A mass attached to a spring is free to oscillate, with angular velocity omega, in a horizontal plane without friction or damping. It is pulled to a distance x_0 and pushed towards the centre with a velocity v_0 at time t=0. Determine the amplitude of the resulting oscillaters omega, x_(0)" and "v_(0). [Hint : Start with the equation x= a cos (omega t+theta) andnote that the initial velocity is negative.]

A mass attached to a spring is free to oscillate, with angular velocit

1.	A mass attached to a spring is free to oscillate, with angular velocity omega, in a horizontal plane without friction or damping. It is pulled to a distance x_0 and pushed towards the centre with a velocity v_0 at time t=0. Determine the amplitude of the resulting oscillaters omega, x_(0)" and "v_(0). [Hint : Start with the equation x= a cos (omega t+theta) andnote that the initial velocity is negative.]
Answer» Solution :Displacement at time t in SHM, `x = a cos (omega t+phi)"""…….."(1)` Where `phi` = initial phase, `omega` = angular FREQUENCY and a= amplitude DIFFERENTIATING equation (1) w.r.t. .t., `V= (dx)/(dt)` `=(d)/(dt) [a cos (omega t+phi)]` `therefore v = -a omega sin (omega t+ phi)""".........."(2)` When `t=0, x =x_(0)" and "(dx)/(dt)= -v_(0)` From equation (1), `x_(0) = a cos (omega xx 0+phi)` `x_(0) = a cos phi ""........."(3)` and from equation (2), `-v_(0) = a omega sin (0+phi)""[t=0]` `therefore v_(0) = a omega sin phi` `therefore (v_0)/(omega) = a sin phi"""........."(4)` Squaring and ADDING equation (3) and (4), `x_(0)^(2)+(v_(0)^(2))/(omega^2)= a^(2)cos^(2) phi+ a^(2) sin^(2) phi` `=a^(2) (cos^(2) phi + sin^(2) phi)` `therefore x_(0)^(2) +(v_(0)^(2))/(omega^2) = a^(2)""[therefore cos^(2) phi+ sin^(2) phi= 1]` `therefore a= sqrt(x_(0)^(2) +(v_(0)^(2))/(omega^2))`.