When  Approximations  Fail    Purpose:  To  compare  the  theoretical  and  experimental  relationship  between  the  angular  speed  and  angle  from  which  the  pendulum  was  dropped.      Procedures:      

1. Set  up  Vernier  rotary  sensor  and  logger  pro.    2. Drop  the  pendulum  from  a  certain  angle  and  observe  its  amplitude  and  

angular  speed.  Amplitude  is  equivalent  to  the  angle  from  which  it  was  dropped.  Testing  smaller  angle  values  is  preferable.    

3. Experimentally  determine  the  point  in  which  the  angular  speed  seems  to  deviate  from  the  theoretical  formula,  given  by  the  formula,  

 𝜔 ≡ !!  

4. Remember  to  record  the  distance  from  the  pendulum’s  axis  of  rotation  to  the  mass  of  the  pendulum.  

 Pre-­‐lab  Questions:      

1. Calculate the expected frequency of oscillation for small angle oscillations of the pendulum. Assume that g is exactly known to be 9.81m/s2. 𝜔 = 3.50  rad/s  

 2. Based on the results of the fit. What is the observed amplitude of the pendulum

and theAssociated uncertainty (expressed in degrees)? What is the observed angular frequency and associated uncertainty (expressed in radians per second)?

Amplitude Value (including uncertainty): .4.516 +/-­‐  .012  degrees  Angular  Frequency  (including  uncertainty):  3.501  +/-­‐  .0  rad/s  

 3. Assume that the data presented in Figure 3.4 were collected using the same

pendulum as the one described in Exercise 1. How does the observed frequency compare to the frequency calculated in the previous exercise? Are the two frequencies compatible with one another?

Compatible. The uncertainty of exercise 1 is .02 rad /  s,  and  the  experimental  value  3.501  rad/s  fall  within  the  uncertainty  range.  As  a  result,  the  frequencies  are  compatible        ‘      

Data:    Distance  from  the  pendulum’s  axis  of  rotation  to  the  mass:    0.725  +/-­‐  0.0001    

   

   

3.664  

3.666  

3.668  

3.67  

3.672  

3.674  

3.676  

3.678  

3.68  

3.682  

0   0.05   0.1   0.15   0.2   0.25  

Angular  Frequency  (rad/sec)  

Amplitude  (rad)  

Angular  Frequency  plotted  against  Amplitude  

Error Propagation Formula: 𝛿𝜔 = !!

𝑔  𝐿!!!

Theoretical Angular Frequency: 𝜔 ≡ !!=  3.678+/-­‐  .0025  rad/s  

 Conclusion:           In  this  lab,  we  observed  the  variation  in  angular  frequency  depending  on  the  amplitude.  In  theory,  since  angle  is  not  part  of  the  equation  in  finding  angular  frequency,  we  can  assume  that  the  angular  frequency  will  remain  consistent  at  all  angles.  However,  as  our  experiment  proves,  the  angular  frequency  decreases  as  the  angle  increases.       In  this  lab,  we  experimentally  determined  the  point  at  which  we  observed  a  significant  change  in  the  angular  frequency.  This  occurred  when  we  increased  the  angle  from  approximately  .07  radians  to  .08  radians,  where  we  observed  an  angular  frequency  drop  from  3.679  to  3.670  rad/s.  The  trials  prior  to  this  point  did  not  demonstrate  a  significant  drop,  which  led  us  to  conclude  that  the  exact  point  which  the  angular  frequency  deviates  from  the  model  must  exist  between  .08  -­‐  .70  radians.     According  to  the  uncertainty  of  our  theoretical  angular  frequency  value,  3.678+/-­‐  .0025  rad/s,  we  should  expect  to  observe  a  significant  change  in  angular  frequency  when  the  angle  is  3.678  -­‐  .0025  =  3.6755  rad/s,  which  is  very  close  to  our  observed  value,  3.670  rad/s.  This  shows  that  our  data  is  very  accurate.     Furthermore,  we  observed  that  the  angular  frequency  continued  to  decrease  as  the  angle  was  increased,  which  does  not  obey  the  theoretical  model.       A  significant  random  error  in  this  experiment  was  the  inconsistent  path  of  the  pendulum,  which  often  went  off  course.  To  alleviate  this  error,  we  could  choose  

to  use  an  expensive  pendulum  that  stays  on  course  at  all  times,  or  use  a  creative  release-­‐mechanism  that  will  ensure  a  consistent  path.  One  systematic  error  in  this  lab  is  the  possible  stretch  on  the  string,  which  when  released  may  vary  the  length  at  which  the  pendulum  moves.  In  order  to  alleviate  this  error,  we  could  use  a  specialized  string  that  is  designed  not  to  stretch.