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PHY 111: Module 9A Student Note Packet (Chapter 7) Read the OpenStax College Physics textbook and reference other sources, as indicated, to fill in these notes. Take time to complete these notes thoughtfully. Done well, these note packets will serve as your notes, your study guide, and problem-solving tutorial for each chapter. Unless indicated otherwise, use your textbook to answer the questions. VIDEO 9.00- Introduction to this module

Introduction This is an exciting chapter, because it gives us a way to solve problems that would take a long time to do with our knowledge thus far and it also allows us to solve problems that would be impossible to solve with our current knowledge, from chapters 1-6. Work is defined as force over a distance and can be calculated as: W=Fdcos(θ) where F is the magnitude of the force, d is the magnitude of the displacement, and θ is the angle between the force and displacement vectors. The other important equation to know for this unit is: Ei + Wnc = Ef where Ei is the initial mechanical energy of the system, Wnc is the work done by non-conservative forces, and Ef is the final mechanical energy in the system. Open the PhET skatepark simulation. Click on "Intro." Check the box for the bar graph. Note that the skater has no energy at this time. Grab the skater with your mouse and move him upward. Notice what happens to his energy. Now, drop him about three-quarters of the way up the ramp and watch him skate back and forth. Click on slow motion and watch the bar graph. What is happening to his total amount of energy?

What is happening to his gravitational potential energy (listed on the graph as simply 'potential')? When is it a maximum? minimum?

VIDEO 9.01- Energy Skate Park

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What is happening to his kinetic energy? When is it a maximum? minimum?

Think about the above equation and how it could apply if we define "initial" as before you picked him up and "final" as after you let go. He had zero initial energy. Did you do work on the skater during the time between initial and final, as they were just defined? Explain.

While you may still have a lot to learn about work and energy, this exercise gives you a foundation for learning more.

Section 7.1 Work: The Scientific Definition Looking at Figure 7.2, Answer the following 2 questions: 1. How much work does the person do on the briefcase? Explain.

2. Can work be negative? Explain.

• • •

Positive work tells us that the system gains energy. ___________ work tells us that the system loses energy. Zero work tells us that the total energy of the system does not change.

What are the units of Energy? What are the units of Work? What are three different ways to express this unit?

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More things to know about work: •

Work is a scalar quantity, and it can be positive or negative. F and d will ALWAYS be positive; the sign of W is determined by the cos(θ) result.

•

If many forces, F1 , F2 , F3 , .... act on an object, the net work is:

G G

G

Wnet = W1 + W2 + W3 + W4 + ..... •

Since the equation for work has cosine in it, it is good to learn a little more about the cosine function. The graph of cosine as a function of angle looks like this: cos(0°) = +1 cos (90°) = 0 cos(180°) = -1

Which means that the closer the angle between the force and the displacement is to 0 degrees, the more work the force will do. If the angle between the force and the displacement vector is between 90 and 270 degrees, the force will do negative work and remove energy from the system.

Work Diagrams Work diagrams are useful in helping you to determine the correct angle to use in the work diagram and will be a required part of your solution when calculating work! Steps to drawing a work diagram: • • • •

Draw an arrow to represent the displacement vector, making sure to draw the arrow in the actual direction the object moves Draw an arrow to represent the force, making sure to draw the arrow in the actual direction of the force, just as it would appear on an FBD Label the angle between the two vectors you drew, whether it is 0°, 90°, 180°, or some other angle Repeat for each force for which you are calculating the work

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Ranking Task 1: Equal Forces on Boxes In the figures below, identical boxes of mass 10 kg are moving at the same initial velocity to the right on a flat surface. The same magnitude force, F, is applied to each box for the distance, d, indicated in the figures. Rank these situations in order of the work done on the box by F while the box moves the indicated distance to the right. Negative values of work rank lower than positive works.

Least 1______ 2______ 3______ 4______ 5______ 6______ Greatest OR, all of the boxes have the same work done on them by the force, F. _______ OR, none of the boxes have work done on them by the force, F. _____________ Explain the reasoning for your ranking.

Ranking Tasks are adapted from A Selection of Ranking Tasks, T.L. O’Kuma, D.P. Maloney and C.J. Hieggelke, ed. SOLUTION 6.01- Ranking Task Solution

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Work Example 1: A block of mass m is acted on by applied constant forces F1 and F2 and moves a distance s across a rough horizontal surface. Find the net work done by all the forces acting on the block. Ignore air resistance. F1 F1 = 100N φ = 30° φ F2 F2 = 200N d = 30 m d

fk = 50N Strategy:

First draw a FBD to determine what forces act on the block. Then calculate the work done by each force. To make finding the angle between vectors apparent, draw a Work Diagram (NOT FBD) for each force. Add together the work done by each force to find the net work.

* *

*

Force

30°

●

Work W F1 = (100N)(30m) cos(30 )

Work Diagram (NOT FBD) F1

G F1

FBD

W F1 = +2598 J

d

Wnet = W1 + W2 + W3 + W4 + .....

VIDEO 9.02- Work Diagrams and work calculations

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Example 2: A 50 kg skier is being pulled 50 m up a 30° slope at constant speed by a tow rope. The tension in the cable is 280N and a frictional force of 35N opposes the motion. Calculate the net work done by all the forces acting on the skier

•

Draw a FBD and determine what forces act on the block.

•

Calculate the work done by each force and then add.

Force

Work

Work Diagram (NOT FBD)

Wnet = W1 + W2 + W3 + W4 + .....

●

VIDEO 9.03- Work Diagrams and work calculations

W net =

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7.2 Kinetic Energy (and the Work-Energy Theorem) Write the equation that defines kinetic energy and explain what each symbol means, as well as listing the SI unit for each quantity.

This section also discusses the Work-Energy Theorem, but we will wait and use the Conservation of Mechanical Energy equation. These two ideas (and their corresponding equations) are consistent with one another, but it is sometimes difficult for beginning students to understand their relationship, so we will focus on the Conservation of Mechanical Energy equation.

7.3 Gravitational Potential Energy Write the equation that defines the change in gravitational potential energy and explain what each symbol means, as well as listing the SI unit for each quantity.

We can also define a certain height to have a gravitational potential energy of zero. This allows us to define the gravitational potential energy (and not just the change in gravitational potential energy: PEg = mgh

7.4 Conservative Forces and Potential Energy Explain how the textbook defines a conservative force.

Write the equation that defines the spring potential energy and explain what each symbol means, as well as listing the SI unit for each quantity.

VIDEO 9.04- Discussion of where the spring potential energy equation comes from

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The two types of conservative forces that are studied in this course are the force of gravity and the spring force. Note that the spring force can also be called the elastic force, which could describe the force exerted by a something that stretches (e.g. fabric, a rubber band) or bends (e.g. a plastic spoon made into an improvised catapult).

Good to remember!

Based on the prior paragraph, we can deduce, for a force we see in this

course, if the force is not the force of ____________ or the ___________ force, it must be a non-conservative force. Put another way, we can list these two conservative forces in the left column and then every other force from this course in the right column:

Conservative forces ____________ And _____________

Non-conservative forces Friction, applied, tension, normal, air resistance (and any other force you’ll see in PHY 111, other than the two in the first column)

What are the two equations the textbook uses to describe the conservation of energy in equation 7.48?

7.5 Nonconservative Forces and Friction What a strange and potentially confusing title to section 7.5! Which of the following is most like the title? a. animals and flowers b. fruit and apples c. countries and Maryland Friction is a conservative force! A better title would be “Friction and other nonconservative forces.” How does the textbook define a nonconservative force?

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The textbook does a good job connecting the Work-Energy theorem to equation 7.58. However, in place of the work-energy theorem and in place of the equations shown in 7.48, you can just use equation 7.58, which says:

And your equation sheet breaks apart the two different types of potential energy on both sides, to give a 7-term equation:

VIDEO 9.05 Conservative and Nonconservative Forces

7.5 Conservation of Energy The law of conservation of energy states: a. The total energy in the universe is constant in any process. b. Energy may change form and or move from one system to another, but the total remains the same. c. You should turn off your lights when you leave the room. d. Both (a) and (b) While it’s good to turn off the lights when you are not using them to conserve energy, that’s not what the law of conservation of energy is all about. Hopefully, you answered (d) to the above equation.

The above 7-term equation, applies to mechanical energy. There are 3 types of mechanical energy in this equation: • • •

gravitational potential energy spring (or elastic) potential energy kinetic energy

Your textbook takes the above equation, from your equation sheet, one step further. It adds in all other types of energy, using the symbol OE. This equation, presented by the book as equation 7.67, is the most complete version, though we will stick to our 7-term version for now.

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Applying the Law of Conservation of Energy to Various Situations 1. Draw a sketch of the situation (not a FBD or a Work diagram, just a sketch). 2. Identify and label the point that you will be using as the initial state with an ‘i’. Identify and label the point that you will be using as the final state with an ‘f’. 3. Draw a horizontal line and label it as h=0 and/or PEgravity = 0. It’s up to you where you pick, but usually it is best to pick the height of the lower of the initial and final states. 4. For the initial point, list hi, Δxi, and vi and put numerical values if you know them and put a question mark if you do not know one. For the final point, list hf, Δxf, and vf and put numerical values if you know them and put a question mark if you do not know one. 5. Draw a FBD (if the forces change between your chosen initial and final states, you’ll need to draw more than one). Circle any non-conservative forces acting on the object of interest between the initial and final states. 6. Draw a work diagram for each force you circled in the previous step. Calculate the work or write the symbolic equation, with as much information as you know, for each non-conservative force. 7. Write out the seven-term conservation of energy equation. Cross out terms that are equal to zerosee the values you wrote down in Step 4 to help you. 8. Fill in known information and solve for the unknown value.

How to choose the h=0 line wisely If we were to drop a 1 kg hammer from the top of the mirror, shown on the next page, and wanted to find out how fast it is going when it gets to the top of the couch, we could define our h=0 line to be the floor. When we define our initial and final states, then: Initial = moment of release, at top of mirror; hi = 2.1 m Final = when it gets to the top of the couch; hf = 0.8 m When we apply our 7-term conservation of energy equation, both PEg,i and PEg,f will remain in the equation. However, if we draw our h=0 line as a horizontal line at couch height, then : Initial = moment of release, at top of mirror; hi = 1.3 m Final = when it gets to the top of the couch; hf = 0 m And we would have one less term (because PEg,f = 0) and it would make the math easier! And easier is better, as long as the physics is correct and it still leads us to the correct value!

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PEg=mgh 1. Floor as h= 0 m

2. Top of Sofa as h =0m

2.1 m

1.3 m

1.2 m

0.4 m

0.8 m

0.0 m

0.4 m

-0.4 m

0.0 m

-0.8m

With the floor as the place where h=0 and PEg=0, the 7-term equation simplifies to: PEg,i = PEg,f + KEf which we can use to solve for vf: (1 kg)(9.8 m/s2)(2.1 m) = (1 kg)(9.8 m/s2)(0.8 m) + ½(1 kg)(vf)2 which, when solved, yields: vf = 5.05 m/s OR, using the top of the couch as the place where h=0 and PEg=0, the 7-term equation simplifies to: PEg,i = KEf which we can use to solve for vf: (1 kg)(9.8 m/s2)(1.3 m) = ½(1 kg)(vf)2 which, when solved, yields: vf = 5.05 m/s (same answer either way!!)

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Ranking Task : Sliding Masses on Incline I Rank, in order from least to greatest, the final kinetic energies of the sliding masses the instant before they reach the bottom of the incline. All surfaces are frictionless. All masses start from rest.

Least 1______ 2______ 3______ 4______ 5______ 6______ Greatest OR All masses have the same final kinetic energy. __________ Explain the reasoning for your ranking.

Ranking Tasks are adapted from A Selection of Ranking Tasks, T.L. O’Kuma, D.P. Maloney and C.J. Hieggelke, ed. VIDEO 9.06- Ranking Task Explanation

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Ranking Task : Sliding Masses on Incline II Rank, in order from least to greatest, the change in gravitational potential energy of the sliding masses from the top of the incline to the bottom of the incline. All surfaces are frictionless. All masses start from rest at the top of the incline.

Least 1______ 2______ 3______ 4______ 5______ 6______ Greatest OR All masses have the same change in gravitational potential energy._______ Explain the reasoning for your ranking.

Ranking Tasks are adapted from A Selection of Ranking Tasks, T.L. O’Kuma, D.P. Maloney and C.J. Hieggelke, ed. VIDEO 9.07- Ranking Task Explanation

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What questions do you have about this module? You are welcome to ask questions by email, Canvas message, and/or in video conference office hours. However, please write in at least one question here. It can be a question about a concept, a problem-solving concept, or how concepts apply to a particular situation in your life, in the news, etc.

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