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Obtain the differential equation of forced oscillation.

Answer» <html><body><p></p>Solution :The force act on <a href="https://interviewquestions.tuteehub.com/tag/oscillator-1140010" style="font-weight:bold;" target="_blank" title="Click to know more about OSCILLATOR">OSCILLATOR</a> to sustain the oscillations external periodic force <br/> `F(t)= F_(0) cos omega_(d) t` <br/> Here, F(t) is the amplitude `F_(0)` of external periodic force dependently on time. <br/> Three forces act on oscillator <br/> (1) Restoring force `""F(x) = -<a href="https://interviewquestions.tuteehub.com/tag/kx-1064991" style="font-weight:bold;" target="_blank" title="Click to know more about KX">KX</a>(t)` <br/> (2) Resistive force `""F_(S) = -bv(t)` <br/> External periodic force `F_(d) = F_(0) cos omega_(d)t` <br/> Where `F_(0) cos omega_(d) t` is external periodic force and `omega_(d)= (2pi)/(T) implies T= (2pi)/(omega_d)` <br/> Period of external periodic force depends on the <a href="https://interviewquestions.tuteehub.com/tag/angular-11524" style="font-weight:bold;" target="_blank" title="Click to know more about ANGULAR">ANGULAR</a> frequency of drivingforce. <br/> From Newton.s second law of motion, if the net force is `F(t)= ma(t)` then <br/> `F(t) = F(x) +F_(S)+F_(d)"""......."(1)` <br/> `ma(t) = -kx(t) -bv(t) +F_(0)cos omega_(d)t` <br/> but `a(t) = (d^(2)x(t))/(<a href="https://interviewquestions.tuteehub.com/tag/dt-960413" style="font-weight:bold;" target="_blank" title="Click to know more about DT">DT</a>^2), v(t) = (dx(t))/(dt)` <br/> From equation (1) <br/> `m(d^(2)x(t))/(dt^2)= -b(dx(t))/(dt) -kx(t) -bv(t) +F_(0)cos omega_(d)t` <br/> `therefore m(d^(2)x(t))/(dt^2)+ b(dx(t))/(dt) +kx(t) = F_(0)cos omega_(d)t` <br/> `therefore (d^(2)x(t))/(dt^2)+(b)/(m)*(dx(t))/(dt) +(k)/(m) x(t)= (F_0)/(m) cos omega_(d)t` <br/> is the <a href="https://interviewquestions.tuteehub.com/tag/differential-953008" style="font-weight:bold;" target="_blank" title="Click to know more about DIFFERENTIAL">DIFFERENTIAL</a> equation of forced oscillation. <br/> m is the mass of oscillator. <br/> This oscillator initially oscillates with its natural frequency `omega`. When we apply the external periodic force, the oscillations with the natural frequency die out, and then the body oscillates with the angular frequency of the external force. <br/> Its displacement, after the natural oscillations die out, is given by `x(t) = A cos (omega_(d)t + phi)` where t is the time measured from the moment when the periodic force is applied. <br/> The amplitude of forced oscillation, `A= (F_0)/([m^(2)(omega^(2)-omega_(d)^(2))^(2)+(omega_(d)^(2)b^(2))^(1/2)])` and phase `phi = tan^(-1) ((v_0)/(omega_(d)x_(0)))` <br/> Where `v_(0)` is the velocity of the particle at time `t=0" and "x_(0)` is the displacement of the particle at time t=0.</body></html>


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