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» <html><body><p></p>Solution :Displacement at time t in SHM, <br/> `x = a cos (omega t+phi)"""…….."(1)` <br/> Where `phi` = initial phase, `omega` = angular <a href="https://interviewquestions.tuteehub.com/tag/frequency-465761" style="font-weight:bold;" target="_blank" title="Click to know more about FREQUENCY">FREQUENCY</a> and a= amplitude <br/> <a href="https://interviewquestions.tuteehub.com/tag/differentiating-953151" style="font-weight:bold;" target="_blank" title="Click to know more about DIFFERENTIATING">DIFFERENTIATING</a> equation (1) w.r.t. .t., <br/> `<a href="https://interviewquestions.tuteehub.com/tag/v-722631" style="font-weight:bold;" target="_blank" title="Click to know more about V">V</a>= (dx)/(dt)` <br/> `=(d)/(dt) [a cos (omega t+phi)]` <br/> `therefore v = -a omega sin (omega t+ phi)""".........."(2)` <br/> When `t=0, x =x_(0)" and "(dx)/(dt)= -v_(0)` <br/> From equation (1), <br/> `x_(0) = a cos (omega xx 0+phi)` <br/> `x_(0) = a cos phi ""........."(<a href="https://interviewquestions.tuteehub.com/tag/3-301577" style="font-weight:bold;" target="_blank" title="Click to know more about 3">3</a>)` <br/> and from equation (2), <br/> `-v_(0) = a omega sin (0+phi)""[t=0]` <br/> `therefore v_(0) = a omega sin phi` <br/> `therefore (v_0)/(omega) = a sin phi"""........."(4)` <br/> Squaring and <a href="https://interviewquestions.tuteehub.com/tag/adding-2399902" style="font-weight:bold;" target="_blank" title="Click to know more about ADDING">ADDING</a> equation (3) and (4), <br/> `x_(0)^(2)+(v_(0)^(2))/(omega^2)= a^(2)cos^(2) phi+ a^(2) sin^(2) phi` <br/> `=a^(2) (cos^(2) phi + sin^(2) phi)` <br/> `therefore x_(0)^(2) +(v_(0)^(2))/(omega^2) = a^(2)""[therefore cos^(2) phi+ sin^(2) phi= 1]` <br/> `therefore a= sqrt(x_(0)^(2) +(v_(0)^(2))/(omega^2))`.</body></html>


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