A mass attached to a spring is free to oscillate, with angular velocity omega in a horizontal plane without friction or dampong. It is pulled to a distance x_(0) and pushed towards the centre of with a velocity v_(0) at time t=0. Determine the amplitude of the resulting oscillations in terms of the parameters omega, x_(0) and v_(0).

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 dampong. It is pulled to a distance x_(0) and pushed towards the centre of with a velocity v_(0) at time t=0. Determine the amplitude of the resulting oscillations in terms of the parameters omega, x_(0) and v_(0).
Answer» Solution :SHM ALONG x-axis is `x=Acos (omegat+varphi_(0))` and velocity `V=-A omega SIN (omega t+varphi_(0))` when `t=0,x=x_(0)` and `v=v_(0)` `:.x_(0)=acos (varphi_(0))`………1 and `V_(0)=-omega sin varphi_(0)` (or) `-(V_(0))/(omega)=A sin varphi_(0)`............2 SQUARING and adding 1 and 2 `x_(0)^(2)+(V_(0)^(2))/(omega^(2))=A^(2)cos^(2)varphi_(0)+A^(2)sin^(2)varphi_(0)` `A^(2)=x_(0)^(2)+(V_(0)^(2))/(omega^(2))""A=sqrt(x_(0)^(2)+(V_(0)^(2))/(omega^(2)))`