A description of how
forces acting on a system result in motion of that system.
A ball of mass m is held 20 m off the ground. The
force acting on the ball is the force of
gravity: f=-mg where
If we drop the ball, its dynamics are describe by:
Dynamics of a single joint system with mass m, joint
viscosity b, and length l.
Imagine a point mass that is
at position x1 at
time t1 and
ends up at position x2 at
for example: a ball falling from a height.
The trajectory that it follows to get to x2 is
only one of an infinite number of pathways that it could have
followed. But the point
mass will always follow that same trajectory x(t),
given the same initial conditions.
What is so special about the trajectory x(t)
that it actually does follow?
The trajectory x(t)
minimizes the following cost function:
the functional for a point mass
dynamics of a point mass
If there are external forces
(from motors, muscles) acting on the system:
The primary problem in
dynamics is to find an expression for the kinetic energy of the
Vectors a and b
define a plane.
Vector n is
perpendicular to that plane.
The angle between a and
b is q.
The magnitude of n
increases like a screw rising out of the plane when we rotate vector
a to reach vector b.
Note that when a and b
are parallel, a x b =0
Assume that an object is
composed of small “particles” of mass call mi.
Total mass of the object is:
Position of particle i
is specified by vector xi
Center of mass xc is
the weighted average position of the particles.
Once an object starts moving,
linear momentum describes the tendency for objects to continue moving
in the same direction.
Momentum refers to mass times
velocity of a particle.
Here, because m is a
scalar, velocity and momentum are vectors that are in the same
Linear momentum of an object
is equal to the total mass times the velocity at the center of mass
of the object.
If an object has momentum p,
it will continue to have that momentum unless a force acts on it.
The rate of change in linear
momentum is equal to the force.
The particle with mass m
is being affected by force f and has velocity
is the rate of change of linear momentum.
Angular momentum is a measure
of an object’s rotational motion.
It describes the tendency for objects to continue spinning
about their particular axis.
If an object has angular
momentum h, it will continue to have that momentum unless a
torque acts on it.
is the rate of change of angular momentum.
Imagine that the particle is
connected to point b with weightless rod of length r.
Note that if point b
moves, torque on that point will depend on how we define velocity of
a) If you start with a ball
that is not spinning,
(b) and you twist it with a
(c) the ball will have an
angular velocity that is in the same direction as the torque
The torque on the seesaw
obeys the “right hand rule”.
If the index finger points
along the lever arm and the middle finger points along the force,
the thumb points along the torque.
When you cut cardboard with
a pair of scissors, it is best to move the cardboard as close as
possible to the scissors’ pivot.
In an amusement park ride, a
large “cup” can hold a child as it rotates about a center point.
Objective: to describe the forces that act on the cup.
linear velocity due to angular
due to angular velocity
What it means: If I
would like the cup to rotate with velocity
, I need
to produce force
The child in the cup feels the force “produced” by his mass:
she feels a centripetal force
pushing her outward as she rotates.
Gravity also produces a force on our body.
Simulators use gravity to fool the brain into thinking
it is feeling a force due to motion.
KE due to translation
due to rotation
KE due to rotation
caused by translation