Please refer to the picture to help answer the questions. A ice skater goes into a spin standing vertically with his arms outstretched horizontally. We can crudely model his torso and legs as a cylinder with radius 0.1m and mass 52 kg and each arm is a rod with mass 4 kg and length 0.9 m. The skater will initially spin at 2.0 rad/s.  Question 1: what is the moment of inertia of the skater's torso and legs. Please answer in kg m^2. Question 2: What is

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Please refer to the picture to help answer the questions.

A ice skater goes into a spin standing vertically with his arms outstretched horizontally. We can crudely model his torso and legs as a cylinder with radius 0.1m and mass 52 kg and each arm is a rod with mass 4 kg and length 0.9 m. The skater will initially spin at 2.0 rad/s. 

Question 1: what is the moment of inertia of the skater's torso and legs. Please answer in kg m^2.

Question 2: What is the moment of inertia of one of the skater's arms? Please answer in kg m^2.

Question 3: What is the total moment of inertia of the skater? Please answer in kg m^2

 

### Rotational Inertia

#### Diagram Explanation

This image illustrates two different formulas for the moment of inertia \( I \), which is a measure of an object's resistance to changes in its rotational motion. 

1. **Left Diagram: Rod Rotating About an End**
   - **Formula:**
     \[
     I = \frac{1}{3} ML^2
     \]
   - **Description:**
     - The diagram shows a rod of length \( L \) with its rotational axis at one end. The moment of inertia is defined by the formula \( \frac{1}{3} ML^2 \), where \( M \) is the mass of the rod and \( L \) is its length.

2. **Right Diagram: Solid Cylinder Rotating About Its Central Axis**
   - **Formula:**
     \[
     I = \frac{1}{2} MR^2
     \]
   - **Description:**
     - The diagram depicts a solid cylinder with radius \( R \), rotating about its central axis. The moment of inertia for this shape is given by the formula \( \frac{1}{2} MR^2 \), where \( M \) is the mass of the cylinder and \( R \) is its radius.

These formulas are essential for understanding how the distribution of mass affects an object's rotational dynamics.
Transcribed Image Text:### Rotational Inertia #### Diagram Explanation This image illustrates two different formulas for the moment of inertia \( I \), which is a measure of an object's resistance to changes in its rotational motion. 1. **Left Diagram: Rod Rotating About an End** - **Formula:** \[ I = \frac{1}{3} ML^2 \] - **Description:** - The diagram shows a rod of length \( L \) with its rotational axis at one end. The moment of inertia is defined by the formula \( \frac{1}{3} ML^2 \), where \( M \) is the mass of the rod and \( L \) is its length. 2. **Right Diagram: Solid Cylinder Rotating About Its Central Axis** - **Formula:** \[ I = \frac{1}{2} MR^2 \] - **Description:** - The diagram depicts a solid cylinder with radius \( R \), rotating about its central axis. The moment of inertia for this shape is given by the formula \( \frac{1}{2} MR^2 \), where \( M \) is the mass of the cylinder and \( R \) is its radius. These formulas are essential for understanding how the distribution of mass affects an object's rotational dynamics.
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Can you please help me answer the last three parts of this same problem. 

Question 4: What is the skater's angular momentum?please answer in kg m^2 /s

Question 5: The skater pulls his arms in to his chest and now he can be approximted as just a cylinder. What is his new moment of inertia? Please answer in kg m^2

Question 6: How fast is he spinning now that his arms are pulled in? Assume no external torque acts on him as he spins. Please answer in rad/s

 

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