Dynamics of Point Masses and
Laws of Motion
Dr. Troy Henderson
Review of Dynamics
• Homework: Read the following
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•
•
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2
Ch. 1.2: Vectors
Ch. 1.3: Kinematics
Ch. 1.6: Time derivatives of moving vectors
Ch. 1.7: Relative motion
Dynamics/Mechanics
• Kinematics studies motion without cause
• Vector methods to describe position, velocity, and
accelerations
• The geometry of motion
• Kinetics relates forces and torques to motion
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Vectors
• The sum is defined by the parallelogram rule
• A+B=B+A
Figure 1.2 Parallelogram rule of vector addition.
Vectors
Magnitude of A:
Unit vector in the direction of A:
where
Figure 1.3 Three-dimensional,
right-handed Cartesian
coordinate system.
Vectors
The dot product of two vectors is a scalar defined as
where θ is the angle between the heads of the
vectors.
The dot product is zero if θ is 900.
Figure 1.5 The angle
between two vectors brought
tail to tail by parallel shift.
Vectors
• Scalar projection of B onto A
• Scalar projection of A onto B
Figure 1.6 Projecting the
vector B onto the direction
of A.
Vectors
• The cross product AxB=A B sinθ
• The cross product is not commutative and produces
a vector normal to both A and B
Vectors
• BAC-CAB rule
• Interchange of the dot and the cross
Kinematics
Figure 1.8 Position, velocity and acceleration
vectors.
Kinematics
• The distance of P from the origin
• The velocity v and acceleration a ,
• It is convenient represent the time derivative by
means of an overhead dot. For example,
Time Derivatives of Moving Vectors
Time Derivatives of Moving Vectors
Time derivative of a
rotating vector of
fixed magnitude:
Figure 1.15 Displacement of a rigid body.
Time Derivatives of Moving Vectors
• Inertial time derivative of vector B
Polar Coordinates
• Review polar coordinates and how to derive velocity
and acceleration from position vector!
15
Rotational Motion
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Kepler’s Laws (1609)
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Kepler’s Laws (1609)
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Proof of Kepler’s 2nd Law
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Kepler’s Laws (1619)
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Newton’s Laws (1687)
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Universal Gravitation
m1
orbital path
cm
r
r1
22

rcm

r

r2
m2

m1m2
F  G 2 rˆ
r
 20
3
-1 -2
G  6.674  10 km kg s
Example (Gravity on Earth)
Go
Eagles!
 GmM
F
rˆ
2
r
 GmM
 mgrˆ 
rˆ
2
r
GM 
g
2
r
3.986 E 5
2
g
 9.8 m/s
2
6378
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