Pendulum

A gravity pendulum is a weight on the end of a rigid rod (or a string/rope), which, when given an initial push, will swing back and forth under the influence of gravity over its central (lowest) point. A torsion pendulum consists of a body suspended by a fine wire or elastic fiber in such a way that it executes rotational oscillations as the suspending wire or fiber twists and untwists. Another variety of a torsion pendulum is a fixed elastic coil connected to a rod-like object; once moved off its resting position, the coil will set the rod into an oscillatory motion. Galileo Galilei discovered that pendula exhibit regular periodic motion, a feature which he correctly speculated would make them useful for timekeeping devices such as clocks. This has led to their frequent metaphorical use as a way of describing the passage of time, or the experience of it. The mass of the weight on the pendulum does not affect the outcome, just as it does not affect the outcome of dropped objects. For smaller displacements, the movement of an ideal pendulum can be described mathematically as simple harmonic motion, as the change in potential energy at the bottom of a circular arc is nearly proportional to the square of the displacement. Real pendulums do not have infinitesimal displacements, so their behaviour is actually of a non-linear kind. Real pendulums will also lose energy as they swing, and so their motion will be damped, with the size of the oscillation exponential decay with time. The case of a pendulum with a point mass swinging on a massless rigid rod of length ''l'' is the simple gravity pendulum (simple pendulum). If $\backslash theta$ is the angle between the rod and the vertical, the acceleration is given by $-g\backslash cdot\backslash sin\backslash theta$ and is equal to the angular acceleration multiplied by the length of the rod. So we have the following differential equation: :$\backslash ell\backslash cdot\backslash frac\{d^2\backslash theta\}\{dt^2\}=-g\backslash cdot\; \backslash sin\backslash theta$ When the amplitude of the swing is small, $\backslash sin\backslash theta\; \backslash approx\; \backslash theta$. If the pendulum is initially still at an angle $\backslash theta\_0$ which is also the maximum angle, the function which solves the differential equation above is the following harmonic law: :$\backslash theta\; =\; \backslash theta\_0\backslash cdot\; \backslash cos\backslash left(\backslash sqrt\{\backslash frac\{g\}\{\backslash ell\}\}\backslash cdot\; t\backslash right)$ The term $\backslash sqrt\{\backslash frac\{g\}\{\backslash ell\}\}$ is a Simple_harmonic_motion, which is equal to $\backslash frac\{2\backslash pi\}\{T\_0\}$, where $T\_0$ is the period of a complete oscillation (outward and return). Since $\backslash omega\; =\; \backslash sqrt\{\backslash frac\{g\}\{\backslash ell\}\}\; =\; \backslash frac\{2\backslash pi\}\{T\_0\}$, the period of a complete oscillation can be easily found, and it is given by Christiaan_Huygens's law: :$T\_0\; =\; 2\backslash pi\backslash sqrt\{\backslash frac\{\backslash ell\}\{g\}\}$ When the amplitude of the swing is no more negligible, the differential equation cannot be simplified anymore, and the period ''T'' must be corrected with the following expression, which is the result of a complete elliptic integral of the first kind: :$T\; =\; T\_0\; \backslash left(\; 1+\; \backslash left(\; \backslash frac\{1\}\{2\}\; \backslash right)^2\; \backslash sin^2\backslash left(\backslash frac\{\backslash theta\_0\}\{2\}\backslash right)\; +\; \backslash left(\; \backslash frac\{1\; \backslash cdot\; 3\}\{2\; \backslash cdot\; 4\}\; \backslash right)^2\; \backslash sin^4\backslash left(\backslash frac\{\backslash theta\_0\}\{2\}\backslash right)\; +\; \backslash left(\; \backslash frac\; \{1\; \backslash cdot\; 3\; \backslash cdot\; 5\}\{2\; \backslash cdot\; 4\; \backslash cdot\; 6\}\; \backslash right)^2\; \backslash sin^6\backslash left(\backslash frac\{\backslash theta\_0\}\{2\}\backslash right)\; +\; \backslash cdots\; \backslash right)$ where $\backslash theta\_0$ is the semi-amplitude of the oscillation, that is the maximum angle between the rod of the pendulum and the vertical. For a swing of $180^\backslash circ$ the bob is balanced over its pivot point and so $T=\backslash infty$. Keep in mind the pendulum is made of a rigid rod. The following table shows the correction of the period ''T'' for wide amplitudes. {| border="1" cellpadding="2" |$\backslash theta\_0$ |$T/T\_0$ |- |0 |1.0000 |- |15 |1.0043 |- |30 |1.0174 |- |45 |1.0399 |- |60 |1.0732 |- |75 |1.1189 |- |90 |1.1803 |- |105 |1.2622 |- |120 |1.3729 |- |135 |1.5279 |- |150 |1.7622 |- |165 |2.1854 |- |180 |$+\backslash infty$ |- |} Notice that the equation for ''T'' is independent of the bob's mass. This can be considered a consequence of the equivalence of gravitational mass and inertial mass: a heavier bob has a stronger restoring force, but on the other hand has more inertia and the two effects cancel. As first explained by M. Schuler in his classic 1923 paper, a pendulum whose period exactly equals the orbital period of a hypothetical satellite orbiting just above the surface of the earth (about 84 minutes) will tend to remain pointing at the center of the earth when its support is suddenly displaced. This is the basic principle of Schuler tuning that must be included in the design of any inertial guidance system that will be operated near the earth, such as in ships and aircraft. The presence of ''g'' in the equation means that the pendulum frequency is different at different places on earth. So for example if you have an accurate pendulum clock in Glasgow (''g'' = 9.815 63 m/s²) and you take it to Cairo (''g'' = 9.793 17 m/s²), you must shorten the pendulum by 0.23%. Two coupled pendulums form a double pendulum. ==Torsion pendulums== If ''I'' is the moment of inertia of a body with respect to its axis of oscillation, and if ''K'' is the torsion coefficient of the fiber (torque required to twist it through an angle of one radian), then the period of oscillation of a torsion pendulum is given by :$T\; =\; 2\; \backslash pi\; \backslash sqrt\{\backslash frac\{I\}\{K\}\}$ Both ''I'' and ''K'' may have to be determined by experiment. This can be done by measuring the period ''T'' and then adding to the suspended body another body of known moment of inertia ''I''', giving a new period of oscillation ''T''' :$T\text{'}\; =\; 2\backslash pi\; \backslash sqrt\{\backslash frac\{I+I\text{'}\}\{K\}\}$ and then solving the two equations to get :$K\; =\; \backslash frac\{4\backslash pi^2I\text{'}\}\{T\text{'}^2\; -\; T^2\}$ :$I\; =\; \backslash frac\{T^2I\text{'}\}\{T\text{'}^2\; -\; T^2\}$ The oscillating balance wheel of a watch is in effect a torsion pendulum, with the suspending fiber replaced by hairspring and pivots. The watch is regulated, first roughly by adjusting ''I'' (the purpose of the screws set radially into the rim of the wheel) and then more accurately by changing the free length of the hairspring and hence the torsion coefficient ''K''. ==Damped pendulum== The pendulum equation does not take into account the effects of friction and dissipation. While these effects can be very complicated to model, a good approximation is to add a term prortional to the velocity: :$\backslash ell\backslash cdot\backslash frac\{d^2\backslash theta\}\{dt^2\}=-g\; \backslash cdot\; \backslash sin\backslash theta\; -\; \backslash gamma\; \backslash cdot\; \backslash frac\{d\backslash theta\}\{dt\}$ The positive constant γ is the viscous damping parameter. A system described by this equation is called a damped pendulum. ==Pendulums for Divination/Dowsing== Pendulums (usually a crystal suspended on a chain) are often used for divination and dowsing. The user will first determine which direction (left-right, up-down) determines "yes" and which "no," before proceeding to as the pendulum specific questions (always yes-no questions). ==See also== * Pendulum clock * Simple harmonic motion * Foucault pendulum * Spherical pendulum * Double pendulum * Kater's pendulum * Harmonograph * Metronome Classical mechanics ms:Bandul

Pendulum

I changed the acceleration formula from g sin(theta) to -g sin(theta). To minimize miscommunication, here's what I was thinking in detail: 1. Wikipedia defines acceleration as a vector. So unlike speed, acceleration can be negative. 2. The acceleration has to mean acceleration within the pendulum's arc, that is, sometimes the vector points up a little and sometimes down a little. 3. For definiteness, imagine theta=5 degrees, which is conventionally on the right side. The pendulum will be accelerated to the left side. So it should be negative. 4. Now that the acceleration is -g sin(theta), it becomes part of an English sentence that matches the formula that comes afterwards, where the g sin(theta) correctly has a minus sign. User:Art LaPella 23:30, Aug 26, 2004 (UTC) ==Things that affect the period of a pendulum== Can someone add a little section about what affects the period of a pendulum? Like mass (which doesn't affect it) and amplitude and length of the string (which both do affect it). ~User:Pie4all88 And info about horizontal pendulum and a link "horizontal pendulum" - redirect? Thanks very much. --User:Eleassar777 17:40, 18 Mar 2005 (UTC)

---

See other meanings of words starting from letter:

P

PA | PB | PC | PD | PE | PF | PG | PH | PI | PJ | PK | PL | PM | PN | PO | PR | PS | PT | PU | PW | PX | PY | PZ |

Words begining with Pendulum:

Pendulum
Pendulum
Pendulums
Pendulum_(album)
Pendulum_(album)
Pendulum_(Broadcast_album)
Pendulum_(disambiguation)
Pendulum_clock
Pendulum_clock
Pendulum_Governor
Pendulum_governor
Pendulum_Instruments
Pendulum_Music

These materials are based on Wikipedia and licensed under the GNU FDL