1. b. The center of the hitting end is closer to the center of mass, so it must have more mass.

2. b. Rotational kinetic energy depends on the angular velocity and the moment of inertia. Both have the same angular velocity, but a cylindrical shell has a larger moment of inertia than a solid cylinder with the same mass and radius.

3. $I{}_{\text{A}}>I{}_{\text{D}}>I{}_{\text{B}}>I{}_{\text{C}}$. The moment of inertia is smaller when the mass is more concentrated near the rotation axis.

4. $\tau {}_{\text{E}}>{\tau }_{\text{A}}={\tau }_{\text{D}}>{\tau }_{\text{B}}>{\tau }_{\text{C}}$. The tangential component in E is larger than 2 N.

5. $\alpha {}_{\text{B}}>{\alpha }_{\text{A}}>{\alpha }_{\text{C}}={\alpha }_{\text{D}}.$ Angular acceleration is proportional to torque and inversely proportional to the moment of inertia. Torque is proportional to the radius but moment of inertia is proportional to the square of the radius. Thus angular momentum is proportional to $F/mr$.

6. c. The scale exerts an upward force at half the distance from the pivot as the weight’s downward force. To exert an equal but opposite torque for static equilibrium requires twice the force.

7. b. The velocity at the dot is the vector sum of ${\vec{v}}_{\text{cm}}$ to the right and an upward velocity vector ${\vec{v}}_{i,\text{ rel}}$ of equal magnitude due to rotation about the center of mass.

8. d. There is no net torque on the bucket $+$ rain system, so the angular momentum is conserved. The addition of mass on the outer edge of the circle increases $I$, so $\omega$ must decrease. Mechanical energy is not conserved because the raindrop collisions are inelastic.

1.

1. $4.2\times {10}^2\text{rad}\text{/}{\text{s}}^2$

2. 8.3 rev

3.

1. 1.5 m/s

2. 13 rev

5. 23 g

7. $(1.7\text{cm,}4\text{.}7\text{cm})$

9. $2.57\times {10}^{29}\text{J}$

11.

1. $0.032\text{kg}{\text{m}}^2$

2. 16 J

13.

1. 0.063 m, 0.050 m

2. $0.0082\text{kg}{\text{m}}^2$

15.

1. $(0\text{.}060\text{m}\text{,}0\text{.}040\text{m})$

2. $0.0020\text{kg}{\text{m}}^2$

3. $0.0013\text{kg}{\text{m}}^2$

17.

1. $3.8\times {10}^{-5}\text{kg}{\text{m}}^2$

2. $1.14\times {10}^{-4}\text{kg}{\text{m}}^2$

19. 8.0 cm

21. $0.56\text{N}\text{m}$

23.

1. $\tau =31.0\text{N}\text{m}$

2. $\tau =21.9\text{N}\text{m}$

25. 0.75 rad/s

27. $1.7\times {10}^{-3}\text{N}\text{m}$

29. $F_1=120\text{N, }{F}_2=80\text{N}$

31. $F_1=0.75\text{ kN, }{F}_2=1.0\text{ kN}$

33.

1. $6.4\times {10}^2\text{rpm}$

2. 40 m/s

3. 0 m/s

35. 0.38 J

37.

1. $\vec{A}\times \vec{B}=(21\text{,}\text{i}\text{n}\text{t}\text{o}\text{t}\text{h}\text{e}\text{p}\text{a}\text{g}\text{e})$

2. $\vec{C}\times \vec{D}=(24\text{,}\text{o}\text{u}\text{t}\text{o}\text{f}\text{t}\text{h}\text{e}\text{p}\text{a}\text{g}\text{e})$

39.

1. $\vec{D}=nî$, where $n$ could be any real number

2. $\vec{E}=2ĵ$

3. $\vec{F}=1\widehat{k}$

41. $-0.35\text{N}\text{m}\widehat{k}$

43. $-0.025î\text{kg}\text{m}\text{/}{\text{s}}^2$

45. 50 rpm

47. 83 rpm

49.

1. 15 cm

2. 1.9 J

51. $x_{\text{cm}}=20\text{cm, }{y}_{\text{cm}}=0\text{cm}$

53. ${\displaystyle \frac{M}{3L}}[{(L-d)}^3+d^3]$

55. $I={\displaystyle {\int }_{}^{}}{r}^{2}dm$

57. $51\text{°}$

59. 15,300 N

61. 1.0 m

63.

1. 25 N

2. 25 N

65. 1.6 N

67. 0.52 N

69. Disk: 67 cm, ring: 89 cm

71.

1. 22 rad/s

2. 6.6 rad/s

73.

1. ${\omega }_1=\sqrt{3g\text{/}L}$

2. $v_{\text{tip}}=({\omega }_1)L=\sqrt{3gL}$

75. $\alpha ={\displaystyle \frac{\tau }{I}}={\displaystyle \frac{Tr}{(13M{R}^2\text{/}20)}}={\displaystyle \frac{20Tr}{13M{R}^2}}$

77.

1. No

2. 2000 m/s

3. 4000 m/s

79. 50 rpm

81. 22 rpm

83.

1. 137 km

2. $8.6\times {10}^6\text{m}\text{/}\text{s}$

85. ${\tau }_{\text{drag}}={\displaystyle \frac{\rho {\omega }^2{C}_{\text{D}}a{L}^4}{64}}$

87. 5.8 rev/s

89. 4.0 rpm

91.

1. $2.9\times {10}^{-5}\text{N}\text{m}$

2. 7.0 m/s