PHYS 2325: Rotational Kinematics & Dynamics Study Notes

Arc Length:
- Definition: The length of the arc described by a rotation.
- Formula: $s = r \, riangle heta$
Angular Position:
- Symbol: $heta$ (in radians)
Angular Displacement:
  - Symbol: $riangle heta$ (in radians)
  - Definition: The change in angular position.
  - Formula: $riangle heta = heta_f - heta_i$
Average Angular Velocity:
  - Symbol: $\bar{\beta}$
  - Definition: The ratio of the angular displacement to the time interval.
  - Formula: $\bar{\beta} = \frac{ riangle heta}{ riangle t} = \frac{ heta_f - heta_i}{t_f - t_i}$ (measured in rad/s)
Tangential Velocity:
  - Symbol: $v_t$
  - Definition: The linear velocity along the edge of the rotating path.
  - Formula: $v_t = r \, \bar{\beta}$
Centripetal Acceleration:
  - Symbol: $a_c$
  - Definition: The acceleration directed towards the center of the circular path that keeps an object in circular motion.
  - Formula: $a_c = \frac{v^2}{r} = r \, \bar{\beta}^2$ (m/s²)

When Tangential Velocity Isn’t Constant:
  - Similar to linear speed, angular speed can change.
  - Angular acceleration (average):
    - Formulae:
      - $\bar{\beta} = \frac{v_t}{r}$
      - $riangle\beta = \frac{ riangle v_t}{r}$
      - $\bar{\beta} = \frac{ riangle \beta}{ riangle t} = \frac{ riangle v_t}{t}$
      - Angular acceleration is defined as: $\bar{\beta}_{avg} = \frac{ riangle \beta}{ riangle t}$
Components of Acceleration:
  - Total acceleration has both tangential and centripetal components, denoted as:
    - $a_{t}$ (tangential) and $a_{c}$ (centripetal) are perpendicular.
    - The question arises whether both can be constant at the same time.

In a rotational context, the instantaneous angular quantities are defined as follows:
  - Instantaneous angular displacement:
    - d heta = ext{lim}_{ riangle t
ightarrow 0} rac{d heta}{dt}
    - Definition: The time derivative of angular position $heta$ .
Instantaneous Angular Velocity:
  - Symbol: $\beta$
  - Definition: The time derivative of angular position $heta$ .
  - Formula: $\beta = \frac{d heta}{dt}$
Instantaneous Angular Acceleration:
  - Symbol: $\bar{\beta}$
  - Definition: The time derivative of angular velocity $\beta$ .
  - Formula: $\bar{\beta} = \frac{d\beta}{dt} = \frac{d^2 heta}{dt^2}$

Summary of relations:

  | Linear Quantity | Angular Quantity | Relation
  |----------------------|------------------|---------------------------------|
  | $x$ | $heta$ | $heta = \frac{s}{r}$
  | $\frac{dx}{dt}$ | $\frac{d heta}{dt}$ | $\frac{d heta}{dt} = \bar{\beta}$
  | $v$ | $\beta$ | $\beta = \frac{v_t}{r}$
  | $\frac{dv}{dt}$ | $\frac{d\beta}{dt}$ | $\frac{d\beta}{dt} = \bar{\beta}$

Angular Velocity and Angular Acceleration as Vectors:
  - Both $\beta$ and $\bar{\beta}$ are vectors and their directions differ from linear counterparts.
  - Right-Hand Rule:
    - Direction of vectors points perpendicular from the plane of rotation along the axis. If you curl the fingers of the right hand around the axis in the direction of rotation, the thumb gives the direction of $\beta$ .

Definition: Torque ( $au$ ) is the measure of the force that produces or tends to produce rotation. It is defined as the cross product of the position vector $extbf{r}$ and the force vector $extbf{F}$ .
Formula:
- $au = extbf{r} imes extbf{F} = r \, F \, ext{sin}( heta)$
Direction of Torque:
- Determined via the right-hand rule or indicated as counterclockwise (-) and clockwise (+).

Steps:
  - Identify the point of axis of rotation
  - Determine where the force is applied
  - Measure the perpendicular distance from the line of the force to the point of rotation
  - Angle Measurement:
    - Angle $heta$ between force vector and position vector determines torque’s magnitude.

Problem Statement: A small ball with mass 0.75 kg at the end of a 1.25 m massless rod and the rod hangs from a pivot at an angle of 30° from the vertical.
Finding Gravitational Torque:
- Torque, $au = r imes F_{gravity} = r imes mg \, ext{sin}( heta)$

Definition: Moment of inertia ( $I$ ) quantifies how much torque is needed for a desired angular acceleration about a rotational axis.
Formula:
- $I = ext{sum of all } m_i r_i^2$ where $m_i$ is mass and $r_i$ is the distance from the axis of rotation.
Axis of Rotation: Example shapes and their moments of inertia include:
  - Hoop: $I = mR^2$
  - Solid Cylinder or Disk: $I = \frac{1}{2}mR^2$
  - Thin Rod (about center): $I = \frac{1}{12}mL^2$
  - Solid Sphere: $I = \frac{2}{5}mR^2$

Definition: The moment of inertia about any axis parallel to the axis through the center of mass can be found using the parallel-axis theorem:
- $I_{ ext{parallel}} = I_{ ext{cm}} + md^2$ where $m$ is mass and $d$ is the distance between the axes.
Example Calculation: Each axis shift can be used to re-calculate the moment of inertia with given mass distribution.

For composite objects sharing the same rotational axis, the total inertia is the sum of individual inertias:
- $I_{total} = I_1 + I_2 + … + I_n$
Consideration is needed when the axes are offset: apply parallel-axis theorem accordingly.

If a child hops onto a merry-go-round, the total moment of inertia changes. The new moment of inertia can be calculated as:
- $I_{new} = \frac{1}{2}MR^2 + mr^2$ with $m$ being the mass of the child and $r$ being the distance from the axis.

Work in Rotational Dynamics: Angles and rotational work can be calculated by:
- $W = au heta$ where $au$ is torque and $heta$ is angular displacement.
Rotational Work:
- Total work done on the system can be expressed in terms of initial and final kinetic energy:
- $W = K_f - K_i = \frac{1}{2}I\beta^2 - \frac{1}{2}I\beta_0^2$
Problem Solving: These principles can be applied to real-world scenarios, including the effects of forces and motion on rotating objects, such as the forces on a bicycle wheel during braking.

Analyze hang time for rolling objects, applying forces, acceleration, torque, and rotational motion to illustrate various concepts within rotational dynamics.