Class 091012

Submitting the qa's for this course is recommended, but is more or less optional:

• If you aren't making A's on your tests you probably need to do at least most of the q_a_'s.
• If you can read through the entire qa, do the problems pretty much in your head (with ballpark estimates on the arithmetic--no need for calculator-punching) or with a few scratches on paper, see that you understand the process and the units of the given solutions, and if necessary jot down enough notes to be sure you'll retain what you understand and use it when needed, then you don't need to waste time submitting them.
• If there is any problem you aren't completely sure of, then you should submit including a self-critique, for my feedback.  You don't need to answer every question; only the ones you aren't sure of. 
• If you don't answer a question I will assume you understand it thoroughly.  If you don't submit a q_a_ I will make the same assumption.
• If you find later that you need to work through a previously assigned q_a_ (perhaps because you didn't really understand what you thought you did), it will always be accepted and I will always post my response.
• Don't delude yourself into thinking you understand the given solution when all you're doing is recognizing most of the words.  If you can't work a problem without looking at the solution, you don't understand it.  If you're skipping problems or qa's, you should periodically 'quiz' yourself by going back and working a few of the problems without looking at the solutions.

Energy Conservation in terms of `dW_ON_NC, `dPE and `dKE

We have previously seen the following:

Using F_net = m a and vf^2  = v0^2 + 2 a `ds on an interval between two events, we obtain

F_net * `ds = 1/2 m vf^2 - 1/2 m v0^2.

Defining `dW_net as F_net * `ds, and KE = 1/2 m v^2, the work done on the interval by the net force thus becomes

• `dW_net = `d(KE).

F_net is the work done ON the mass m by the net force.  The distinction between the force and work ON the system vs. force and work done BY the system is very important.  We express the situation here explicitly by writing F_net_ON and `dW_net_ON.  Our statement is thus expressed as follows:

The force F_net_ON does work on a system of mass m as it moves through displacement `ds in the direction of F_net, the work being equal to F_net_ON * `ds.  We designate this as `dW_net_ON.  Then, for this interval,

• `dW_net_ON = `d(KE),

the work done ON the system by the net force is equal to the change in its kinetic energy.  This is the work-kinetic energy theorem.

We have also defined potential energy as being equal and opposite to the work done ON the system by conservative forces. 

It's pretty straightforward to break the work done by the net force into the work done by conservative forces (which will be associated with the change in PE) and the work done by nonconservative forces.  This will give us the more general work-energy theorem:

Any net force is a combination of conservative and nonconservative forces (one or both of which can be zero).  So F_net_ON can be expressed as

• F_net_ON = F_net_ON_cons + F_net_ON_noncons,

The work done by the net force is therefore equal to the work done by conservative forces plus the work done by nonconservative forces.  In symbols we have

`dW_net_ON = F_net_ON * `ds

= (F_net_ON_cons + F_net_ON_noncons) * `ds

= F_net_ON_cons * `ds + F_net_ON_noncons * `ds

= `dW_net_ON_cons + `dW_net_ON_noncons,

where we make the obvious definitions

`dW_net_ON_cons = F_net_ON_cons * `ds and

`dW_net_ON_noncons = F_net_ON_noncons * `ds.

Since

• `dW_net_ON = `dW_net_ON_cons + `dW_net_ON_noncons,

the work-kinetic energy theorem `dW_net_ON = `dKE becomes

• `dW_net_ON_cons + `dW_net_ON_noncons = `dKE.

We call this the work-energy theorem.

Now note that `dW_net_ON_cons is the work done by conservative forces on the system, which our definition of `dPE says to be equal and opposite the change in PE. 

So `dW_Net_ON = - `dPE and our work-energy theorem `dW_net_ON_cons + `dW_net_ON_noncons = `dKE becomes

-`dPE + `dW_net_ON_noncons = `dKE.

Let's abbreviate `dW_net_ON_noncons a bit as `dW_NC_ON, keeping in mind that this expression represents the total work done by all nonconservative forces acting on our system.  Our restatement is

-`dPE + `dW_ON_NC = `dKE.

We add `dPE to both sides to get a more intuitive form of the statement:

• `dW_ON_NC = `dKE + `dPE.

This says that the total work done on the system by nonconservative forces is equal to the sum of the changes in KE and PE.

Our focus for today will be to understand the three quantities, `dW_ON_NC, `dKE and `dPE in terms of systems we have observed in class.

Two-incline system

Students responding to the 09107 Class Notes clearly understood the following:

When the ball rolls down one ramp, we have seen that it loses gravitational PE and gains KE (it goes lower and speeds up). 

When it travels up the other ramp it gains gravitational PE and loses KE.

However its PE when it reaches its maximum displacement on the second ramp is less than it was when it was released on the first ramp.

With every roll down one ramp and up the other, the gravitational PE decreases, as does the maximum KE of the ball when it reaches the point between the two ramps.

We now ask where that PE goes.

• The simple answer is that it is dissipated, mostly by rolling friction between the ball and the ramp.
• Let's look into some of the details:

`q001.  Consider the ball rolling from rest down one ramp then rolling to rest as it travels up the other.

On this interval

• Does its gravitational PE increase or decrease, and why?
• Does its KE increase or decrease, and why?

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`q002.  The work-energy theorem says that  `dW_ON_NC = `dKE + `dPE. 

The right-hand side of this expression is `dKE + `dPE.  Based on your answers to the preceding question, is `dKE + `dPE positive, negative or zero?

Do you therefore conclude that `dW_ON_NC is positive, negative, or zero?

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`q003.  Assume that `dW_ON_NC is the result of rolling friction, so that `dW_ON_NC = F_frict_ON * `ds, where F_frict_ON is the force of rolling friction ON the system and `ds the displacement of the ball along the ramp.  (there is a slight problem with this wording due to the fact that the ramps are not parallel, but we're going to 'gloss over' this for now)

From the sign of `dW_ON_NC, do you conclude that F_frict_ON is in the direction of `ds or opposite the direction of `ds?

Based on your understanding of frictional forces, do you think that F_frict_ON acts in the direction of the ball's motion, or opposite this direction?

Are you answers to these questions consistent?

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Ball down ramp and to floor

The ball rolls from point A at the top of the ramp to point B at the bottom of the ramp then fall to point C, at which it makes its very first contact with the floor.

`q004.  What three events occur at these three points?

What happens during the interval between the first and second events (we will call this 'interval 1')?

What happens during the interval between the second and third events (we will call this 'interval 2')?

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`q005.  The only conservative force doing work during the entire interval from the first to the third event is the gravitational force.  (An excellent question that came up in one of the groups is whether the normal force is conservative.  For the steel ramp it turns out that it is, but the normal force acts in the direction perpendicular to incline, so there is no significant displacement in the direction of this force, and it has no significant effect on this energy situation.  In some situations the normal force could be associated with a significant amount of energy, but analysis of these situations are beyond the scope of the course, except for the general statement that the slight flexing of the ramp does create some thermal energy (heat) at the expense of the ball's KE.)

It is obvious that the KE of the ball increases during both intervals.

What nonconservative forces (if any) act during interval 1?

What nonconservative forces (if any) act during interval 2?

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`q006.  How does the presence of nonconservative forces during interval 1 affect the KE change during that interval (e.g., is the KE change different than it would be if the nonconservative forces were not present, and if so would the KE change be greater or less in the absence of these forces)?

How does the presence of nonconservative forces during interval 1 affect the PE change during that interval?

You may answer according to your intuition, but also attempt to identify `dW_NC_ON, `dPE and `dKE.  Specify whether each is positive or negative, and reconcile your conclusions with the equation `dW_NC_ON = `dPE + `dKE.

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`q007.  Suppose that for interval 1, `dW_NC_ON is -1000 ergs and `dPE is -100 000 ergs.  What then would be `dKE on this interval?  (you don't need to know what an erg is to answer this question, but you're probably wondering; an erg is a unit of energy, it take 10 000 000 ergs to make a Joule, and a Cheerio contains something like a thousand Joules of chemical energy, maybe 15% of which you could use to actually perform work).

What would you estimate to be the PE change between the end of the ramp and the floor?  What therefore do you think would be the KE change between event 1 and event 3?

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Rubber band block across table

Let event 1 be the release of the rubber band.

Let event 2 occur at the instant the rubber band loses contact with the block.

Let event 3 occur at the instant the block comes to rest.

`q008.  What happens during the interval between the first and second events (we will call this 'interval 1')?

What happens during the interval between the second and third events (we will call this 'interval 2')?

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`q009.  Is there any change in gravitational PE between event 1 and event 3?

Is there any change in elastic PE on interval 1?

Is there any change in elastic PE on interval 2?

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`q010.  Answer for each interval:

Is there any nonconservative force acting on the system, and if so what?

Does the KE of the system increase or decrease?

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`q011.  Assume that the rubber band forces are completely conservative (not so, but assume the ideal case).

Assume that frictional forces dissipate .0002 Joule of energy for every centimeter the block slides.

Suppose that the rubber band initially stores .01 Joule of potential energy, and that the block slides 10 cm while the rubber band is in contact with it. 

How much KE will the block have at the end of interval 1?

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`q012.  Continuing the preceding question, what is the change in KE during interval 2, what is the change in PE during this interval, and what therefore is the work done by the nonconservative force?

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`q013.  Continuing the preceding two questions, assuming that the frictional force is the only nonconservative force acting during interval 2, how far will the block therefore slide during this interval?

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Pendulum-projectile

`q014.  If a pendulum of length 10 cm is at its equilibrium position, then it is 10 cm vertically below the fixed point at which it is being held.

If it is then pulled back 6 cm in the horizontal direction, then since the string is still 10 cm long, the mass is still 10 cm from that fixed point.

How far, in the vertical direction, is the pulled-back pendulum below that fixed point?  (Hint:  sketch the figure, identify the necessary right triangle and use the Pythagorean Theorem)

How far does it therefore descend in the vertical direction, between release and return to its equilbrium position?

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`q015.  If the pendulum has a weight of .3 Newton, then how much work is done on it by the gravitational force, as it swings back to equilibrium?

By how much does its gravitational PE therefore change?

What therefore will be its KE at the equilibrium position?

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Homework:

Your label for this assignment: 

ic_class_091012

Copy and paste this label into the form.

Report your results from today's class using the Submit Work Form.  Answer the questions posed above.

You should know everything in the first six problems of Set 5 in the Introductory Problem Set, which will give you a good, and not difficult, introduction to working with vectors.  A link that should get you there is at http://vhmthphy.vhcc.edu/ph1introsets/default.htm .