash \; frac\; \{T\}\; \{T\_0\}\; =\; \backslash \; sqrt\; \{\backslash \; frac\; \{L\}\; \{L\_0\}\}$

By raising squared the report/ratio of the periods, one obtains the report/ratio lengths of the pendulums.

The Clock S inhabitants of Franche-Compt3e, or pendulums (female name), use a beam which is a heavy pendulum. One regulates the period of oscillation while making move a mass along the beam.

See too

• the simple Pendulum and its developments
• the heavy Pendule made up
• the Pendule of Foucault is a heavy pendulum in a revolving reference mark.
• the Cycloïde is the curve which the pendulum should follow so that the period of its oscillations does not depend on the amplitude. (cf cycloidal Pendulum)
• History of the gravitation

External bonds

• simple Pendulum deadened (simulation)
• Pendulum made up
• simple Pendulum (simulation)
• skeletal Simulation (animation Flash)
• Period with all amplitudes (animation Flash)
• Study of the tension of the wire (animation Flash)

Random links:

85 (number) | Battle To fight | Joseph J. Lilley | Perpète-the-geese | List stars of Cancer | Parc_national_d'îles_de_Smith

© 2007-2008 speedlook.com; article text available under the terms of GFDL, from fr.wikipedia.org