simple pendulum

2010-01-12 11:16 pm
in a simple pendulum, it can be shown that
a= -(g/l)x
g is the graviational acceleration
l is the length of rope tiding the weight

compare the equation to that of a simple harmonic oscillation
a= -w^2 x

what does w in this situation mean? why?
更新1:

[w] in a spring SHM is the angular speed of the arbitrary cycle of the phasor what is the geometric meaning of [w] here, the angular frequency??

更新2:

http://hk.knowledge.yahoo.com/question/question?qid=7007080705365 angular acceleration = - (g/l) angular displacement Does the [w] you suggest mean d(theta)/dt ??

回答 (2)

2010-01-13 4:34 am
✔ 最佳答案
[w] in the equation a = -w^2.x indicates the [angular frequency] of the harmonic oscillator. The [angular frequency] is related to the [linear frequency] by the equation: w = 2.pi.f, where f is the linear frequency and pi is the mathematical constant 3.14159 .....

The general equation describing an harmonic oscillator can be written as,
x = A.sin(wt+p)
where A is the amplitude of oscillation, w is the angular frequency, t is the time, and p is the phase angle

The velocity v = dx/dt = w.A.cos(wt+p)
and acceleration a = dv/dt = -w^2.[Asin(wt+p)] = -w^2.x

[Be aware not to mix the angular frequency of a simple pendulum with the angular velocity of the pendulum. They are different physical quantities. The angular frequency of a simple harmonic oscillator, e.g. a simple pendulum, is a constant, but the angular velocity of the pendulum varies with the position (or time) of the pendulum.
In fact, the angular velocity of a simple pendulum is
= v/L = (w.A/L).cos(wt+p)
where L is the length of the pendulum
The max angular velocity = w.A/L

You can see from the equation that the angular velocity varies with time of oscillation, but the angular frequency is not.]


2010-01-13 6:50 am
Actually, according to what i have learnt, w indicates the number of complete oscillations in the time interval of 2(pi) seconds.


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