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What is dynamics?

A description of how forces acting on a system result in motion of that system.


Example:  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 g=9.8 m/s2.  If we drop the ball, its dynamics are describe by:

Example:  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 t2, 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:





Solving the functional for a point mass


Example: 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 system.


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.

Momentum of particle i :

Here, because m is a scalar, velocity and momentum are vectors that are in the same direction.


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

Force 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.

Torque 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 point b.


a) If you start with a ball that is not spinning,

(b) and you twist it with a torque

(c) the ball will have an angular velocity that is in the same direction as the torque vector.


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.  Why?



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 velocity

linear acceleration due to angular velocity

What it means:  If I would like the cup to rotate with velocity     and acceleration         , 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 KE due to rotation

KE due to rotation 

caused by translation