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Unformatted text preview: Husain, Zeena – Homework 1 – Due: Sep 4 2003, 4:00 am – Inst: H L Berk 1 This printout should have 20 questions, check that it is complete. Multiplechoice questions may continue on the next column or page: find all choices before making your selection. The due time is Central time. 001 (part 1 of 1) 10 points One cubic meter (1.0 m 3 ) of aluminum has a mass of 2700 kg, and a cubic meter of iron has a mass of 7860 kg. Find the radius of a solid aluminum sphere that has the same mass as a solid iron sphere of radius 3 . 82 cm. Correct answer: 5 . 45443 cm. Explanation: Let : m Al = 2700 kg , m Fe = 7860 kg , and r Fe = 3 . 82 cm . Basic Concept Density is ρ = m V Solution: Since the masses are the same, ρ Al V Al = ρ Fe V Fe ρ Al µ 4 3 π r 3 Al ¶ = ρ Fe µ 4 3 π r 3 Fe ¶ µ r Al r Fe ¶ 3 = ρ Fe ρ Al r Al = µ ρ Fe ρ Al ¶ 1 3 ( r Fe ) = µ 7860 kg 2700 kg ¶ 1 3 (3 . 82 cm) = 5 . 45443 cm 002 (part 1 of 2) 10 points This problem shows how dimensional analysis helps us check our work and sometimes even help us find a formula. A rope has a cross sec tion A = 9 m 2 and density ρ = 1710 kg / m 3 . The “linear” density of the rope μ , is de fined to be the mass per unit length, in the form μ = ρ x A y . Based on dimensional analysis, find the powers x and y . 1. x = 1 , y = 1 2. x = 1 , y = 2 3. x = 2 , y = 2 4. x = 2 , y = 1 5. x = 1 , y = 1 6. x = 1 , y = 1 7. x = 2 , y = 1 8. x = 1 , y = 2 9. x = 1 , y = 1 correct Explanation: Basic Concepts: Kilogram (kg): a unit of mass (M). Meter (m): a unit of length (L). [ x ] means ”the units of x ”. The units of both sides of any equation must be the same for the equation to make sense. Solution: The units of the left hand side (LHS) are given as [ μ ] = M L = ML 1 and the right hand side has [ ρ x A y ] = µ M L 3 ¶ x × ( L 2 ) y = M x L 3 x L 2 y = M x L 2 y 3 x The powers on the units of mass and length need to be the same as for the LHS above, so x = 1 2 y 3 x = 1 2 y = 1 + 3 = 2 y = 1 Husain, Zeena – Homework 1 – Due: Sep 4 2003, 4:00 am – Inst: H L Berk 2 Thus the answer is ( x, y ) = (1 , 1). 003 (part 2 of 2) 10 points A simple pendulum is made out of a string with length L and a mass m attached to one end of the string. Its period of oscillation T may depend on the gravitational acceleration g , and also depend on L and m . Based on dimensional analysis, which of the following expressions is dimensionally accept able, where k is a dimensionless constant? 1. T = k g L 2. T = k s L g correct 3. T = k m g L 4. T = k r g L 5. T = k s L m g 6. T = k m L g 7. T = k L g 8. T = k r m g L Explanation: Here we proceed in the same way: a pe riod is a measure of time, thus the correct expression must have units of time....
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This note was uploaded on 01/27/2009 for the course PHY 303K taught by Professor Turner during the Fall '08 term at University of Texas at Austin.
 Fall '08
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