MECH2004 - Lecture 1

← Back to Home

Unit Outline - Slides

Assessments

• Quizzes (fortnightly) $-$ 25%
• Labs $-$ 15%
• Exam (closed book) $-$ 60%

---

Kinematics of Particles - Slides

Kinematics and Kinetics

• Kinematics $-$ describes motion of bodies without considering forces
• Kinetics $-$ studies relationship between motion and forces

Relating Displacement, Velocity & Time

For when acceleration is a function of position:
$vdv=\frac{ds}{dt}dv=\frac{dv}{dt}ds=ads$

Useful trigonometric integral identity:
$\int \frac{1}{A^2-(kx{)}^2}dx=\frac{1}{k}\arcsin (\frac{kx}{A})$

General solution for $\frac{d^{2}s}{d{t}^2}+k^{2}s=0$ is:
$s(t)=C_1\sin (kt)+C_2\cos (kt)$

• $C_1=\frac{v_0}{k}$ -> starting velocity
• $C_2=s_0$ -> starting displacement

See Wk1: Sample Problem 1 for application of above formulae

Coordinate Systems

Rectangular Coordinates

• Velocity Magnitude:
  $v=\sqrt{v_x^2+v_y^2}$
• Velocity Direction:
  $\theta ={\tan }^{-1}\frac{v_y}{v_x}$

Normal and Tangential Coordinates

Normal and Tangential Components

• When a particle moves along a curved path, it is convenient to describe its motion using normal ($n$) and tangential ($t$) coordinates
  • The origin ($O$) is located on the particle, thus it alongside the coordinate system moves with the particle

• The centre of curvature ($O^{\prime }$) lies on the concave side of the curve
  • The $n$-axis has positive direction towards $O^{\prime }$
• The radius of curvature ($\rho$) is perpendicular distance from curve to $O^{\prime }$

The position of the particle at any instant is defined by $s$

• $s$ is the distance along the curve from a fixed reference point

Link to original

$\rho$ $-$ radius of curvature
$\dot{\beta }$ $-$ angular velocity

• Velocity:
  $\vec{v}=v_t{e}_t=\rho \dot{\beta }{e}_t$
• Acceleration:
  $\vec{a}=\frac{v^2}{\rho }{e}_n+\dot{v}{e}_t$

Types of Motion

Relative Motion

Relative Position

• The absolute positions of two particles A and B with respect to the fixed $x,y,z$-reference frame are given by $r_A$ and $r_B$
• The position of B relative to A is represented by:
  $r_{B/A}=r_B-r_A$

Relative Velocity

• The velocity of B relative to A is represented by:
  $v_{B/A}=v_B-v_A$
  OR:
  $v_B=v_A+v_{B/A}$
  Note: $v_{B/A}=-v_{A/B}$

Relative Acceleration

• The acceleration of B relative to A is represented by:
  $a_{B/A}=a_B-a_A$
  OR:
  $a_B=a_A+a_{B/A}$

Link to original

Dependent Motion

• In kinematics problems, motion of one object can depend on another

To find an equation relating velocities, determine the relation in position coordinates and then derive the expression

• You don’t need to consider line segments that don’t change
  As they derive to 0!
• See examples in slides

Full Method:

1. Define position coordinates from fixed datum lines, along the path of each particle
2. Relate the position coordinates to the cord length
3. If a system contains more than one cord, relate position of a point on one cord to a point on another cord
   • Separate equations are written for each cord
4. Differentiate the position coordinate equation(s) to relate velocities and accelerations

Helpful video:

---

ENSC2004 - Lecture 13

Link to original

---

MECH2004 - Lecture 2