81.

(a) The rotational inertia relative to the specified axis is

I =

m

i

r

2

i

= (2M )L

2

+ (2M )L

2

+ M (2L)

2

which is found to be I = 4.6 k g

·m

2

. Then, with ω = 1.2 rad/s, we obtain the kinetic energy from

Eq. 11-27:

K =

1

2

Iω

2

= 3.3 J .

(b) In this case the axis of rotation would appear as a standard y axis with origin at P . Each of the

2M balls are a distance of r = L cos 30

◦

from that axis. Thus, the rotational inertia in this case is

I =

m

i

r

2

i

= (2M )r

2

+ (2M )r

2

+ M (2L)

2

which is found to be I = 4.0 k g

·m

2

. Again, from Eq. 11-27 we obtain the kinetic energy

K =

1

2

Iω

2

= 2.9 J .

Document Outline

• 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