DynamicsDependent and Relative Motion
Dependent Motion
• When two or more masses are connected so that the motion of one
of them results in the predictable motion of the others.
• Multiple dimensions but a single degree of freedom 1DOF vs 1D
• Use Geometry to find the relationship between the masses. Normally
a total length or distance.
• Take derivative of distance to find dependent velocities and
dependent acceleration.
• Notice we don’t need vectors, it is just like the 1D kinematic analysis
• Useful for pulley and linkage systems.
Dependent Motion Example
Relative Motion of two Particles
• We assume that the two particles are
translating only relative to each other!
• This means the two coordinate systems do
not rotate relative to each other. Cartesian
Coordinates!
• 𝑟𝐵 = 𝑟𝐴 + 𝑟𝐵/𝐴
• 𝑣𝐵 = 𝑣𝐴 + 𝑣𝐵/𝐴
• 𝑎𝐵 = 𝑎𝐴 + 𝑎𝐵/𝐴
• 𝑟𝐵/𝐴 is the relative position of B to A
• 𝑣𝐵/𝐴 is the relative velocity of B to A
• 𝑎𝐵/𝐴 is the relative acceleration of B to A
A
𝑟𝐵/𝐴
B
𝑟𝐴
𝑟𝐵
Notes on Relative Motion
• We can only use the conservation of momentum F=ma on an ‘inertial’
coordinate system. (This means the coordinate system is not
accelerating)
• It is often easier or even only possible to measure from a moving
coordinate system.
• Assuming A is the inertial coordinates (usually relative to the earth),
we first know our own motion rA, vA, aA
• We then measure the motion of our target rB/A, vB/A, aB/A from our
perspective.
• We can then find the absolute motion of B from the difference.
• In 2D, we usually end up with a system of two equations and two
unknowns.
Relative Motion and the Great Heliocentric
Debate
• Relative Motion from a moving platform appears very complex.
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What is the motion of the moon relative to the earth?
What is the motion of mars relative to the sun?
What is the motion of mars relative to earth?
The great Greek/Roman Deities
• Aristarchus (3rd Century BC) proposed a Heliocentric model of the solar system.
• Multiple Indian and Islamic Heliocentric models (3rd Century AD)
• Nicolaus Copernicus 1543 was the first person to create a mathematical
Heliocentric model.
• Tycho Brahe 1587-Weird Steam Punk type that measured astronomical motion
very accurately
• Johannes Kepler 1609-(Opportunistic Thief/Conman/My Hero) of Brahe’s work
and the first measurement of planetary motion as ellipse. First planetary motion
tables. The horoscope scam.
• Galileo Galilei 1616-First saw moons of Jupiter and Saturn’s rings. Didn’t believe
it until he showed his work to Kepler and the two matched.
Homework
• Problems 12-195, 201, 211, 214, 218, 221, 234