Study Notes on Dynamics and Kinematics

12.1 - RECTILINEAR KINEMATICS: INTRODUCTION

  • Motion of large objects like rockets and cars is often analyzed as particles.
    • Example: Determine velocity and acceleration of a rocket based on its altitude over time.
    • Example: Analyze a sports car's motion along a straight road as a particle.
    • Dynamics involves the study of:
    • Statics: Study of bodies in equilibrium.
    • Kinematics: Concerned with geometric aspects of motion (displacement, velocity, acceleration).
    • Kinetics: Concerned with the forces causing motion.
    • Mechanics: Study of how bodies react to forces acting upon them.

12.2 - RECTILINEAR KINEMATICS: CONTINUOUS MOTION

  • A particle travels along a straight-line path defined by coordinate axis s.
    • Total distance traveled by the particle sT is a positive scalar representing the total length of the path.
    • Position vector r and scalar s represent the particle's position relative to the origin O.
    • Typical units for r and s include meters (m) or feet (ft).
    • Displacement of the particle is defined as the change in position.
    • Vector form: (\Delta r = r' - r)
    • Scalar form: (\Delta s = s' - s)
  • Velocity: Measure of the rate of change in position, a vector quantity.
    • Magnitude of velocity is known as speed (units: m/s or ft/s).
    • Average velocity during time interval ( ext{Δt}): (v_{avg} = \frac{\Delta r}{\Delta t})
    • Instantaneous velocity is the time derivative of position: ( v = \frac{dr}{dt} )
    • Speed is the magnitude of velocity: ( v = \frac{ds}{dt} )
    • Average speed is total distance traveled divided by elapsed time: ((v{sp}){avg} = \frac{sT}{\Delta t})

12.3 - ACCELERATION

  • Acceleration: Rate of change in velocity, also a vector quantity.
    • Typical units: m/s² or ft/s².
    • Derivative equations for velocity and acceleration can be manipulated:
    • ( ds = v dv )
    • Instantaneous acceleration is the time derivative of velocity.
    • Vector form: ( a = \frac{dv}{dt} )
    • Scalar form: ( a = \frac{dv}{dt} = \frac{d^{2}s}{dt^{2}} )
    • Acceleration can be positive (speed increasing) or negative (speed decreasing).
    • Average acceleration is given by ( a_{avg} = \frac{\Delta v}{\Delta t} = \frac{(v' - v)}{\Delta t} ).
    • Key equations include:
    • Differentiating position to get velocity: ( v = \frac{ds}{dt} )
    • Differentiating velocity to get acceleration: ( a = \frac{dv}{dt} ) or equivalently ( a = v \frac{dv}{ds})
    • Integrate acceleration to find velocity and position:
      • ( \int v dv = \int a dt )
      • ( \int s ds = \int v dt )

SUMMARY OF KINEMATIC RELATIONS: RECTILINEAR MOTION

  • Key equations for rectilinear motion under constant acceleration (acceleration is constant, (a = a{c})): 1) Motion equations relate:\n - Velocity: ( v = v{o} + a_{c} t )
    • Position: ( s = s{o} + v{o} t + \frac{1}{2} a_{c} t^2 )
    • Using calculus for relative motion and accumulated displacement as functions of time.

RECTILINEAR MOTION (cont’d)

  • Example problems include calculating:
    1) Average acceleration from a velocity-time graph.
    2) Position of a particle given initial velocity and constant acceleration: ( s = s{o} + v{o} t + \frac{1}{2} a_{c} t^2 ).
    3) Using integration to determine overall distance covered as velocity changes over time.

12.4 - GENERAL CURVILINEAR MOTION

  • The path of motion over a three-dimensional space may be tracked using coordinates.
  • Motion often involves the use of vectors to describe position, velocity, and acceleration.
  • Position vector is defined as ( r = x \hat{i} + y \hat{j} + z \hat{k} ).
    • Velocity vector is defined as the time-derivative of the position vector: ( v = \frac{dr}{dt} ).

CURVILINEAR MOTION: POSITION

  • Instantaneous velocity is tangent to the path at that position. Magnitude can be calculated by ( v = \frac{ds}{dt} ).

CURVILINEAR MOTION: VELOCITY

  • Acceleration vector represents the rate of change of the velocity vector and can be derived similarly through time differentiation: ( a = \frac{dv}{dt} = \frac{d^2r}{dt^2} ).

CURVILINEAR MOTION: ACCELERATION

  • Components may be resolved into x, y, z coordinates for analysis relative to fixed reference.
  • Acceleration magnitude follows the equation ( a = \sqrt{(ax)^2 + (ay)^2 + (a_z)^2} ).

12.5 - CURVILINEAR MOTION: RECTANGULAR COMPONENTS

  • Position defined in three-dimensional coordinates can apply in motion analysis to derive velocity and acceleration:
    • Position vector is expressed as ( \vec{r} = x \hat{i} + y \hat{j} + z \hat{k} ).
    • Velocity can also be derived with respect to each axis respectively.

12.6 - MOTION OF A PROJECTILE

  • Projectile motion treated as two dimensional motion.
    • Horizontal direction is uniform (constant velocity): ( x = xo + v{ox} t ).
    • Vertical direction influences motion via acceleration due to gravity: ( y = yo + v{oy} t - \frac{1}{2} g t^2 ).

12.7 - CURVILINEAR MOTION: NORMAL AND TANGENTIAL COMPONENTS

  • N-t coordinates describe particle motion along a curved trajectory.
    • Position and motion defined relative to curvature radius and tangential speed at any point along the path.

12.8 - CURVILINEAR MOTION: CYLINDRICAL COMPONENTS

  • Cylindrical coordinates apply when considering the radial and transverse components of motion respectively.

12.9 - ABSOLUTE DEPENDENT MOTION

  • Cable and pulley systems modify the relative motion of each participating object.

12.10 - RELATIVE-MOTION ANALYSIS OF TWO PARTICLES USING TRANSLATING AXES

  • Analyzing relative motion may require translation of coordinate systems relative to moving objects.
  • Position and velocity can be defined relative to fixed points or frames of reference involving multiple particles or objects in motion.