The moment of inertia of a solid sphere that is rotating about any diameter is equal to I0 = 2/5 MR2 . If the mass of the sphere is doubled but the radius is reduced in half, how does the moment of inertia change?It becomes 1/16 of the initial moment of inertiaIt becomes 1/4 of the initial moment of inertiaIt becomes 1/2 of the initial moment of inertiaIt becomes 1/8 of the initial moment of inertia How does the rotational energy of a body change if the angular velocity is doubled?the rotational energy increases by a factor of 8.the rotational energy increases by a factor of 2.the rotational energy does not change.the rotational energy increases by a factor of 4. ### Moments of Inertia for Various Objects**1. Hoop about Cylinder Axis**- Diagram: A hoop with radius \( R \).- Moment of Inertia: \( I = MR^2 \).**2. Annular Cylinder (or Ring) about Cylinder Axis**- Diagram: A ring with inner radius \( R_1 \) and outer radius \( R_2 \).- Moment of Inertia: \( I = \frac{M}{2} (R_1^2 + R_2^2) \).**3. Solid Cylinder (or Disk) about Cylinder Axis**- Diagram: A solid cylinder with radius \( R \).- Moment of Inertia: \( I = \frac{1}{2} MR^2 \).**4. Solid Cylinder (or Disk) about Central Diameter**- Diagram: A solid cylinder with radius \( R \).- Moment of Inertia: \( I = \frac{MR^2}{4} + \frac{ML^2}{12} \).**5. Thin Rod about Axis through Center ⊥ to Length**- Diagram: A thin rod with length \( L \).- Moment of Inertia: \( I = \frac{ML^2}{12} \).**6. Thin Rod about Axis through One End ⊥ to Length**- Diagram: A thin rod with length \( L \).- Moment of Inertia: \( I = \frac{ML^2}{3} \).**7. Solid Sphere about Any Diameter**- Diagram: A solid sphere with diameter \( 2R \).- Moment of Inertia: \( I = \frac{2}{5} MR^2 \).**8. Thin Spherical Shell about Any Diameter**- Diagram: A thin spherical shell with diameter \( 2R \).- Moment of Inertia: \( I = \frac{2}{3} MR^2 \).**9. Hoop about Any Diameter**- Diagram: A hoop with radius \( R \).- Moment of Inertia: \( I = \frac{1}{2} MR^2 \).**10. Slab about ⊥ Axis through Center**- Diagram: A rectangular slab with dimensions \( a \) and \( b \).- Moment of Inertia: \( I = \frac{M(a^2 + b^2)}{12} \).These formulas are essential for calculating the

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College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

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The moment of inertia of a solid sphere that is rotating about any diameter is equal to I₀= 2/5 MR². If the mass of the sphere is doubled but the radius is reduced in half, how does the moment of inertia change?

It becomes 1/16 of the initial moment of inertia

It becomes 1/4 of the initial moment of inertia

It becomes 1/2 of the initial moment of inertia

It becomes 1/8 of the initial moment of inertia

How does the rotational energy of a body change if the angular velocity is doubled?

the rotational energy increases by a factor of 8.

the rotational energy increases by a factor of 2.

the rotational energy does not change.

the rotational energy increases by a factor of 4.

### Moments of Inertia for Various Objects

**1. Hoop about Cylinder Axis**
- Diagram: A hoop with radius \( R \).
- Moment of Inertia: \( I = MR^2 \).

**2. Annular Cylinder (or Ring) about Cylinder Axis**
- Diagram: A ring with inner radius \( R_1 \) and outer radius \( R_2 \).
- Moment of Inertia: \( I = \frac{M}{2} (R_1^2 + R_2^2) \).

**3. Solid Cylinder (or Disk) about Cylinder Axis**
- Diagram: A solid cylinder with radius \( R \).
- Moment of Inertia: \( I = \frac{1}{2} MR^2 \).

**4. Solid Cylinder (or Disk) about Central Diameter**
- Diagram: A solid cylinder with radius \( R \).
- Moment of Inertia: \( I = \frac{MR^2}{4} + \frac{ML^2}{12} \).

**5. Thin Rod about Axis through Center ⊥ to Length**
- Diagram: A thin rod with length \( L \).
- Moment of Inertia: \( I = \frac{ML^2}{12} \).

**6. Thin Rod about Axis through One End ⊥ to Length**
- Diagram: A thin rod with length \( L \).
- Moment of Inertia: \( I = \frac{ML^2}{3} \).

**7. Solid Sphere about Any Diameter**
- Diagram: A solid sphere with diameter \( 2R \).
- Moment of Inertia: \( I = \frac{2}{5} MR^2 \).

**8. Thin Spherical Shell about Any Diameter**
- Diagram: A thin spherical shell with diameter \( 2R \).
- Moment of Inertia: \( I = \frac{2}{3} MR^2 \).

**9. Hoop about Any Diameter**
- Diagram: A hoop with radius \( R \).
- Moment of Inertia: \( I = \frac{1}{2} MR^2 \).

**10. Slab about ⊥ Axis through Center**
- Diagram: A rectangular slab with dimensions \( a \) and \( b \).
- Moment of Inertia: \( I = \frac{M(a^2 + b^2)}{12} \).

These formulas are essential for calculating the

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