Elementary Mechanics and Thermodynamics SOLUTIONS MANUAL - J. Norbury

# Elementary Mechanics and Thermodynamics SOLUTIONS MANUAL - J. Norbury

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SOLUTIONS MANUAL for elementary mechanics & thermodynamics Professor John W. Norbury Physics Department University of Wisconsin-Milwaukee P.O. Box 413 Milwaukee, WI 53201 November 20, 2000

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Contents 1 MOTION ALONG A STRAIGHT LINE 5 2 VECTORS 15 3 MOTION IN 2 & 3 DIMENSIONS 19 4 FORCE & MOTION - I 35 5 FORCE & MOTION - II 37 6 KINETIC ENERGY & WORK 51 7 POTENTIAL ENERGY & CONSERVATION OF ENERGY 53 8 SYSTEMS OF PARTICLES 57 9 COLLISIONS 61 10 ROTATION 65 11 ROLLING, TORQUE & ANGULAR MOMENTUM 75 12 OSCILLATIONS 77 13 WAVES - I 85 14 WAVES - II 87 15 TEMPERATURE, HEAT & 1ST LAW OF THERMODY- NAMICS 93 16 KINETIC THEORY OF GASES 99 3

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4 CONTENTS 17 Review of Calculus 103
Chapter 1 MOTION ALONG A STRAIGHT LINE 5

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6 CHAPTER 1. MOTION ALONG A STRAIGHT LINE 1. The following functions give the position as a function of time: i) x = A ii) x = Bt iii) x = Ct 2 iv) x = D cos ωt v) x = E sin ωt where A, B, C, D, E, ω are constants. A) What are the units for A, B, C, D, E, ω ? B) Write down the velocity and acceleration equations as a function of time. Indicate for what functions the acceleration is constant . C) Sketch graphs of x, v, a as a function of time. SOLUTION A) X is always in m . Thus we must have A in m ; B in m sec - 1 , C in m sec - 2 . ωt is always an angle, θ is radius and cos θ and sin θ have no units. Thus ω must be sec - 1 or radians sec - 1 . D and E must be m . B) v = dx dt and a = dv dt . Thus i) v = 0 ii) v = B iii) v = Ct iv) v = - ωD sin ωt v) v = ωE cos ωt and notice that the units we worked out in part A) are all consistent with v having units of m · sec - 1 . Similarly i) a = 0 ii) a = 0 iii) a = C iv) a = - ω 2 D cos ωt v) a = - ω 2 E sin ωt
7 i) ii) iii) x t v a x x v v a a t t t t t t t t C)

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8 CHAPTER 1. MOTION ALONG A STRAIGHT LINE iv) v) 0 1 2 3 4 5 6 t - 1 - 0.5 0 0.5 1 x 0 1 2 3 4 5 6 t - 1 - 0.5 0 0.5 1 x 0 1 2 3 4 5 6 t - 1 - 0.5 0 0.5 1 v 0 1 2 3 4 5 6 t - 1 - 0.5 0 0.5 1 v 0 1 2 3 4 5 6 t - 1 - 0.5 0 0.5 1 a 0 1 2 3 4 5 6 t - 1 - 0.5 0 0.5 1 a
9 2. The figures below show position-time graphs. Sketch the correspond- ing velocity-time and acceleration-time graphs. t x t x t x SOLUTION The velocity-time and acceleration-time graphs are: t v t t v t a t a t a v

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10 CHAPTER 1. MOTION ALONG A STRAIGHT LINE 3. If you drop an object from a height H above the ground, work out a formula for the speed with which the object hits the ground. SOLUTION v 2 = v 2 0 + 2 a ( y - y 0 ) In the vertical direction we have: v 0 = 0, a = - g , y 0 = H , y = 0. Thus v 2 = 0 - 2 g (0 - H ) = 2 gH v = p 2 gH
11 4. A car is travelling at constant speed v 1 and passes a second car moving at speed v 2 . The instant it passes, the driver of the second car decides to try to catch up to the first car, by stepping on the gas pedal and moving at acceleration a . Derive a formula for how long it takes to catch up. (The first car travels at constant speed v 1 and does not accelerate.) SOLUTION Suppose the second car catches up in a time interval t . During that interval, the first car (which is not accelerating) has travelled a distance d = v 1 t . The second car also travels this distance d in time t , but the second car is accelerating at a and so it’s distance is given by x - x 0 = d = v 0 t + 1 2 at 2 = v 1 t = v 2 t + 1 2 at 2 because v 0 = v 2 v 1 = v 2 + 1 2 at t = 2( v 1 - v 2 ) a

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12 CHAPTER 1.
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