LEARNING OBJECTIVES:

1) I will be able to explain the concept of a 'dot' product to a bright, articulate algebra 2 student after today's class.

WORDS O' THE DAY:

• Work (Force through displacement)
• Dot Product ("scalar product")
• Work ("Newton meter (Nm)" or "Joule (J) ")

FORMULAE OBJECTUS:

• Work = fdcosθ (including the integral form)

WORK O' THE DAY:

We had a question on variable forces -- here's an interesting example. There *appears* to be a problem with the logic here.... see if you can identify it and see if you can see how the author works around it.

I'm gonna call a wee bit of an audible here... for your second homework problem, calculate the normal force acting on the box (that should be interesting).

Take a few moments and make some written comments on what you learned from doing the homework.

Be prepared to speak articulately to the class on this point.

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Let's start recording our formulae objectus. You MUST be able to use and recite (on deman) any and all formulae we have used THIS SEMESTER!!!

Recall that our definition of work for an object undergoing a constant force was:

W = Fxcos(θ)

Where F = the force being the applied force and

x = the displacement the object experiences

NOTE: The book sometimes uses 'r' to represent the displacement of the object for expedience.

• The "r" in that case can be thought of some radial distance emanating from the point of the source and
• θ = the angle formed between the displacement and the applied force.

 

We will start using 'r' now as we move away from motion in just the 'x' or 'y' directions.

So...

W = Frcos(θ)

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Notice also that the definition of a scalar ("dot") product is:

ABcos(θ)

It is certainly no accident that the definition of a scalar product is the same form as the definition of work (since work is a scalar quantity)

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Let's take a look at dot product math HERE

Now let's take a look at how our book introduces dot products: (example 7.2 on page 182)

1. Take a few moments to review that problem with your groupies... take particular note of the part of that example where the 'dot' products of i and j vectors = 0.

2. Now take a few more minutes to follow through with that math...

3. What are the real world implications of that math?

Now let's take a look at part 2 of that problem which gets a tad more difficult:

• The problem tells us that A(vector) = 2i + 3j
• The problem tells us that B(vector) = -i +2j

It's fairly straightforward to calculate the the lengths of the hypotenuses' of each vector since the 'i' and 'j' quantities are basically the two legs of a triangle:

root(2² + 3²⁾ = root(13)

Similarly

root(-1² + 2²) = root(5)

Here's where (I think at least) it gets a tad tricky:

Consider the definition of a dot product:

A·B = ABcosθ

if we want to find the angle θ, we have to realize that there are two vector quantities present (Force and displacement) AND THE ANGLE BETWEEN THOSE TWO VECTORS is θ

so... now let's do some basic rearranging:

A·B/AB = cosθ

Cos^-1(A·B/AB) = θ

so now we calculate the 'dot' product of A and B and then divide that quantity by the product of the magnitude of A and B

*YIKES*

Now do 7.3--- oh and add in a part where you have to find the angle that the force makes to the displacement (OUCH)

So...

1. write down the vector equations
2. Close the book
3. Do as much as you can (using your notes is ok, using the book is a no-no)
4. Fight with the odd parts as best you can, working with your group to *UNDERSTAND* what you need to do
5. take a few peeks at the book if you need to.

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HOMEWORK

• Do homework problems 11 and 14 on page 205
• Read the next section on work done by a spring *lightly*. The derivations get a little bit over the top (inho). However, make sure you *ARE* comfortable with the definition of work done by a spring *AND* why that value is negative.
• Do example 7.5:
  • Read the problem
  • Sketch the situation (I know it's in the book...sketch it anyway from just inspecting the problem)
  • Write a qualitative analysis using what you've learned about work done by a spring
  • Solve the problem *without* peeking at the book for as long as you can
  • Open the book... look for CONCEPTS that stumped you...
  • Close the book
  • repeat