Under what condition does a particle experiencing electric force perform simple harmonic motion?
1 Answer
The equation describing Simple Harmonic Motion contains complete description of the motion, and is represented as
y(t)=Asinomegaty(t)=Asinωt ....(1)
whereyy is position of particle in theyy -axis,AA is maximum amplitudeyy can achieve andomegaω is angular frequency of the particle executing SHM and is a system constant.
It acceleration is given as
|veca|=ddoty(t)=-omega^2Asin omegat .....(2)
Now a particle of mass
vecF_e=qvecE
From Newton's second Law of motion, acceleration of the particle is given as
veca=q/mvecE ......(3)
For the charge particle to execute SHM equation (2) and (3) must match. Mass of the particle being constant, we have
q|vecE|=-ksin omegat .....(4)
wherek is an other constant dependent of system parameters.
Equation (4) lays down conditions for the particle to execute SHM.
Needless to say that the particle cannot execute SHM if both charge and electric field are constants.