Pendulum

A pendulum is a mechanical device consisting of a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting mechanical equilibrium position, it is subject to a restoring force due to gravity, which will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, or frequency, is commonly known as its "period". The period depends both on the length of the pendulum and, somewhat, on the amplitude (the width of the pendulum's swing). The SI unit of the period of a pendulum is the second (s). The regular motion of a pendulum lends itself to timekeeping. The pendulum clock was invented by Christiaan Huygens in 1656. It became the world's standard manner of timekeeping for several centuries. The Shortt-Synchronome clock achieved accuracy of about one second per year before it was superseded as a time standard by the quartz clock in the 1930s. Pendulums are also used in scientific instruments such as accelerometers and seismometers. Historically, pendulums were used as gravimeters to measure the acceleration of gravity in geo-physical surveys, and as a standard of length. The word pendulum is Neo-Latin, from the Latin , meaning .

A pendulum is a mechanical device consisting of a weight suspended from a pivot so that it can swing freely. When a pendulum is displaced sideways from its resting mechanical equilibrium position, it is subject to a restoring force due to gravity, which will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum's mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, or frequency, is commonly known as its "period". The period depends both on the length of the pendulum and, somewhat, on the amplitude (the width of the pendulum's swing). The SI unit of the period of a pendulum is the second (s). The regular motion of a pendulum lends itself to timekeeping. The pendulum clock was invented by Christiaan Huygens in 1656. It became the world's standard manner of timekeeping for several centuries. The Shortt-Synchronome clock achieved accuracy of about one second per year before it was superseded as a time standard by the quartz clock in the 1930s. Pendulums are also used in scientific instruments such as accelerometers and seismometers. Historically, pendulums were used as gravimeters to measure the acceleration of gravity in geo-physical surveys, and as a standard of length. The word pendulum is Neo-Latin, from the Latin , meaning . For larger amplitudes, the period increases gradually with amplitude so it is longer than given by equation (1). For example, at an amplitude of θ 0 = 0.4 radians (23°) it is 1% larger than given by (1). The period increases asymptotically (to infinity) as θ 0 approaches π radians (180°), because the value θ 0 = π is an unstable equilibrium point for the pendulum. The true period of an ideal simple gravity pendulum can be written in several different forms (see pendulum (mechanics)), one example being the infinite series: T = 2\pi \sqrt{\frac{L}{g}} \left[ \sum_{n=0}^\infty \left( \frac{\left(2n\right)!}{2^{2n} \left(n!\right)^2} \right)^2 \sin^{2n} \left(\frac{\theta_0}{2}\right) \right] = 2\pi \sqrt{\frac{L}{g}} \left( 1 + \frac{1}{16}\theta_0^2 + \frac{11}{3072}\theta_0^4 + \cdots \right) where \theta_0 is in radians. The difference between this true period and the period for small swings (1) above is called the circular error. In the case of a typical grandfather clock whose pendulum has a swing of 6° and thus an amplitude of 3° (0.05 radians), the difference between the true period and the small angle approximation (1) amounts to about 15 seconds per day. For small swings the pendulum approximates a harmonic oscillator, and its motion as a function of time, t, is approximately simple harmonic motion: \theta (t) = \theta_0 \cos \left( \frac{2\pi}{T}\, t +\varphi \right) where \varphi is a constant value, dependent on initial conditions. For real pendulums, the period varies slightly with factors such as the buoyancy and viscous resistance of the air, the mass of the string or rod, the size and shape of the bob and how it is attached to the string, and flexibility and stretching of the string. In precision applications, corrections for these factors may need to be applied to eq. (1) to give the period accurately. A damped, driven pendulum is a chaotic system. \ell^\mathrm{eq} = \frac{I_O}{mr_\mathrm{CM}} where I_O is the moment of inertia of the pendulum about the pivot point O , m is the total mass of the pendulum, and r_\mathrm{CM} is the distance between the pivot point and the center of mass. Substituting this expression in (1) above, the period T of a compound pendulum is given by T = 2\pi \sqrt\frac{I_O}{mgr_\mathrm{CM}} for sufficiently small oscillations. For example, a rigid uniform rod of length \ell pivoted about one end has moment of inertia I_O = \frac{1}{3}m\ell^2 . The center of mass is located at the center of the rod, so r_\mathrm{CM} = \frac{1}{2}\ell Substituting these values into the above equation gives T = 2\pi\sqrt{\frac{\frac{2}{3}\ell}{g}} . This shows that a rigid rod pendulum has the same period as a simple pendulum of two-thirds its length. Christiaan Huygens proved in 1673 that the pivot point and the center of oscillation are interchangeable. This means if any pendulum is turned upside down and swung from a pivot located at its previous center of oscillation, it will have the same period as before and the new center of oscillation will be at the old pivot point. In 1817 Henry Kater used this idea to produce a type of reversible pendulum, now known as a Kater pendulum, for improved measurements of the acceleration due to gravity. Pendulums in clocks (see example at right) are usually made of a weight or bob (b) suspended by a rod of wood or metal (a) . To reduce air resistance (which accounts for most of the energy loss in precision clocks) the bob is traditionally a smooth disk with a lens-shaped cross section, although in antique clocks it often had carvings or decorations specific to the type of clock. In quality clocks the bob is made as heavy as the suspension can support and the movement can drive, since this improves the regulation of the clock (see Accuracy below). A common weight for seconds pendulum bobs is . Instead of hanging from a pivot, clock pendulums are usually supported by a short straight spring (d) of flexible metal ribbon. This avoids the friction and 'play' caused by a pivot, and the slight bending force of the spring merely adds to the pendulum's restoring force. The highest precision clocks have pivots of 'knife' blades resting on agate plates. The impulses to keep the pendulum swinging are provided by an arm hanging behind the pendulum called the crutch, (e) , which ends in a fork, (f) whose prongs embrace the pendulum rod. The crutch is pushed back and forth by the clock's escapement, (g,h) . Each time the pendulum swings through its centre position, it releases one tooth of the escape wheel (g) . The force of the clock's mainspring or a driving weight hanging from a pulley, transmitted through the clock's gear train, causes the wheel to turn, and a tooth presses against one of the pallets (h) , giving the pendulum a short push. The clock's wheels, geared to the escape wheel, move forward a fixed amount with each pendulum swing, advancing the clock's hands at a steady rate. The pendulum always has a means of adjusting the period, usually by an adjustment nut (c) under the bob which moves it up or down on the rod. Moving the bob up decreases the pendulum's length, causing the pendulum to swing faster and the clock to gain time. Some precision clocks have a small auxiliary adjustment weight on a threaded shaft on the bob, to allow finer adjustment. Some tower clocks and precision clocks use a tray attached near to the midpoint of the pendulum rod, to which small weights can be added or removed. This effectively shifts the centre of oscillation and allows the rate to be adjusted without stopping the clock. The pendulum must be suspended from a rigid support. During operation, any elasticity will allow tiny imperceptible swaying motions of the support, which disturbs the clock's period, resulting in error. Pendulum clocks should be attached firmly to a sturdy wall. The most common pendulum length in quality clocks, which is always used in grandfather clocks, is the seconds pendulum, about long. In mantel clocks, half-second pendulums, long, or shorter, are used. Only a few large tower clocks use longer pendulums, the 1.5 second pendulum, long, or occasionally the two-second pendulum, which is used in Big Ben. The Q is related to how long it takes for the oscillations of an oscillator to die out. The Q of a pendulum can be measured by counting the number of oscillations it takes for the amplitude of the pendulum's swing to decay to 1/e = 36.8% of its initial swing, and multiplying by π. In a clock, the pendulum must receive pushes from the clock's movement to keep it swinging, to replace the energy the pendulum loses to friction. These pushes, applied by a mechanism called the escapement, are the main source of disturbance to the pendulum's motion. The Q is equal to 2π times the energy stored in the pendulum, divided by the energy lost to friction during each oscillation period, which is the same as the energy added by the escapement each period. It can be seen that the smaller the fraction of the pendulum's energy that is lost to friction, the less energy needs to be added, the less the disturbance from the escapement, the more 'independent' the pendulum is of the clock's mechanism, and the more constant its period is. The Q of a pendulum is given by: Q = \frac{M \omega}{\Gamma} where M is the mass of the bob, is the pendulum's radian frequency of oscillation, and Γ is the frictional damping force on the pendulum per unit velocity. ω is fixed by the pendulum's period, and M is limited by the load capacity and rigidity of the suspension. So the Q of clock pendulums is increased by minimizing frictional losses (Γ). Precision pendulums are suspended on low friction pivots consisting of triangular shaped 'knife' edges resting on agate plates. Around 99% of the energy loss in a freeswinging pendulum is due to air friction, so mounting a pendulum in a vacuum tank can increase the Q, and thus the accuracy, by a factor of 100. The Q of pendulums ranges from several thousand in an ordinary clock to several hundred thousand for precision regulator pendulums swinging in vacuum. A quality home pendulum clock might have a Q of 10,000 and an accuracy of 10 seconds per month. The most accurate commercially produced pendulum clock was the Shortt-Synchronome free pendulum clock, invented in 1921. Its Invar master pendulum swinging in a vacuum tank had a Q of 110,000 and an error rate of around a second per year. Their Q of 10 3 –10 5 is one reason why pendulums are more accurate timekeepers than the balance wheels in watches, with Q around 100–300, but less accurate than the quartz crystals in quartz clocks, with Q of 10 5 –10 6 . g \propto L. 1792: To define a pendulum standard of length for use with the new metric system, in 1792 Jean-Charles de Borda and Jean-Dominique Cassini made a precise measurement of the seconds pendulum at Paris. They used a -inch (14 mm) platinum ball suspended by a iron wire. Their main innovation was a technique called the "method of coincidences" which allowed the period of pendulums to be compared with great precision. (Bouguer had also used this method). The time interval Δt between the recurring instants when the two pendulums swung in synchronism was timed. From this the difference between the periods of the pendulums, T 1 and T 2 , could be calculated: \frac {1}{\Delta t} = \frac {1}{T_1} - \frac {1}{T_2} 1821: Francesco Carlini made pendulum observations on top of Mount Cenis, Italy, from which, using methods similar to Bouguer's, he calculated the density of the Earth. He compared his measurements to an estimate of the gravity at his location assuming the mountain wasn't there, calculated from previous nearby pendulum measurements at sea level. His measurements showed 'excess' gravity, which he allocated to the effect of the mountain. Modeling the mountain as a segment of a sphere in diameter and high, from rock samples he calculated its gravitational field, and estimated the density of the Earth at 4.39 times that of water. Later recalculations by others gave values of 4.77 and 4.95, illustrating the uncertainties in these geographical methods. Kater built a reversible pendulum (see drawing) consisting of a brass bar with two opposing pivots made of short triangular "knife" blades (a) near either end. It could be swung from either pivot, with the knife blades supported on agate plates. Rather than make one pivot adjustable, he attached the pivots a meter apart and instead adjusted the periods with a moveable weight on the pendulum rod (b,c) . In operation, the pendulum is hung in front of a precision clock, and the period timed, then turned upside down and the period timed again. The weight is adjusted with the adjustment screw until the periods are equal. Then putting this period and the distance between the pivots into equation (1) gives the gravitational acceleration g very accurately. Kater timed the swing of his pendulum using the "method of coincidences" and measured the distance between the two pivots with a micrometer. After applying corrections for the finite amplitude of swing, the buoyancy of the bob, the barometric pressure and altitude, and temperature, he obtained a value of 39.13929 inches for the seconds pendulum at London, in vacuum, at sea level, at 62 °F. The largest variation from the mean of his 12 observations was 0.00028 in. representing a precision of gravity measurement of 7×10 −6 (7 mGal or 70 μm/s 2 ). Kater's measurement was used as Britain's official standard of length (see below) from 1824 to 1855. Reversible pendulums (known technically as "convertible" pendulums) employing Kater's principle were used for absolute gravity measurements into the 1930s. ==Further reading== G. L. Baker and J. A. Blackburn (2009). The Pendulum: A Case Study in Physics (Oxford University Press). M. Gitterman (2010). The Chaotic Pendulum (World Scientific). Michael R. Matthews, Arthur Stinner, Colin F. Gauld (2005) The Pendulum: Scientific, Historical, Philosophical and Educational Perspectives, Springer Schlomo Silbermann,(2014) "Pendulum Fundamental; The Path Of Nowhere" (Book) L. P. Pook (2011). Understanding Pendulums: A Brief Introduction (Springer).

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