Is dfrac 1 omega omega 2

Equation for a harmonic oscillation

All vibrating systems are called Oscillators designated. Oscillators whose path-time function corresponds to a sine function are called harmonic oscillators.

Application example

What can you do with the oscillation equation?

With the oscillation equation we can calculate the deflection of a harmonic oscillator at any point in time t if the oscillation duration or frequency is known and for a known amplitude.

Depending on which of the variables ω, T or f is known, we select one of the three above-mentioned variants of the oscillation equation.

Sample task:

A harmonic oscillator oscillates with an oscillation period of 1.2 seconds. The maximum deflection is 12cm.

At time t = 0s the oscillator is in the rest position on the way up in the positive y-direction.

Question: Where is the oscillator at the following times

a) t = 0.6s

b) t = 1s

c) t = 1.5s?

Solution:

The following values ​​are given:

T = 1.2s

yMax = 12cm

We put into the oscillation equation for harmonic oscillations

enter the given values ​​and calculate the respective deflection

(Danger: Calculator on WHEEL to adjust!):

a) For t = 0.6s surrendered

The Sinusterm results in 0, so the value y = 0 is also obtained for the deflection.

So the oscillator is located in the rest position. That is also logical, because the time t = 0.6s corresponds exactly to half the period of oscillation.

b) For t = 1s surrendered

The Sinusterm now gives the value -0.866. Multiplied by the amplitude of 12cm you get the value for the deflection y = -10.39cm.

The oscillator is therefore at y = -10.39cm, that is 10.39cm below the resting positionbecause in the task "above" as positive y-direction was specified.

c) For t = 1.5s surrendered

The Sinusterm results in the value 1.

The deflection therefore corresponds to the amplitude: y = yMax.

The oscillator is at the maximum deflection12cm above the rest position, so in the upper one Turning point.

Note:

The deflection can be values ​​between yMax and -yMax accept. The sine term by which the amplitude is multiplied varies between 1 and -1.

Important:

The calculator must be set to RAD for all calculations, as the phase angle Radians is specified!