**1. What is the center of mass of a two-particle system?**

**Answer:** The center of mass of a two-particle system is the point where the total mass of the system can be considered to be concentrated. It is found by the formula

where m_{1} and m_{2} are the masses and r_{1} and r_{2} are their respective positions.

**2. How is the center of mass of a rigid body defined?**

**Answer:** The center of mass of a rigid body is the point at which the total mass of the body can be considered to be concentrated for the purpose of analyzing translational motion.

**3. What is the moment of a force?**

**Answer:** The moment of a force, or torque, is a measure of the force’s tendency to cause rotational motion about an axis. It is given by

where **r** is the position vector from the axis to the point of application of the force **F**.

**4. Define torque in the context of rotational motion.**

**Answer:** Torque is the rotational equivalent of force. It measures the tendency of a force to rotate an object around an axis. Torque is calculated as the cross product of the position vector and the force vector:

**5. What is angular momentum?**

**Answer:** Angular momentum is the rotational analog of linear momentum. For a particle, it is defined as

where **r** is the position vector and **p** is the linear momentum of the particle. For a rigid body, it is the sum of the angular momenta of all the particles in the body.

**6. State the law of conservation of angular momentum.**

**Answer:** The law of conservation of angular momentum states that if no external torque acts on a system, the total angular momentum of the system remains constant.

**7. Provide an example application of the conservation of angular momentum.**

**Answer:** An example is a figure skater spinning: as the skater pulls in their arms, they reduce their moment of inertia and spin faster to conserve angular momentum.

**8. What is the moment of inertia?**

**Answer:** The moment of inertia is a measure of an object’s resistance to changes in its rotational motion. It depends on the mass distribution relative to the axis of rotation and is given by

for discrete masses or

for continuous mass distributions.

**9. Define the radius of gyration.**

**Answer:** The radius of gyration k is defined as the distance from the axis of rotation at which the total mass of the body could be concentrated without changing its moment of inertia. It is given by

where I is the moment of inertia and M is the total mass.

**10. What are the values of moments of inertia for simple geometrical objects?**

**Answer:** For a thin rod rotating about an end:

For a solid cylinder rotating about its central axis:

For a solid sphere rotating about its diameter:

**11. Explain the parallel axis theorem.**

**Answer:** The parallel axis theorem states that the moment of inertia of a body about any axis parallel to an axis through its center of mass is given by

where **I**cm is the moment of inertia about the center of mass axis, M is the mass of the body, and d is the distance between the axes.

**12. Explain the perpendicular axis theorem.**

**Answer:** The perpendicular axis theorem states that for a planar object lying in the xy-plane, the moment of inertia about an axis perpendicular to the plane (the z-axis) is the sum of the moments of inertia about the x and y axes:

**13. What is the equilibrium of rigid bodies?**

**Answer:** Equilibrium of rigid bodies occurs when the body is in a state of rest or moves with constant velocity, implying that the net force and net torque acting on the body are zero.

**14. What are the equations of rotational motion for a rigid body?**

**Answer:** The equations of rotational motion are analogous to the linear motion equations and include:

where θ is the angular displacement, ω is the angular velocity, and α is the angular acceleration.

**15. How do you compare linear and rotational motions?**

**Answer:** Linear and rotational motions are analogous: displacement corresponds to angular displacement, velocity to angular velocity, acceleration to angular acceleration, mass to moment of inertia, force to torque, and momentum to angular momentum.

**16. What is the moment of inertia of a solid disk rotating about its central axis?**

**Answer:** The moment of inertia of a solid disk rotating about its central axis is

where M is the mass and R is the radius of the disk.

**17. How is torque related to angular acceleration?**

**Answer:** Torque (τ) is related to angular acceleration (α) through the equation

where I is the moment of inertia of the object.

**18. What happens to the angular velocity of a rotating body if its moment of inertia changes but no external torque is applied?**

**Answer:** If no external torque is applied and the moment of inertia changes, the angular velocity will adjust to conserve angular momentum

If the moment of inertia decreases, the angular velocity increases, and vice versa.

**19. What is the role of the radius of gyration in calculating the moment of inertia?**

**Answer:** The radius of gyration helps simplify the calculation of the moment of inertia. It represents the distance from the axis of rotation at which the mass of the body could be concentrated to yield the same moment of inertia:

**20. Why is the conservation of angular momentum important in rotational motion?**

**Answer:** Conservation of angular momentum is crucial because it allows for the prediction of rotational behavior in isolated systems, explaining phenomena such as gyroscopic effects and the behavior of rotating bodies in the absence of external torques.