(e) 40 rad s 6. During a pirouette a figure skater reduces her moment of inertia from I = 1.24 kg m2 to I2 = 0.25 kg m2 by pulling in her arms. If she rotates with an angular speed of 0.5 rad s-1, before pulling in her arms, what is her final angular speed when she %3D Wi %3D finished the process? (a) 2.48 rad s-2 (b) 0.10 rad s-2 (c) 0.40 rad s-2 (d) 9.92 rad s-2 (e) 0.50 rad s-2 of the same mass and radius roll down an incline,

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### Educational Content on Moment of Inertia

**1. Determine the moment of inertia \( I \) of the Earth on its path around the Sun. Use the following values:**

- **Mass of the Earth:** \( M_E = 6 \cdot 10^{24} \, \text{kg} \)
- **Radius of the Earth's orbit around the Sun:** \( R_{ES} = 150 \cdot 10^6 \, \text{km} \)

Choices for \( I \):

\( a) \, 1.35 \cdot 10^{41} \, \text{kg} \cdot \text{m}^2 \)

\( b) \, 5.4 \cdot 10^{46} \, \text{kg} \cdot \text{m}^2 \)

\( c) \, 6.75 \cdot 10^{46} \, \text{kg} \cdot \text{m}^2 \)

\( d) \, 1.35 \cdot 10^{47} \, \text{kg} \cdot \text{m}^2 \)

**2. Consider problem 1 above. How would the value you found for \( I \) change if the Earth's year were only half as long (i.e., 182.5 days instead of 365 days)?**

Choices:

\( a) \) The value of \( I \) would double.

\( b) \) The value of \( I \) would be half.

\( c) \) The value of \( I \) would not change.

\( d) \) The value of \( I \) would quadruple.

\( e) \) The value of \( I \) would be one quarter.

**3. Consider problem 1 above. How would the value you found for \( I \) change if the radius of the Earth's path around the Sun would be only half as large (i.e., \( 75 \cdot 10^6 \, \text{km} \) instead of \( 150 \cdot 10^6 \, \text{km} \)?**

- Explore how changes in Earth's orbit affect its moment of inertia.

This content assists learners in understanding how the moment of inertia is influenced by mass distribution and orbit radius in celestial mechanics.
Transcribed Image Text:### Educational Content on Moment of Inertia **1. Determine the moment of inertia \( I \) of the Earth on its path around the Sun. Use the following values:** - **Mass of the Earth:** \( M_E = 6 \cdot 10^{24} \, \text{kg} \) - **Radius of the Earth's orbit around the Sun:** \( R_{ES} = 150 \cdot 10^6 \, \text{km} \) Choices for \( I \): \( a) \, 1.35 \cdot 10^{41} \, \text{kg} \cdot \text{m}^2 \) \( b) \, 5.4 \cdot 10^{46} \, \text{kg} \cdot \text{m}^2 \) \( c) \, 6.75 \cdot 10^{46} \, \text{kg} \cdot \text{m}^2 \) \( d) \, 1.35 \cdot 10^{47} \, \text{kg} \cdot \text{m}^2 \) **2. Consider problem 1 above. How would the value you found for \( I \) change if the Earth's year were only half as long (i.e., 182.5 days instead of 365 days)?** Choices: \( a) \) The value of \( I \) would double. \( b) \) The value of \( I \) would be half. \( c) \) The value of \( I \) would not change. \( d) \) The value of \( I \) would quadruple. \( e) \) The value of \( I \) would be one quarter. **3. Consider problem 1 above. How would the value you found for \( I \) change if the radius of the Earth's path around the Sun would be only half as large (i.e., \( 75 \cdot 10^6 \, \text{km} \) instead of \( 150 \cdot 10^6 \, \text{km} \)?** - Explore how changes in Earth's orbit affect its moment of inertia. This content assists learners in understanding how the moment of inertia is influenced by mass distribution and orbit radius in celestial mechanics.
**Educational Website: Understanding Rotational Motion**

---

**Exercises: Rotational Motion Concepts**

**3. Hypothetical Changes in Moment of Inertia**

- *(a)* The value of \( I \) would double.
- *(b)* The value of \( I \) would be half.
- *(c)* The value of \( I \) would not change.
- *(d)* The value of \( I \) would quadruple.
- *(e)* The value of \( I \) would be one quarter.

**4. Calculating Torque on a Pulley**

Consider the sketch of the setup for this experiment. Assume that the tension in the string running over the pulley is \( T = 10 \, \text{N} \) and the radius of the pulley is \( R = 10 \, \text{cm} \). The torque acting on the pulley is:

- *(a)* 100 Nm
- *(b)* 10 Nm
- *(c)* 1 Nm
- *(d)* 0.1 Nm
- *(e)* 0.01 Nm

**5. Angular Acceleration of a Pulley**

Consider problem 4 above. Assuming the pulley is a cylindrical, solid disk of mass \( M = 5 \, \text{kg} \), the angular acceleration of the pulley is closest to:

- *(a)* \( 0.2 \, \text{rad/s}^2 \)
- *(b)* \( 2 \, \text{rad/s}^2 \)
- *(c)* \( 4 \, \text{rad/s}^2 \)
- *(d)* \( 20 \, \text{rad/s}^2 \)
- *(e)* \( 40 \, \text{rad/s}^2 \)

**6. Figure Skater's Angular Speed**

During a pirouette, a figure skater reduces her moment of inertia from \( I_1 = 1.24 \, \text{kg m}^2 \) to \( I_2 = 0.25 \, \text{kg m}^2 \) by pulling in her arms. If she rotates with an angular speed of \( \omega_1 = 0.5 \, \text{rad/s} \) before pulling in her arms, what is her final angular speed when she finishes the process?

- *(a)*
Transcribed Image Text:**Educational Website: Understanding Rotational Motion** --- **Exercises: Rotational Motion Concepts** **3. Hypothetical Changes in Moment of Inertia** - *(a)* The value of \( I \) would double. - *(b)* The value of \( I \) would be half. - *(c)* The value of \( I \) would not change. - *(d)* The value of \( I \) would quadruple. - *(e)* The value of \( I \) would be one quarter. **4. Calculating Torque on a Pulley** Consider the sketch of the setup for this experiment. Assume that the tension in the string running over the pulley is \( T = 10 \, \text{N} \) and the radius of the pulley is \( R = 10 \, \text{cm} \). The torque acting on the pulley is: - *(a)* 100 Nm - *(b)* 10 Nm - *(c)* 1 Nm - *(d)* 0.1 Nm - *(e)* 0.01 Nm **5. Angular Acceleration of a Pulley** Consider problem 4 above. Assuming the pulley is a cylindrical, solid disk of mass \( M = 5 \, \text{kg} \), the angular acceleration of the pulley is closest to: - *(a)* \( 0.2 \, \text{rad/s}^2 \) - *(b)* \( 2 \, \text{rad/s}^2 \) - *(c)* \( 4 \, \text{rad/s}^2 \) - *(d)* \( 20 \, \text{rad/s}^2 \) - *(e)* \( 40 \, \text{rad/s}^2 \) **6. Figure Skater's Angular Speed** During a pirouette, a figure skater reduces her moment of inertia from \( I_1 = 1.24 \, \text{kg m}^2 \) to \( I_2 = 0.25 \, \text{kg m}^2 \) by pulling in her arms. If she rotates with an angular speed of \( \omega_1 = 0.5 \, \text{rad/s} \) before pulling in her arms, what is her final angular speed when she finishes the process? - *(a)*
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