1. Energy Chapter 10 What is Energy?
We all have a concept: Heat? Ability to work?

2. What is Energy?
"It is important to realize that in physics today, we have no knowledge of what energy is. We do not have a picture that energy comes in little blobs of a definite amount." -Richard Feynman "Lectures on Physics"
Richard Feynman is one of the most famous modern day lecturers on physics.
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3. What is Energy? Con’t
Energy - A measure of being able to do work…..–NASA.gov
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4. Matter and Energy
The combination of matter and energy makes up the universe. Matter is substance and energy is the mover of substance. -Paul Hewitt
Hewitt is a noted author of physics texts
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5. Definition
Energy is an abstract quantity with the ability to effect physical change in matter. Energy is the substance from which all things in the Universe are made up.
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6. Forms of Energy
Two broad categories of energy: Potential: Stored Energy Chemical – Stored in chemical bonds Mechanical – Stored in objects by tension Nuclear – Stored in nucleus of atoms Gravitational – Stored based on height and weight Electrical – Stored in batteries Kinetic: Energy due to motion Radiant – electromagnetic Thermal - Heat Motion – Moving objects Sound – motion of air and other media by sound waves
One of many categorizations of forms energy can take. The word “type” is often used and is misleading. There is only one energy, which can take many forms.
Categories are also overlapping and confusing eg. What is the difference between chemical and electrical energy as defined above? Very little if any to me. If we talk about the ability of the motion of electrons in a wire to do work isn’t that kinetic????
Key is to recognize energy and the different forms it can take and the ability to convert from one to another.
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7. Conservation of Energy
Energy Conservation during freefall:
Conservation is one of the fundamental concepts of physics today.
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First equation is the basic kinematics equation relating velocities, acceleration and distance
With a little manipulation we can see that there is a basic equality between the status at the beginning and at the end: something is conserved.
If we multiply both sides by m/2we get a similar equation
If we look at the units of the two terms of the equation we can see that they are Nm, force times distance, which is work.
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8. Calculus Approach
Lets look at the freefall of a mass, m, from N2L and calculus:
We can also take a calculus based approach to the same situation.
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We know that the force on an object in free fall is ma, which is m(dv/dt) which is –mg
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Using the chain rule we can show that dv/dt is equal to v(dv/dy)
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Substituting into the first equation for dv/dt gives
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If we multiply by dy we get
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If we integrate from initial to final conditions we get
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Integrating and manipulating gives the same equation
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9. Kinetic Energy
Obviously an object in motion possesses energy, if you don’t think so, go catch a 100 mph fastball. What gave that baseball so much energy? It was the work done by the pitcher applying a force to it over a fairly short distance. Simplistically the chemical energy in the body of the pitcher was converted to mechanical energy by the muscles in his body which applied the force to the ball.
Think of kinetic energy as the force necessary to obtain a given velocity over a distance.
Fd = mad = m(v/t)(v/2)t = ½ mv2
Assuming a frictionless surface and a constant force, application of a net force results in acceleration
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10. Kinetic Energy
Kinetic energy can be defined as the work it takes to get a mass moving at a given velocity.
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11. Gravitational Potential Energy
Similar concept to kinetic energy except that the orientation is vertical and the force is gravity. Think KE which has not been created. M d mg
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12. Mechanical Energy
Mechanical energy is the sum of the kinetic and potential energies of a system.
This statement says that Mechanical Energy of a system does not change; however, kinetic and potential energy can be converted from one to other.
Law of Conservation of Mechanical Energy
Only valid in frictionless situations. Friction uses the kinetic energy to generate heat which is a different form of energy.
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13. Problem 1
A boy reaches out of a window and tosses a ball straight up with a speed of 10 m/s. The ball is 20 m above the ground as he releases it. Use energy to find: a. The ball’s maximum height above the ground. b. The ball’s speed as it passes the window on its way down. c. The speed of impact on the ground.
Analysis: Use Kf + Uf = Ki + Ui to solve for y1, v2 and v3. 5
a. At y1, v1 = 0; at y0, v0 = 10 and y0 = 0
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14. Problem 1 con’t b. At y0, v0 = 10; at y2=0, find v2
Speed up = speed down.
c. At y0 = 0, v0 = 10; At y3 = -20, find v3.
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15. Energy Bar Charts
Bar chart modeling a rock thrown upward and returning to same elevation:
A way of showing the relation ship between kinetic and gravitational potential energy is the bar chart
At the instant it is released the rock only has kinetic energy. As it moves up it looses velocity and gains altitude eg looses kinetic energy and gains potential energy.
At its highest point it has zero velocity and zero kinetic energy and all mechanical energy is converted to potential.
As if falls it gains velocity/kinetic energy and looses altitude/potential energy.
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16. Zero of Potential Energy
Our earlier calculus based derivation of
The following expression represented the change in potential energy:
The initial position was considered to be zero and resulted in U = mgyf
U will vary based on where the zero potential energy level is placed.
ΔU will always be the same regardless of the location of the zero level.
It should be obvious with a little thought that gravitational potential energy depends upon the frame of reference selected. A weight five feet above the floor of an airplane flying at 40,000 ft has potential energy reference to the floor of the plane and a much greater potential energy relative to the surface of the earth. Very similar to where we select the axes in a projectile motion problem and the answer for maximum height.
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17. Non-Freefall Ug
Horizontal surface: no change in gravitational potential energy Slanted/undulating surface: define s-axis parallel to surface of movement.
Newtons’ Second law along the s-axis is….
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Using the chain rule we can rewrite the above as …..
You can see from figure b that the net force along the s-axis is …
Thus N2L becomes…
Multiplying both sides by ds gives…
You can see from the figure that (sin theta ds) is dy so substituting we get….
This is identical to the equations we found for freefall. Integrating this equation from initial to final conditions gives….
Conclusion: It is only the change in height reference to the earth’s surface that determines gravitational potential energy. Y-axis should always be perpendicular to the earth’s surface.
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18. Ballistic Pedulum
A 10 g bullet is fired into a 1200 g wood block hanging from a 150 cm long string. The bullet embeds itself into the block and the block swings out to an angle of 40°. What was the speed of the bullet?
Analysis: Two part problem: The impact of the bullet with the block is inelastic. Momentum is conserved. After the collision the block swings as a pendulum. The sum of the kinetic and potential energies before and after do not change as the block swings to its largest angle.
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19. Ballistic Pendulum con’t
Collision/Momentum calculations:
If we can calculate v1x from swing/energy relationships, we can calculate the speed of the bullet.
The momentum conservation equation applied to inelastic collision gives…..
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The wood block is initially at rest, with (v0x)w = 0, so the bullets velocity is…..
(v1x) is the velocity of the block and bullet immediately after collision…
Now we need to use the energy relationship to calculate the speed of the v1….
The standard kinetic/potential equation applies….
We divide through by (mW + mB) and the velocity at the end of the swing is zero…..
Or…
We can use the triangle to compute y2 as m and v1 as being 2.62 m/s
V2 = 0 and dividing by (mW + mB) gives: or
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20. Restoring Forces Restoring Force Elastic Equilibrium Length, L0
Displacement, Δs
Δs = L - L0
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Restoring force: A force that returns a system to its equilibrium state. Eg springs, rubber, beams, boards, any medium which flexes under load and returns to its original state when the load is removed is considered elastic.
Elastic: Systems that exhibit restoring forces. We will use a spring as an example of an elastic system.
Equilibrium length: Length in equilibrium state
Displacement: Change in length of the system. Lengthening is positive. Shortening is negative
Plot of Fsp vs Δs is a line with a slope k.
K is called the spring constant.
Fsp = k Δs
Spring constant, k
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21. Hooke’s Law
The Restoring force of a spring is opposite in direction of the change in length, Δs; hence, the negative sign.
The (Fsp)s – Δs relationship is shown graphically to the right
Not really a law; better expressed as a model of restoring force.
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22. Hooke’s Law Problem
You need to make a spring scale for measuring mass. You want each 1.0 cm length along the scale to correspond to a mass difference of 100 g. What should be the value of the spring constant? 15
Restoring force for a delta-s of 1.0 cm is 100(9.8)
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23. Elastic Potential Energy
Is the force applied to the ball constant?
Describe the mechanical energy in both situations; before and after.
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Characterize the mechanical energy before and after restorative force is applied
Since Δs is negative the restorative force is positive. Until released all energy is potential energy.
After release the ball accelerates until equilibrium length is reached; Restorative force decreases as Δs decreases. When Δs = 0, energy is all kinetic energy in the ball.
Net force on the ball can be calculated with calculus
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24. Elastic Potential Energy con’t
N2L for the ball is:
By Hooke’s law, (Fnet)s = -k(s – se), substituting gives:
Using the chain rule:
Substituting gives:
Integrating from initial to final conditions:
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25. Elastic Potential Energy con’t
From previous slide:
Substituting and rewriting gives:
½ mv2 is obviously the kinetic energy.
is the elastic potential energy, Us
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26. Elastic Potential Energy Problem
How far must you stretch a spring with k = 1000 N/m to store 200 J of energy?
Elastic potential energy is defined as:
Solving for ∆s:
#20
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27. Elastic Collisions
Perfectly Elastic Collision: A collision in which mechanical energy is conserved.
Must conserve momentum and mechanical energy
Not possible where friction is involved
If only one object, m1, initially moving and all motion is along a line:
Solving the first equation for (vfx)1 and substituting into the second gives:
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28. Elastic Collisions con’t
Squaring and simplifying gives:
There are two solutions, (vfx)2 = 0 which is trivial and:
Substituting this into the momentum equation gives
These allow us to compute the final velocity of each object in terms of the initial velocity of m1 and the relative masses of each object
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29. Elastic Collision Problem
A 50 g marble moving at 2.0 m/s strikes a 20 g marble at rest. What is the speed of each marble immediately after the collision?
Analysis: Expect that v1 will decrease and v2 will significantly increase. Laws conservation of Momentum and ME will be observed. 25
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30. Using Reference Frames
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31. Energy Diagrams
Energy Diagram: A graph showing a system’s potential energy and total energy as a function of position.
The diagram is the energy diagram for a particle in free fall. Note that the horizontal axis is altitude, the vertical axis is energy and the TE line represents the total energy of t he system. At the origin all of the mechanical energy is kinetic. As the object moves upward it looses kinetic energy and gains potential energy. When it reaches the total energy line, all energy is potential and the object is at zero velocity. As it falls it gains speed and gains kinetic energy and looses altitude and looses potential energy.
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32. Energy Diagram for a Spring
Us =.5k(x-xe)2 The vertex of the parabola is at xe. As the spring is stretched to the right or compressed to the left, the potential energy increases and the kinetic energy decreases. Upon release, restoring forces moves the end of the spring toward xe and potential energy is converted to kinetic energy. The sprig
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33. Generalized Energy Diagram
In a generalized diagram minima represent high speed areas and maxima represent low speed areas. Distance from curve to TE line is kinetic energy.
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34. Stable and Unstable Equilibrium
X2, x3 and x4 are special points on a PE curve. Consider x2 if E2 is the total energy. If a particle is at x2 it has no velocity or kinetic energy and cannot move away from x2; it is in static equilibrium.
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