Particle Kinematics

1
Particle Kinematics
Inertial frame non accelerating, non rotating
reference frame Particle point mass at some
position in space
Position Vector
Velocity Vector

• Direction of velocity vector is parallel to path
• Magnitude of velocity vector is distance
  traveled / time

Acceleration Vector
2
Particle Kinematics

• Straight line motion with constant acceleration

• Simple Harmonic Motion

• Circular Motion at const speed

3
Visualizing Velocity and Acceleration

• Direction of velocity vector is parallel to path
• Magnitude of velocity vector is distance
  traveled / time

• Acceleration component parallel to path is
  caused by change in speed of particle. This
  component has magnitude
• Acceleration component perpendicular to path is
  caused by change in direction of particle.
  This component has magnitude

4
Newtons laws

• For a particle
• For a rigid body in motion without rotation, or a
  particle on a massless frame

You MUST take moments about center of mass
5
Calculating forces required to cause prescribed
motion of a particle

• Idealize system
• Free body diagram
• Kinematics
• Fma for each particle.
• (for rigid bodies or frames
  only)
• Solve for unknown forces or accelerations

6
Deriving Equations of Motion for particles

• 1. Idealize system
• 2. Introduce variables to describe motion(often
  x,y coords, but we will see other examples)
• 3. Write down r, differentiate to get a
• 4. Draw FBD
• 5.
• 6. If necessary, eliminate reaction forces
• 7. Result will be differential equations for
  coords defined in (2), e.g.
• 8. Identify initial conditions, and solve ODE

7
Motion of a projectile
8
Work and Energy relations
Rate of work done by a force (power developed by
force)
Total work done by a force (power developed by
force)
Rate of work done by a moment (power developed by
moment)
Kinetic energy
Power-kinetic energy relation
Work-kinetic energy relation
9
Motors and Actuators
Power developed by actuator
Power developed by a motor T is motor
torque!
10
DC brushless Motor Characteristics
Rated voltage No load speed No load current Stall
torque Stall current
Motor current-voltage-speed eqn
Motor torque-current-speed relation
Motor torque-voltage-speed relation
Motor torque-speed relation
11
Potential energy
Potential energy of aconservative force (pair)
12
Energy relations for conservative systems
subjected to external forces
Internal Forces (forces exerted by one part of
the system on another)
External Forces (any other forces)
System is conservative if all internal forces are
conservative forces (or constraint forces)
Energy relation for a conservative system
Kinetic and potential energy at time
Kinetic and potential energy at time
Work done by external forces
13
Power transmission in machines

• If you can
• Neglect mass of system (so T0)
• Assume all internal forces are workless (no
  deformation, e.g. in springs, no friction) so V0

Total rate of work done by externalforces during
motion of machine is zero
14
Impulse-momentum relations
Impulse-momentum for a single particle
Linear Impulse of a force
Linear momentum of a particle
Impulse-momentum relations
Impulse-momentum for a system of particles
Total external impulse
Total linear momentum
Conservation law
15
Collisions
16
Angular Impulse-Momentum relations(for a single
particle)
Angular Impulse
Angular Momentum
Impulse-Momentum relations