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Phys105_ps1_2009

Phys105_ps1_2009 - Physics 105 Problem Set 1 Due Thursday 3...

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Unformatted text preview: Physics 105 Problem Set 1 Due: Thursday, September 24, 2009, 3 PM Reading: K&K, chapter 1. Students who are interested in enrolling in Physics 105 will solve and hand in Problems 1-5. These will be graded and (except for Problem 6) will count towards your 105 grade. Students who are uncertain whether they will stay in Physics 105 should complete all the Physics 103 work as well. Students who switch to Physics 103 from Physics 105, but have not completed the Physics 103 work on time, will receive zero credit for missed work. Turn this in to the Undergraduate Physics Office in Jadwin 208 by 3:00 PM on Thursday. Please NEATLY write your name, the room (A09 or 103A) and the time (9 or 10 AM) of your class on your homework. Problem 1. Let ˆ a and ˆ b be unit vectors in the x- y plane making angles θ and φ with the x axis, respectively. (a) (K&K 1.7) Show that ˆ a = ˆ i cos θ + ˆ j sin θ and ˆ b = ˆ i cos φ + ˆ j sin φ , and using vector algebra show that cos( θ- φ ) = cos θ cos φ + sin θ sin φ. (Hint: use the dot product – see Kleppner and Kolenkow (K&K), pp. 5, 9, 10.) A unit vector in the x- y plane can be written ˆ a = ˆ i cos θ + ˆ j sin θ . You may also have seen a vector written by listing its components: ˆ a = (cos θ, sin θ ). This vector can also be written in the form of a 2 × 1 matrix, also known as a “column vector:” ˆ a = cos θ sin θ . This is more than just notational intricacy – vectors and matrices are related. Note: don’t worry if you haven’t seen this before, it’s supposed to be new! For a brief intro to matrices, go to the writeup called “Matrices and Matrix Multiplication”, which is posted on Blackboard in the “Notes” folder in Course Materials. We would be happy to discuss it further at office hours. (b) Show that when the column vector representing ˆ a is multiplied by the 2 × 2 matrix R ( α ) = cos α- sin α sin α cos α , using the standard rules of matrix multiplication, the result is a new column vector ˆ c ˆ c = R ( α )ˆ a where ˆ c is just ˆ a rotated by the angle α . See the writeup for the scoop on matrix multiplication, with examples. R is called a rotation matrix. The notion of a matrix as an operator that does something to a vector is widely used in physics, especially in quantum mechanics....
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Phys105_ps1_2009 - Physics 105 Problem Set 1 Due Thursday 3...

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