rotational motion

1. Definition of Rotational Motion
   Rotational motion occurs when an object spins around an axis.

2. Key Concepts

   • Angular Displacement: The angle through which an object has rotated about a fixed axis.
   • Angular Velocity: The rate of change of angular displacement, typically measured in radians per second.
   • Angular Acceleration: The rate of change of angular velocity, represented in radians per second squared.

3. Equations of Rotational Motion

   • For constant angular acceleration, the equations are analogous to linear motion:
   1. 
      heta = heta0 + rac{1}{2} ( ext{ω}0 + ext{ω}) t
   2. 
      ext{ω} = ext{ω}_0 + ext{α} t
   3. 
      ext{ω}^2 = ext{ω}0^2 + 2 ext{α} heta where (\theta) is angular displacement, (\text{ω}0) is initial angular velocity, (\text{ω}) is final angular velocity, and (\text{α}) is angular acceleration.

4. Torque

   • Torque (\tau) is the rotational equivalent of force and is given by the equation:
     \tau = r imes F
     where (r) is the distance from the pivot point to where the force is applied, and (F) is the force applied. Torque is measured in Newton-meters (N·m).

5. Moment of Inertia (I)

   • Moment of inertia is the rotational analog of mass. It measures the resistance to rotational motion and depends on the distribution of mass around the axis of rotation.
   • For common shapes:
     • Solid Cylinder: (I = \frac{1}{2} m r^2)
     • Hollow Cylinder: (I = m r^2)

6. Conservation of Angular Momentum

   • In the absence of an external torque, the angular momentum of a system remains constant.
   • Angular momentum (L) can be expressed as:
     L = I imes ext{ω}

7. Applications of Rotational Motion

   • Concepts of rotational motion apply to various fields, including engineering, astrophysics, and daily life scenarios like the functioning of wheels, gears, and more.

8. Practice Problems

   1. A wheel accelerates from rest to 300 rad/s in 10 seconds. What is the angular acceleration?
   2. Calculate the torque required to rotate a disk with a radius of 0.5 m using a force of 20 N.
   3. If a figure skater pulls in their arms, how does their rotation speed change?

These key points provide a foundational framework for rotational motion, crucial for exam preparation and understanding of