Problem Set 9

Physics 5200

Due 13 Apr. 1999

1. (3 points) Show that the ratio of two successive maxima in the displacement of a damped harmonic oscillator is constant. (Note: The maxima do not occur at the points of contact of the displacement curve with the curve Ae^-gt. You will need to find the local maxima of the diplacement function.)

   
   

2. (2 points) The frequency f_d of a damped harmonic oscillator is 100 Hz, and the ratio of the amplitude of two successive maxima is one half.
   1. What is the undamped frequency f₀ of this oscillator?
   2. What is the resonant frequency f_r?
      
      
3. (3 points) The amplitude of a damped harmonic oscillator drops to 1/e of its initial value after n complete cycles. Show that the ratio of the damped period of oscillation to the undamped period is given by:
   
   T_d/T_o = [1 + (4p²n²)^-1]^1/2 ~ 1 + 1/(8p²n²)
   
   where the approximation in the last expression is valid for large n. (See the formulas in Appendix D).
   
   

4. (2 points) In Section 4.3, an approximate expression is derived for the range of a projectile with linear air drag. For a projectile with initial speed v₀ at an angle a to the horizontal, the range over flat terrain is approximately:
   
   x_max = v₀² sin(2a)/g - 4v₀³g sin(2a) sin(a) / 3g² 
   
   where g is the coefficient of air drag. For a fixed initial speed, and fixed g, will the projectile travel further for an angle a greater than or less than 45°?