Question

1. What equation(s) describe(s) the total mechanical energy of a damped, driven oscillator at resonance?

E=12mv2

E=12mA2?20

E=14mA2?20

E=12kA2

E=12kx2

2. Choose which of the following statements are true concerning driven oscillations, resonance, and standing waves. a.resonance cannot occur if there is any damping in the system b. for a mass on a spring, if the mass increases the natural frequency decreases c. it is not possible to have a system where there is a node at one end of the vibration and an antinode at the other end d. for a longitudinal standing wave in a pipe, where particles are minimally displaced there is a pressure antinode e. if only a damping force is present, the amplitude of oscillation must decrease over time f. if there are no damping forces present, a standing wave does not require a constant input of energy

Answer 1

1. For a driven damped oscillator, the equation of displacement x(t) and corresponding velocity v(t) is given by

                                                     

                                                       

Here is the driver frequency and is the phase difference.

Therefore, the mechanical energy is given by

                                                        

At resonance, (driver frequency is close to natural frequency).

Therefore,                                        

2.

a) Driving is required only when damping is present. So, resonance is a phenomena associated with driven damped oscillator. Resonance can occur when damping is present.

Therefore, Statement is false.

b) Natural frequency is given by

So, if mass increases, natural frequency decreases provided the spring constant or its equivalent in a system is constant.

Therefore, Statement is true.

c) It is possible to have a pipe with one end closed and other end open. So, it is possible to have a node at one end and an antinode at the other.

Therefore, Statement is false.

d) Pressure and displacement are out of phase. So, a displacement node corresponds to a pressure antinode.

Therefore, Statement is true.

e) Damping causes energy loss. So, amplitude must decrease over time.

Therefore, Statement is true.

f) If no damping is present, there is no driving required to sustain a standing wave.

Therefore, Statement is true.