PHYS 210L Final, Fall 2007

Physics 210L Final exam January 12, 2008 Name:____________________________ Section number:___________________ Instructor’s Name:__________________ ID number:_________________________ DO NOT START THE EXAM BEFORE YOU ARE TOLD TO BEGIN Grading I II-1 II-2 II-3 TOTAL The duration of this exam is 60 minutes. No notes or books allowed. Scientific calculators are allowed All results should be given with the exact number of significant figures. _____________________________________________________________________ 1 I. (50%) In the “Standing Waves on a Stretched String” experiment, the string length is changed and the corresponding resonant frequencies are measured. The hanging mass is M=12Kg ± 6% The following results were obtained for the 2nd Harmonic standing waves. L(cm) f(KHz) 50 1.297 60 0.901 70 0.662 80 0.507 90 0.400 You may find the formulae on page 4 useful. a- Write down the necessary equation. (10%) b- Choose your variables such that you obtain a linear relationship between them. (10%) c- Use linear regression to find the slope of your line along with its error. (15%) 2 d- Determine from the slope the linear density of the string along with its error. (15%) 3 Linear Regression The method of least squares is used to fit a curve (find a theoretical equation) to a set of experimental data. First assume that a linear relation exists between y and x y = Ax + B (1) Substitution of x = xi will in general not give the value of yi. The “errors” will be ei = y – yi = Axi + B – yi (2) To determine the best straight line that fits the N, sets of data, A and B have to be chosen so that the sum of the squares of the “errors” is minimized. This means that the simultaneous equations, obtained by equating the partial derivatives of (y – yi)2 with respect to A and B to zero, should be solved. This condition leads then to the following results Δ − = ∑ ∑ ∑ i i i i y x x y N A ) ( (3) and Δ − = ∑ ∑ ∑ ∑ ) ( i i i i i x y x y x B 2 (4) where 2 2 (∑ ) ∑ − Δ = i i x x N (5) The correlation coefficient r provides an indicator of how good a fit the best straight line is. This coefficient is defined as ∑ ∑ ∑ − − − − = 2 2 ) ( ) ( ) )( ( y y x x y y x x r i i i i (6) For r = 0, the values of x and y are independent of one another and there is no linear correlation. The closer r is to +1 or to –1, the better the linear correlation is. Finally, the error in A is given by: Δ − = ∑ 2 2 2 i A e N N σ 4 II. Questions 1. (15%) In the “Michelson Interferometer” experiment, a- Explain why the interference fringes shown on the screen appear to move when we change the inclination of the glass plate placed between the beam splitter and the mirror. (8%) b- In this part of the experiment what variables were measured in order to determine the index of refraction of the glass plate? (7%) 5 2. (20%) In an attempt to determine the gravitational acceleration experimentally, the motion of an object down an inclined frictionless surface was studied. The time needed to cover a certain distance d was measured. The inclination angle is 1.0 % .The following results were obtained: = 17 ± o θ N t(seconds) d(cm) a( ) 0.198 6 0.262 10 0.286 12 0.321 14 mg θ a- Determine the average value of the acceleration down the incline along with its error. (10%) b- Derive the relationship between the acceleration a, the gravitational acceleration g, and the inclination angle θ .(5%) c- Find g along with its error. (5%) 6 7 3. What are the different parts of a spectrometer and what are the functions of each? (15%)