Note on homework due on October 1 (Lecture 7)
The homework due on Monday asks for the Equations of Motion (EOM) on a couple
of occasions. A few words on the matter.
Newton's first two laws, as discussed today, can be summarized as F=ma. In
fact F=ma is an equation of motion. There will be two F=ma equations for
every point. Why is this an equation of motion?
Well, the answer lies in asking what we are trying to do in Dynamics. The
answer is that we are trying to figure out how things move. In other words,
what trajectory is going to be. For example, if you launch a rocket, you would
like to know where it will go (right?).
The location of a rocket can be characterized by its height, h, say. The
velocity h' is the first derivative. The acceleration is the second derivative
h'' (I am using primes because I can't put a dot on top of the h). We want to
solve for h. So we need an equation for h. It will be a differential equation.
If the rocket is just heading straight up, and you know the force of the
thruster, then the equation of motion is simply F=mh''. Solve this
differential equation (also called integrating the DE) and you will get h as a
function of time. Now, the linear acceleration, in say the x direction, of of
the rocket may not in general be just an h''. It might be more complicated
like sin(theta)h'' + thetadoth' or somethine. So the F=ma might be more
complicated, but it is still an equation of motion, a differential equation.
That is your quest.
A note on kinematic constraints. Every kinematic constraint does two things:
it adds an extra equation (the kinematic constraint) and it adds an extra
variable (the unknown force at the joint). Net-net, the number of unknowns
remains equal to the number of equations, and all remains hunky dory....