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Physics 1140 Laboratory name printed:_______________________________ Fall 2001 ID no. _____________________________ Homework Set #2 My lab section number: ________ This homework is due at 5:00 p.m., Wednesday, September 20 in your TAs mailbox in the lab. Please use these sheets of paper. Please round uncertainties in this problem set (all three pages) to one significant figure. 1. Chapter 3.7. Uncertainty in a function of one variable using derivatives. (4 pts) Given that t = 6.0 + 0.5 s. What are the values of the functions below and their uncertainties? a) z = t3 z = 63 = 216 dz/dt = 3 t2 = 108 z = (dz/dt) t = 108 x 0.5 = 54 rounds to 50 z + z = 216 + 54 or 220 + 50 this could be written as (2.2+0.5) x 102 but usually we use scientific notation only for numbers that are hard to write without it, like 1.6 x 1019. b) z = 1 / t2 z = t-2 dz/dt = -2t-3 = -2/t3 = -2/216 = -0.00926 z = 1 /36 = 0.0278 z = (dz/dt) t = -0.00926 x 0.5 = 0.0045 rounds to 0.005 z + z = 0.028 + 0.005 could be (2.8+0.5) x 10-2, but why bother. 1 2. More uncertainties. Find the uncertainty in the quantities below. (2pts) Use the step by step procedure, which for a) below, means applying the rules for differences to the difference in square bracket and then applying the rule for products to the product remaining. a) z = w / (y x) = (12 + 1) / [ (25 + 4) (10 + 3) ] s = y-x = 15 z = 12/15 = 0.80000 etc (z/z)2 =(1/12)2 + (5/15)2 = 0.118 z/z = 0.34 z = 0.34 x 0.8 = 0.27 rounds to 0.3 z + z = 0.8 + 0.3 b) z = x ex with x = 2.0+0.1 dz/dz = x ex + ex = (x+1) ex = 3e2 = 22.2 z = 2e2 = 14.78 z = 22.2 x 0.1 = 2.22 rounds to 2 and the 14.78 rounds to 15 z + z = 15 + 2 2 3. Uncertainty in trig functions: x = 35° + 5°. z = cos [3 (x 10°) ]. What is z? (2 pts) Hint: Convert degrees to radians and use radian measure on your calculator. The derivatives of the trig functions that are in the tables only "work" for radian measure. radians = degrees x (/180) x +10° = 45° ; times (/180) = /4 radians 3 (x 10°) = 3/4 = 2.356 z = cos (3/4) = -0.707 dz/dx = -3 sin (3/4) = -2.12 dx = 5 /180 = 0.087 radians, z = 0.087 x (-2.12) = -0.185 rounds to -0.2 + or - a negative number is still + or -. z + z = -0.7 + 0.2 4. Uncertainty in a function of two variables using derivatives: (2 pts) Evaluate the derivatives and write down the algebraic expression for F in terms of x and y: F(x,y) = y / (x+1). dF/dy = 1/(x+1) dF/dx = -y/(x+1)2 = dF2 = [(dF/dx)x]2 + [(dF/dy)y]2 = [(y x /(x+1)2]2 + [y /(x+1)]2 = { [ y x]2 + [(x+1)y]2 } /(x+1)4 F = square root of all this: { [ y x]2 + [(x+1)y]2 } /(x+1)4 F = square root of {[(y x /(x+1)2]2 + [y /(x+1)]2} 3