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)` and note that the initial velocity is negative.]

Explanation

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))`.