hw1.solns - Physics 318 Homework 1 Solutions February 5,...

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Unformatted text preview: Physics 318 Homework 1 Solutions February 5, 2008 Problem 1 We choose the z axis to be in the vertical direction. The force on the particle is given by F = sign ( z ) z 2 mg, (1) where > 0 is the constant of proportionality. The factor of sign( z ) in the first term makes the friction force be in the direction opposite to the motion. For the way up, z > 0 and Newtons second law gives m z = z 2 mg (2) m dv dt = v 2 mg = m dv dz dz dt = m 2 d ( v 2 ) dz (3) dv 2 dz = 2 m v 2 2 g (4) v ( z ) = radicalbigg Ae 2 z/m gm , (5) where A is a constant of integration. We use the initial condition v = v at z = 0 to solve for A , giving v ( z ) = radicalbigg ( v 2 + gm ) e 2 z/m gm . (6) We can find the maximum height reached z max by setting v ( z max ) = 0. This gives e 2 z max /m = v 2 t v 2 + v 2 t , (7) where v t = radicalBig mg/ is the terminal velocity (found by setting F = 0). For the trip down the differential equation is m z = z 2 mg , and the same method gives the solution v ( z ) = radicalbigg Be 2 z/m + gm (8) 1 for some constant B . It must be that B = v 4 t /A so that v ( z max ) = 0. Now we can solve for v ( z = 0): v (0) = radicalBig B + v 2 t (9) = v | v t | radicalBig v 2 + v 2 t . (10) [Note there was a typo in the homework set, the absolute value sign was omitted from the final answer (10). The answer given in the homework set is correct if v t is interpreted to be terminal speed instead of terminal velocity.] Problem 2 Spherical polar coordinates are defined by the equations x = r sin( ) cos( ) , y = r sin( ) sin( ) , z = r cos( ) (11) a.) Using the chain rule, we can write d r = d ( r e r ) = dr e r + r e r r dr + r e r d + r e r d (12) To proceed, we will need to know e r for each { r, , } . This is best done with an illustration z x x e r e e r e r r + r r r + r z y x r r + r sin( ) e e r e r + e r r e r r = 0 e r = e e r = sin( ) e (13) Thus using Eqn. (13) we find d r = dr e r + rd e...
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This homework help was uploaded on 02/24/2008 for the course PHYS 3318 taught by Professor Flanagan during the Spring '08 term at Cornell University (Engineering School).

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hw1.solns - Physics 318 Homework 1 Solutions February 5,...

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