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The most accurate way to model a pendulum (without air resistance) is as a differential equation in terms of the angle it makes with the vertical, θ, the length of the pendulum, l, and the acceleration due to gravity, g.
d²θ/dt² = -g*sin(θ)/l
There is no easy way to integrate this to get θ as a function of time, but if you assume θ is small, you can use the small angle approximation sin(θ)~θ which makes the equation
d²θ/dt² = -g*θ/l
Which can then be integrated to get the solution
θ(t)=θmax*sin(t*√(g/l))
Using this equation, you can easily derive that the period of the pendulum (time required to go through one full cycle) would be
T=2π*√(l/g)
If air resistance is also accounted for in the original differential equation, the exact equation will be much harder to derive, but in general will involve an exponential decay of a sin function.
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The angle between the string of a pendulum at its equilibrium position and at its maximum displacement is the pendulums?
amplitude
How is a pendulum used in the world today?
Pendulums are often used in clocks to power the gears that move the hands. However, most clocks built today often have pendulums only for show, as those types of clocks are usually inaccurate and require a lot of winding.
What is the period of the pendulum?
The period of a simple pendulum, with very short swings, is approximated byT = 2 pi (L/G)(0.5)More complex pendulums, or pendulums with greater than insignificant swing, have more complex equations, usually to correct for circular error.
How is inertia related to pendulums?
Inertia is a massive object's resistance to change. It should be obvious then that mass is directly proportional to how long the pendulum swings before coming to rest, since more mass means harder to stop. Mathematically, you'll find this dependence on mass in the damping equations of pendulums.
Does a short or a long pendulum have a longer period?
The time of swing of a pendulum is T = 2π √ (l/g) where l is the length of the pendulum. As T ∝√l (Time is directly proportional to the square root of l) then, the longer the pendulum, the greater is the period. Therefore longer pendulums have longer periods than shorter pendulums.
