Week 2

There are two videos to view this week.

Robert Brooks shows some interesting robots that he has built.  They demonstrate that there are fairly sophisticated tasks that robots can perform.

Michael Shermer is a person that examines claims.  Since there are many astounding claims that people report. It is necessary to evaluate these with a skeptical perspective.  The nature of science is to carefully analyze data, models, theories and ideas.  Science only includes things that can be tested. A belief therefore falls out of the realm of science.

GSCI 101 9:30-10:45  [10:10-10:15]

[Force table, spring, ruler (normal force), rope (tension)]

{people pushing and rotating}

Robot

1. brain == microprocessor, computer
2. Structure (static)
3. movement
4. sensing

Structure

• balance or equilibrium
•   sum of forces and sum of the torques=0

Consider what this means.  First notice that mathematics is a language that provides a concise statement. If you learn the language this type of translation is typically straightforward.

Suppose you find five forces that are acting on our robot structure.

Now lets consider something a bit more difficult.  What do the arrows mean?

coordinate systems label POINTS

space

time

Point in space is labeled by three numbers and if it is an event it is labeled by another number time.

(x,y,z) at time t

SCALARS (amount of money, age, number of students in the classroom)

These quantities can be manipulated mathematically to get sums and differences.

Realize that we must add similar quantities.  č UNITS

It does make sense to combine different units: rates, speed, …

A point in space is not a simple quantity but a group of numbers. Can we manipulate complex groups.  We know the answer. If you are at point A and you need to get to point B can you find the information for the path from the points A, B ?

But now the information is more complex and notice that this complexity can be contained by an arrow.  The arrow tells us how far AMOUNT or MAGNITUDE and in which direction.  We need to figure out how do we add these more complex entities.

Stand and push. Notice when there is balance. Push same amount but in opposite directions.

Now push on a third person.

Pick an origin. Label some points and show position vectors. Imply that to get to B from A we can think of this as the path generated by moving AčB.

A is the path from out reference point or origin to A

B ‘’’’’’’                                                                   B

Graphical Method: (show on board).  See appendix C. Further examples can be found online for example at  http://en.wikibooks.org/wiki/FHSST_Physics/Vectors/Addition.

 ASIDE There is a useful trick associated with this idea of the general mathematics.  If I know a set of rules and a classic example then perhaps new problems are trivial.   Suppose you learn that ten pennies add to 10 that two nickels add to 10 …. Now the problem is if I have ten people how many are there? If I have two families of five how many people are there ?   You might look at me and say this is obvious what are you   Lets look at a graph of position vs time do flat line then do a sloped line 1-    You know a lot already PLEASE USE YOU INFORMATION!!! 2-    Get the velocity graph 3-    Get the acceleration graph   If you can apply the same basic rules to two different things then you only need to figure out these rules once.  The application to the second thing will be easy.  Method: change the names. Apply the rules. Change the names back. Example: Two families of five  becomes to groups of 5 pennies. We know 5 pennies + 5 pennies = 10 pennies Change the names back. 5 people + 5 people = 10 people

The power of vectors is derived from the fact that many interesting quantities behave in the same manner.

Force, velocity, acceleration, momentum

I always think that these things need to have a “HOW MUCH” and “WHICH DIRECTION” piece.

To complete this discussion two more details:

1. Simple objects such as money and people that do not require this more complex form of addition are known as SCALARS.
2. If you want to multiply (another standard mathematical operation) you need to extend the rule for vectors.  We will NOT multiply two vectors [These rules exist.]  We will define multiplication of a vector times a scalar.  The result is probably obvious.  I can make the velocity 3 times bigger or I can reduce the force by ˝.  Here we multiply the scalar [3, ˝] times the magnitude and don’t change the direction.  [While it is important to recognize that this type of multiplication is distinct from the multiplication of two simple numbers, you already know how to do the problem so incorporate you intuition.]

NEXT LECTURE č finish chapter 1 by introducing speed velocity acceleration.

NEXT WEEK start chapter 2  Newton’s Laws. What is the effect of a force on an object when NOT in balance.

Chapter 1:

Inertial mass

forces  čpush or pull

weight (gravity) č well known force  (g=9.8 m/s-s

which means: magnitude mg, direction down

difference between inertial mass and gravitational mass

 Inertial mass measures the resistance to motion. Gravitation mass measures how attracted to bodies are under the influence of gravity.   Interesting thing is that they are related.  The heavier the object the harder it is to move.  Of course you say but if that is your point of view then your missing the argument. The statements above are two independent facts. There is no reason for these two quantities to be related.  They are we have intuitively learned this connection and so we assume it is a required connection but it is not.   Einstein puzzled over this and found a way within the theory of gravity to link these two facts.

equilibrium

static č balance  small object ∑ forces=0

extended ∑ Torques=0

rotational force is a torque

speed velocity

instantaneous speed

acceleration

Problem

forward and reverse kinematics for a manipulator.

Forward is you drive the joints and compute where the hand is.

Reverse is that you choose the goal ( where the hand will go)

There is a complicated process that must be carried out.

Let me just explain a bit to you.

There are ways to express vectors component form (x,y,z). We mentioned this when we talked about the complexity of a point as an introduction to the need to introduce vectors.

Lets look at 2 representations of the information in a vector

 Graphical but think length and angle or direction. č Precise

3 DOF for a vector č 3 pieces of information

6 for a rigid body

Discussed rigid bodies,  vectors, numbers, types of forces, equilibrium and graphical addition of forces.

The point is you can introduce other mathematical operations other than addition.

?? Think of any.

Rotation and translation, stretch.

Rotaions are sweet in that you can build a rotation matrix and use a new form of multiplication to arrive at the rotated vector

GSCI 101 9:30-10:45  [10:10-10:15]

[Force table, spring, ruler (normal force), rope (tension)]

{people pushing and rotating}

Review-----------------------

Robot

1. brain == microprocessor, computer
2. Structure (static)
3. movement
4. sensing

Structure

• balance or equilibrium
•   sum of forces and sum of the torques=0

Consider what this means.  First notice that mathematics is a language that provides a concise statement. If you learn the language this type of translation is typically straightforward.

Suppose you find five forces that are acting on our robot structure.

end of review------------------

Force table----------------

If you can identify all of the forces acting on an object at rest and you sum them up using the vector nature of forces then you can ask if the sum is zero or non zero.  The non zero case puts the body in equilibrium [ignoring the most general case of extended objects which still can twist and turn even when the total force is zero].  The result is that the object remains at rest.  ROBOT DOES NOT FALL DOWN.

Is there an easy way to balance forces so that our robot doesn’t fall?

Build it like a car

 Forces to know   [Unit of force is the Newton] gravity mg down Support force or Normal Automatic force of a surface [away from the surface] Friction Force  that resisits motion [surface may exert along its sur.] Net Vocabulary word for the total force Spring force Force that a compressed  or stretched spring exerts Air resistance Type of frictional Tension Stretching force that ropes exert Tension is a complicated phenomena.  Sometimes it is easier to examine ropes used to support or pull something [under dynamic equilibrium or balance] At each point the rope is pulled equally in both directions. The amount or magnitude of this pull is the tension. Tension is the amount of force applied to the end. For balance each end must have a force equal to the tension applied. Rope forces act along the direction of the rope. A pulley or person (on a rope swing). Feels forces equal to the number of contacts to the rope. (On the swing imagine that your hands are supporting you and then you see that each hand will feel a rope force up.)

NEXT LECTURE č finish chapter 1 by introducing speed velocity acceleration.

NEXT WEEK also start chapter 2  Newton’s Laws. What is the effect of a force on an object when NOT in balance.

Chapter 1:

Inertial mass

forces  čpush or pull

weight (gravity) č well known force  (g=9.8 m/s-s

which means: magnitude mg, direction down

difference between inertial mass and gravitational mass

 Inertial mass measures the resistance to motion. Gravitation mass measures how attracted to bodies are under the influence of gravity.   Interesting thing is that they are related.  The heavier the object the harder it is to move.  Of course you say but if that is your point of view then your missing the argument. The statements above are two independent facts. There is no reason for these two quantities to be related.  They are we have intuitively learned this connection and so we assume it is a required connection but it is not.   Einstein puzzled over this and found a way within the theory of gravity to link these two facts.

equilibrium

static č balance  small object ∑ forces=0

extended ∑ Torques=0

rotational force is a torque

speed velocity

instantaneous speed

acceleration