10.2 Rotation with Constant Angular Acceleration

Rotational kinematics deals with the motion of objects rotating around a fixed axis. It's all about understanding how things spin, from wheels to planets, using equations that relate angular displacement, velocity, and acceleration.

Connecting rotational and linear motion is key. By linking angular quantities to their linear counterparts, we can analyze complex motions like a car's wheels rolling down the road or a figure skater's spin on ice.

Rotational Kinematics

Rotational kinematics equations

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• Definition of angular acceleration $\alpha$: rate of change of angular velocity $\omega$ with respect to time $t$ ($\alpha = \frac{d\omega}{dt}$)
• Integrate angular acceleration with respect to time to obtain angular velocity: $\omega = \omega_0 + \alpha t$
  • $\omega_0$ represents the initial angular velocity
• Integrate angular velocity with respect to time to obtain angular displacement $\theta$: $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$
  • $\theta_0$ represents the initial angular displacement
• Combine the equations for $\omega$ and $\theta$ to eliminate time: $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$
  • Useful when time is not explicitly given or required in the problem

Applications of rotational kinematics

• Identify the given quantities and the quantity to be solved for in the problem
• Choose the appropriate kinematic equation based on the given information
  • $\omega = \omega_0 + \alpha t$ (relates angular velocity, angular acceleration, and time)
  • $\theta = \theta_0 + \omega_0 t + \frac{1}{2}\alpha t^2$ (relates angular displacement, initial angular velocity, angular acceleration, and time)
  • $\omega^2 = \omega_0^2 + 2\alpha(\theta - \theta_0)$ (relates angular velocity, initial angular velocity, angular acceleration, and angular displacement)
• Substitute the given values into the chosen equation and solve for the unknown quantity
• Pay attention to units and convert if necessary
  • Angular displacement in radians (rad)
  • Angular velocity in radians per second (rad/s)
  • Angular acceleration in radians per second squared (rad/s²)
• Examples
  • Calculating the final angular velocity of a spinning wheel given initial angular velocity and angular acceleration
  • Determining the time required for a rotating object to reach a specific angular displacement

Analysis of fixed-axis rotation

• Understand the relationships between angular displacement, angular velocity, and angular acceleration
  • Angular velocity is the rate of change of angular displacement ($\omega = \frac{d\theta}{dt}$)
  • Angular acceleration is the rate of change of angular velocity ($\alpha = \frac{d\omega}{dt}$)
• Interpret graphs of angular displacement, angular velocity, and angular acceleration vs. time
  • Angular displacement vs. time
    • Slope of the graph represents angular velocity
    • Curvature of the graph represents angular acceleration
  • Angular velocity vs. time
    • Slope of the graph represents angular acceleration
    • Area under the curve represents angular displacement
  • Angular acceleration vs. time
    • Constant value indicates constant angular acceleration
    • Changing value indicates non-constant angular acceleration
• Use calculus to analyze rotational motion
  1. Differentiate angular displacement to obtain angular velocity ($\omega = \frac{d\theta}{dt}$)
  2. Differentiate angular velocity to obtain angular acceleration ($\alpha = \frac{d\omega}{dt}$)
  3. Integrate angular acceleration to obtain angular velocity ($\omega = \int \alpha dt$)
  4. Integrate angular velocity to obtain angular displacement ($\theta = \int \omega dt$)

Connecting Rotational and Linear Kinematics

Relate linear and angular quantities for objects undergoing fixed-axis rotation

• Understand the relationship between linear and angular quantities
  • Linear displacement $s$ is related to angular displacement $\theta$ by the radius of rotation $r$: $s = r\theta$
  • Linear velocity $v$ is related to angular velocity $\omega$ by the radius of rotation $r$: $v = r\omega$
  • Tangential acceleration $a_t$ is related to angular acceleration $\alpha$ by the radius of rotation $r$: $a_t = r\alpha$
• Apply these relationships to solve problems involving both linear and angular quantities
  • Example: Calculating the linear velocity of a point on a rotating wheel given its angular velocity and radius
• Recognize that the direction of linear velocity is always tangential to the circular path of rotation
• Note the presence of centripetal acceleration $a_c$ in addition to tangential acceleration
  • Centripetal acceleration is directed towards the center of rotation and is given by $a_c = \frac{v^2}{r} = r\omega^2$
  • It is responsible for keeping the object in circular motion

Rotational Dynamics and Energy

Rotational inertia and torque

• Moment of inertia (I) represents an object's resistance to rotational acceleration
  • Depends on the mass distribution of the object relative to the axis of rotation
• Torque (τ) is the rotational equivalent of force, causing angular acceleration
  • τ = r × F, where r is the position vector from the axis of rotation to the point of force application
• Newton's Second Law for rotation: τ = Iα, relating torque, moment of inertia, and angular acceleration

Angular momentum and energy

• Angular momentum (L) is the rotational analog of linear momentum
  • L = Iω, where I is the moment of inertia and ω is the angular velocity
• Conservation of angular momentum applies in the absence of external torques
• Rotational kinetic energy is given by KE = 1/2 Iω²
  • Contributes to the total energy of a rotating system

Parallel axis theorem

• Allows calculation of moment of inertia about any axis parallel to an axis through the center of mass
• I = I_cm + Md², where I_cm is the moment of inertia about the center of mass, M is the total mass, and d is the distance between the parallel axes