A mass attached to a spring is free to oscillate, with an 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 tmet t=0. Determine the amplitude of the resulting oscillations in terms of the parameter omega, x_(0), v_(0) (Hint : Start with the equation x=a cos (omega t+theta) and note that the initial velocity is negative.)

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

1.	A mass attached to a spring is free to oscillate, with an 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 tmet t=0. Determine the amplitude of the resulting oscillations in terms of the parameter omega, x_(0), v_(0) (Hint : Start with the equation x=a cos (omega t+theta) and note that the initial velocity is negative.)
Answer» Solution :Given `x=a cos (omegat+THETA)` velocity `v=(dx)/(dt)=-(a omega)sin(omegat+theta)` `"i.e."x_(0)=a cos theta` `"i.e."v_(0)=-(a omega)sin(0+theta)"……(1)"` `v_(0)=-omega sin(theta)` or `a sin theta=-(v_(0))/(omega)".........(2)"` Solving (1) and (2) and adding we get `a^(2)=x_(0)^(2)+(v_(0)^(2))/(omega^(2))` or `a=sqrt(x_(0)^(2)+((v_(0)^(2))/(omega^(2))))`