Question

Find the angular frequency and the amplitude of harmonic oscillations of a particle if at distances `x_(1)` and `x_(2)` from the equalibrium position its velocity equald `v_(1)` and `v_(2)` respectively.

Answer 1

From the Eqn, `v_(x)^(2)=omega^(2)(a^(2)-x^(2))` (see Eqn. 4 of `4.1 )`
`v_(x)^(2)=omega^(2)(a^(2)-x_(1)^(2))` and `v_(2)^(2)=omega^(2)(a^(2)-x_(2)^(2))`
Solvinig these Eqns simultaneously, we get
`omega=sqrt((v_(1)^(2)-v_2)^(2)//(x_(2)^(2)-x_(1)^(2))), a=sqrt((v_(1)x_(2)^(2)-v_(2)^(2)x_(1)^(2))//(v_(1)^(2)-v_(2)^(2)))`