Chapter 2 – Kinematics of Particles Canvas NOTES

CHAPTER 2 - KINEMATICS OF PARTICLES

• Instructor: Farah Hammami

• Department: Mechanical and Aerospace Engineering, University of Houston

Concepts

• Kinematics focuses on the motion of bodies without considering the forces causing the motion.

• Particle: A body with physical dimensions that can be treated as a point due to its small size relative to the path's curvature.

Choice of Coordinates

• Different coordinate systems include:

  • Rectangular

  • Cylindrical

  • Spherical

  • Normal-tangential

• Reference Frames:

  • Fixed Reference Frame (Absolute Motion)

  • Moving Reference Frame (Relative Motion)

Rectilinear Motion

Definition

• Motion along a straight line.

Properties

• In rectilinear motion, velocity (v) and acceleration (a) are vectors but can be simplified since the direction is easily identified by the path line. Vector notation can often be omitted.

Analytical Integration (1)

• Constant acceleration:

  • Equation: a = f(t)

• Integration can be employed when acceleration is a function of time.

Analytical Integration (2)

• Acceleration can also be expressed in terms of velocity or displacement:

  • a = f(v)

  • a = f(s)

Example 1

• A lake's suitability for aircraft landing is analyzed with:

  • Touchdown speed: 100 mi/hr

  • Desired reduction: from 100 mi/hr to 20 mi/hr in a distance of 1500 ft.

• Need to find the value of K in the equation a = -Kv² and calculate the time (t) during the specified interval.

Example 2

• A train decelerates from 80 mi/hr to 60 mi/hr over a distance of ½ mi while a car attempts to beat the train, requiring calculation of the car's acceleration (a) to win by 4 seconds and the car's velocity at the crossing.

Example 3

• The acceleration of a particle is defined as a = 2t - 10.

  • Need to determine velocity and displacement functions based on initial conditions.

Example 4

• Given velocity function: v = k√s, with k = 0.2 mm^(1/2)s^(-1).

  • Calculate position, velocity, and acceleration functions.

Plane Curvilinear Motion

• Describes particle motion along a curved path in 2D.

• Coordinate systems available:

  • Rectangular

  • Normal-tangential

  • Polar

Special Application – Projectile Motion

• Aerodynamic drag and Earth's curvature are neglected.

• The only acceleration considered is downward acceleration due to gravity (g).

Example 5

• Water nozzle's speed (45 ft/sec): find the angle (θ) for optimal landing distance after clearing a wall.

Constrained Motion of Connected Particles

General Principles

• Essential to identify constraints and differentiate between fixed and moving components.

• 1-D and 2-D freedom equations for constraints concerning length.

Example 17

• For variable y, determine the upward velocity of A in terms of the downward velocity of B, considering the rope's constant length.

Example 18

• Particle A moves in a circular arc while a counterweight B moves vertically. Derive the relationships between their velocities based on angle θ.

Relative Motion (Translating Axes)

• Fixed Reference Frame: Observations made from the ground.

• Moving Reference Frame: Observations made from another moving object.

Example 14

• A jet's velocity is observed from another aircraft to determine actual speed as well as perceived velocity.

Example 15

• Analyze relative speeds and accelerations of a train and car approaching an intersection.