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Unformatted text preview: oldhomewk 31 PAPAGEORGE, MATT Due: Apr 28 2008, 4:00 am 1 Question 1, chap 15, sect 1. part 1 of 2 10 points Two massless springs with spring constants 744 N / m and 5022 N / m are hung from a horizontal support. A block of mass 2 kg is suspended from the pair of springs, as shown. The acceleration of gravity is 9 . 8 m / s . 744 N / m 5022 N / m 2 kg When the block is in equilibrium, each spring is stretched an additional x . Then the block oscillates with an amplitude of 40 m, and it passes through its equilibrium point with a speed of 720 m / s. What is the angular velocity of this system? 1. = 4 rad / s 2. = 7 rad / s 3. = 16 rad / s 4. = 18 rad / s correct 5. = 8 rad / s 6. = 15 rad / s 7. = 13 rad / s 8. = 10 rad / s 9. = 5 rad / s 10. = 20 rad / s Explanation: Let m = 2 kg , A = 40 m , v = 720 m / s , k 1 = 744 N / m , and k 2 = 5022 N / m . Basic Concepts: Hookes law F = k x = ma = d 2 x dt 2 d 2 x dt 2 + k m x = 0 , (1) whose integral form has a sine function x ( t ) = A sin( t + ) , (2) v ( t ) dx dt v ( t ) = A cos( t + ) , and (3) a ( t ) dv dt a ( t ) = 2 A sin( t + ) , where (4) = radicalbigg k m . (5) The angular velocity is the square root of the coefficient of x in Eq. 1. The frequency of oscillation f versus angu lar frequency is f 2 . (6) k 1 k 2 m Consider the forces from a springs point of view. The oscillating mass exerts the same force, F (at some instant in time) on the springs k 1 and k 2 , F = k 1 x 1 x 1 = F k 1 F = k 2 x 2 x 2 = F k 2 . oldhomewk 31 PAPAGEORGE, MATT Due: Apr 28 2008, 4:00 am 2 Now consider the effective spring constant, k series , where x = x 1 + x 2 , therefore k series = F x = F x 1 + x 2 = F F k 1 + F k 2 = 1 1 k 1 + 1 k 2 = k 1 k 2 k 1 + k 2 . (7) = (744 N / m) (5022 N / m) (744 N / m) + (5022 N / m) = 648 N / m . Solution: The question presents the springs in series, Eq. 6 and Eq. 7, therefore series = radicaltp radicalvertex radicalvertex radicalbt k 1 k 2 m bracketleftBig k 1 + k 2 bracketrightBig (8) = radicaltp radicalvertex radicalvertex radicalbt (744 N / m) (5022 N / m) (2 kg) bracketleftBig (744 N / m) + (5022 N / m) bracketrightBig = radicalBigg (648 N / m) (2 kg) = 18 rad / s , and f = series 2 (9) = (18 rad / s) 2 = 2 . 86479 cycles / s , and T = 1 f = 1 (2 . 86479 cycles / s) = 0 . 349066 s ....
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 Spring '09
 KLEINMAN
 Physics, Mass

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