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sol05 - MIT OpenCourseWare http/ocw.mit.edu 6.055J 2.038J...

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MIT OpenCourseWare http://ocw.mit.edu 6.055J / 2.038J The Art of Approximation in Science and Engineering Spring 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms .

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6.055J/2.038J (Spring 2008) Solution set 5 Do the following warmups and problems. Due in class on Friday, 18 April 2008 . Open universe: Collaboration, notes, and other sources of information are encouraged. However, avoid looking up answers until you solve the problem (or have tried hard). That policy helps you learn the most from the problems. Bring a photocopy to class on the due date , trade it for a solution set, and figure out or ask me about any confusing points. Your work will be graded lightly: P (made a reasonable effort), D (did not make a reasonable effort), or F (did not turn in). Warmups 1. Counting dimensionless groups How many independent dimensionless groups are there in the following sets of variables: a. size of hydrogen including relativistic effects: e 2 / 4 π 0 , ~ , c , a 0 (Bohr radius) , m e (electron mass) . According to the Buckingham Pi theorem, five quantities composed of three independent di- mensions make two independent dimensionless groups. The question did not ask you to choose the dimensionless groups. But it is useful to see what they could be. The following pair is a reasonable choice for the two groups: ~ 2 e 2 / 4 π 0 Π 1 m e a 0 ( e 2 / 4 π 0 ) and Π 2 ~ c . The groups are often called Π variables, following a tradition started by Buckingham, author of the Buckingham Pi theorem. In the preceding choice of groups, the second group Π 2 is the fine-structure constant α . b. period of a spring–mass system in a gravitational field: T (period) , k (spring constant) , m , x 0 (amplitude) , g . Five quantities composed of three independent dimensions make two independent dimension- less groups. Here is a reasonable combination: kT 2 kx 0 Π 1 m and Π 2 mg . It turns out that the second group Π 2 does not affect the period T . However, dimensional analysis does not tell you that result; it has to be derived by physical thinking.
Z 2 Solution set 5 / 6.055J/2.038J: Art of approximation in science and engineering (Spring 2008) c. speed at which a free-falling object hits the ground: v , g , h (initial drop height) . Three quantities composed of two dimensions (length and time) produce one independent dimensionless group. A reasonable choice is 2 v . Π 1 gh That ratio – except for a factor of 2 – has a physical interpretation as the ratio of kinetic energy on impact to the potential energy at the start. d. [tricky!] weight W of an object: W , g , m . These three quantities are composed of three dimensions (mass, length, and time), so there should be zero dimensionless groups! However, there is at least one group: W / mg is dimen- sionless. What went wrong is that the three quantities are composed of two independent dimensions: mass M and acceleration LT 2 . So the Buckingham Pi theorem predicts one independent di- mensionless group. A reasonable choice for it is W / mg .

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