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65. We use conservation of mechanical energy. The center of mass is at the midpoint of the cross bar of the H and it drops by L/2, where L is the length of any one of the rods. The gravitational potential energy decreases by MgL/2, where M is the mass of the body. The initial kinetic energy is zero and the final kinetic energy may be written 12Iω 2, where I is the rotational inertia of the body and ω is its angular velocity when it is vertical. Thus 0 = −MgL/2 + 1 2 Iω2 =⇒ ω = √ MgL/I . Since the rods are thin the one along the axis of rotation does not contribute to the rotational inertia. All points on the other leg are the same distance from the axis of rotation, so that leg contributes (M/3)L2, where M/3 is its mass. The cross bar is a rod that rotates around one end, so its contri- bution is (M/3)L2/3 = ML2/9. The total rotational inertia is I = (ML2/3) + (ML2/9) = 4ML2/9. Consequently, the angular velocity is ω = √ MgL I = √ MgL 4ML2/9 = √ 9g 4L . Main Menu Chapter 1 Measurement Chapter 2 Motion Along a Straight Line Chapter 3 Vectors Chapter 4 Motion in Two and Three Dimensions Chapter 5 Force and Motion I Chapter 6 Force and Motion II Chapter 7 Kinetic Energy and Work Chapter 8 Potential Energy and Conservation of Energy Chapter 9 Systems of Particles Chapter 10 Collisions Chapter 11 Rotation 11.1 - 11.10 11.1 11.2 11.3 11.4 11.5 11.6 11.7 11.8 11.9 11.10 11.11 - 11.20 11.11 11.12 11.13 11.14 11.15 11.16 11.17 11.18 11.19 11.20 11.21 - 11.30 11.21 11.22 11.23 11.24 11.25 11.26 11.27 11.28 11.29 11.30 11.31 - 11.40 11.31 11.32 11.33 11.34 11.35 11.36 11.37 11.38 11.39 11.40 11.41 - 11.50 11.41 11.42 11.43 11.44 11.45 11.46 11.47 11.48 11.49 11.50 11.51 - 11.60 11.51 11.52 11.53 11.54 11.55 11.56 11.57 11.58 11.59 11.60 11.61 - 11.70 11.61 11.62 11.63 11.64 11.65 11.66 11.67 11.68 11.69 11.70 11.71 - 11.80 11.71 11.72 11.73 11.74 11.75 11.76 11.77 11.78 11.79 11.80 11.81 - 11.90 11.81 11.82 11.83 11.84 11.85 11.86 11.87 11.88 11.89 11.90 11.91 - 11.100 11.91 11.92 11.93 11.94 11.95 11.96 11.97 11.98 11.99 11.100 11.101 - 11.103 11.101 11.102 11.103 Chapter 12 Rolling, Torque, and Angular Momentum Chapter 13 Equilibrium and Elasticity Chapter 14 Gravitation Chapter 15 Fluids Chapter 16 Oscillations Chapter 17 Waves—I Chapter 18 Waves—II Chapter 19 Temperature, Heat, and the First Law of Thermodynamics Chapter 20 The Kinetic Theory of Gases Chapter 21 Entropy and the Second Law of Thermodynamics Chapter 22 Electric Charge Chapter 23 Electric Fields Chapter 24 Gauss’ Law Chapter 25 Electric Potential Chapter 26 CapacitanceChapter 27 Current and Resistance Chapter 27 Current and Resistance Chapter 28 Circuits Chapter 29 Magnetic Fields Chapter 30 Magnetic Fields Due to Currents Chapter 31 Induction and Inductance Chapter 32 Magnetism of Matter: Maxwell’s Equation Chapter 33 Electromagnetic Oscillations and Alternating Current Chapter 34 Electromagnetic Waves Chapter 35 Images Chapter 36 Interference Chapter 37 Diffraction Chapter 38 Special Theory of Relativity Chapter 39 Photons and Matter Waves Chapter 40 More About Matter Waves Chapter 41 All About Atoms Chapter 42 Conduction of Electricity in Solids Chapter 43 Nuclear Physics Chapter 44 Energy from the Nucleus Chapter 45 Quarks, Leptons, and the Big Bang
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