a=ωt

hello

I know that voltage and current change sinusoidally in alternate circuits

a sinusoidal property changes according to Asin(a)

but without any explanation, my notes say that a=ωt

what is that exactly and where it derives from?

thanks!

Reply 1

Phichi

11

Original post by studos

hello

I know that voltage and current change sinusoidally in alternate circuits

a sinusoidal property changes according to Asin(a)

but without any explanation, my notes say that a=ωt

what is that exactly and where it derives from?

thanks!

If you want to derive this in terms of where the

\omega

comes from. Consider

sin(bx)

.

We know that the period

T

of both sine and cosine is

2\pi

. Thus, when adding a constant, such as in the above example

b

, the period of the function changes. If we let

b=2

, so the function is

sin(2x)

, we can logically deduce that the period of the new function is

\pi

, it is halved from that of

sin(x)

. This can be seen as the factor of two is causing the wave to cycle twice as fast, hence its frequency is doubled. Due to the relation of time period to frequency (

T=\frac{1}{f}

), this causes the time period to be halved, hence,

\pi

.

Now going back and applying this logic to

sin(bx)

, gives us the period of the function to be

T=\frac{2\pi}{b}

and hence

b=\frac{2\pi}{T}

.

Mathematically

\omega

is defined as

\omega=\frac{2\pi}{T}

or

\omega=2\pi f

.

Thus making the function

sin(\omega x)

.

In your case, however, the voltage or current is changing with time, thus

sin(\omega t)

.

\omega

, in this case is the angular frequency. This can be easily comprehended as

\omega=2\pi f

(edited 9 years ago)

Reply 2

studos

OP

3

what is the real meaning of ω?
mathematically, it's just a factor of the sin value along with t
what is the effect of introducing a factor inside sin, what is the effect to the sinusoidal graph?
but what does a factor affecting the t, really mean? what is the physical phenomenon that we tried to express mathematically by introducing a factor together with t?
what is the real meaning of ω=2π/Τ ?
they say that ω is angular velocity
angular velocity as I understand it, means how fast an angle changes
the rate of change of the angle, expressed as sin of that angle, is expressed as ω = angular velocity?
as t flows, there is a change in sin(t), which change is sinusoidal
by introducing ωt, as ωt flows, how differently is sin(ωt) changed, depending on ω?

Reply 3

Phichi

11

Original post by studos

what is the real meaning of ω?
mathematically, it's just a factor of the sin value along with t
what is the effect of introducing a factor inside sin, what is the effect to the sinusoidal graph?
but what does a factor affecting the t, really mean? what is the physical phenomenon that we tried to express mathematically by introducing a factor together with t?
what is the real meaning of ω=2π/Τ ?
they say that ω is angular velocity
angular velocity as I understand it, means how fast an angle changes
the rate of change of the angle, expressed as sin of that angle, is expressed as ω = angular velocity?
as t flows, there is a change in sin(t), which change is sinusoidal
by introducing ωt, as ωt flows, how differently is sin(ωt) changed, depending on ω?

The real meaning of

\omega

in this case in angular frequency, and mathematically its not 'just a factor of the sin value along with t'.

The effect on the graph can be seen by my last post, as

\omega=\frac{2\pi}{T}

, increasing the value of

\omega

will cause the time period of the oscillation to become smaller, hence, a larger frequency. You can't model the alternating current say by just

sin(t)

, as you're always going to have the same frequency associated with it, which of course, will not be

2\pi

in practically every case.

A factor affecting 'the t' will affect the oscillation shown by the graph.

The real meaning of

\omega=\frac{2\pi}{T}

is the rate of rotation, in this case the rate of change of phase for your waveform, whether its current or voltage.

In this case,

\omega

is angular frequency (or speed), not angular velocity. It's the magnitude of the angular velocity

|{\boldsymbol{\omega}}|

Indeed angular velocity is the pseudovector that represents the rate of change of angular displacement.

sin(\omega t)

changes just as a normal

sin(t)

would assuming

\frac{d\omega}{dt}=0

. The

\omega

again affects the frequency of the waveform.

a=ωt

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