3.2

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Videos: Conservation of energy Elastic potential energy Elastic potential energy in your bicep and tricep muscles Energy and work Gravitational potential energy Kinetic energy Physics of wind power Solar flare is a plasma in motion Steam rising out of cooling tower at a nuclear power plant

Investigations: 3A: Work and the force vs. distance graph (p. 113) 3B: Inclined plane and the conservation of energy (p. 118) 3C: Work and energy for launching a paper airplane (p. 122) 3D: Springs and the conservation of energy (p. 124) 3E: Conservation of momentum (p. 128) 3F: Collisions (p. 132) 3G: Shock absorbers and dampers (p. 139) 3H: Specific heat of water and steel (p. 144)

Science: Algebra derivation using the conservation of energy Brownian motion Choosing origin for potential energy reference frame Difference between heat and temperature Einstein's theory of relativity Elastic potential energy Elastic potential energy in your muscles Elastic potential energy in your muscles Energy and power for light bulb Energy and work Energy content of phases of matter Energy flow in open and closed systems Faster than light travel? Forms of energy Friction Galileo and the Leaning Tower of Pisa Gravitational potential energy Hooke's Law for springs Intensity of sunlight at Earth's surface Kelvin temperature scale and absolute zero Kinetic energy Law of conservation of energy Loose springs and stiff springs (BLM) Ordered and random kinetic energy Other conservation laws in physics Plasma in motion during a solar flare Plasma is a phase of matter Potential energy Power and radiant energy Power is the rate of doing work Radiant energy Relationship between momentum conservation and Newton's third law Symmetry in the physics of collisions Units of work and energy van der Waals forces

Technology: Batteries and electrical power Calories on food labels Conserving energy in everyday life Converting units and understanding energy content from food Efficiency in technology Energy content of everyday goods Energy flow in clothing manufacture Energy transformations in an internal combustion (piston) engine Energy transformations in technology usually produce heat Everyday objects that store energy Everyday technology in our age of energy Friction from brake pads in a car's wheel Hydroelectric dam Objects with elastic potential energy Power in everyday technology Power, wattage, and the light bulb Radiant power and the light bulb Technology and agriculture over the last century

Engineering: Calculating energy production for the Hoover Dam Design a wind turbine power plant Efficiency Electric current Electromagnetic spectrum Open and closed systems Siting constraint for power plants Specific heat Springs Volts, amps, and power

Mathematics: Algebra derivation using the conservation of energy Calculating energy production for the Hoover Dam Changes in a quantity represented by Greek symbol Δ Choosing origin for potential energy reference frame Converting units and understanding energy content from food Deducing a physical equation Different rates of change Hooke's Law for springs Non-linear relationships Proportional relationships Using algebra to derive temperature conversion formula Using algebra to solve for a variable

Interactive Simulations Interactive simulation of motion down an inclined plane Motion down an inclined plane Interactive simulation of an oscillating spring Spring-loaded carts simulation Elastic collisions between two balls Inelastic collisions between two balls Interactive simulation of an oscillating spring Interactive simulation of an oscillating spring Wind power design simulation Interactive Calculators Work equation calculator Kinetic energy equation Potential energy equation Elastic potential energy equation Efficiency equation Conservation of momentum in collisions equation Power equation calculator
Electric power equation Temperature conversion equation Specific heat equation

1 - Science of Physics
2 - Forces and Motion 3 - Energy Transformations Chapter 3 at a glance 3.1 - Energy 3.2 - Conservation of energy 3.3 - Flow of energy 3.4 - Temperature and the kinetic theory of matter 3.5 - Chapter review4 - Waves and Sound
5 - Electricity and Magnetism
6 - Light and Optics
7 - The Atom
8 - Linear Motion
9 - Vectors and Forces
10 - Forces
11 - Newton's Laws and Gravitation
12 - Circular Motion and Gravitation
13 - Torque and Static Equilibrium
14 - Momentum and Collisions
15 - Angular momentum
16 - Power and Machines
17 - Harmonic Motion
18 - Waves
19 - Sound
20 - Electricity
21 - Electric and Magnetic Fields
22 - Electromagnetism and Induction
23 - Light and Color
24 - Geometrical Optics
25 - Electromagnetic Radiation
26 - The Properties of Matter
27 - Thermodynamics and Heat Transfer
28 - The Atom
29 - The Quantum Theory
30 - TBD
31 - TBD
32 - TBD
33 - TBD
34 - TBD
35 - TBD
36 - TBD
37 - The atomic nucleus and radioactivity
38 - Nuclear reactions
39 - Relativity
40 - Particle physics and the standard model
41 - Electronics (semiconductors)
42 - Nuclear technology
43 - Lasers and photonics
44 - Metals, alloys and materials science
45 - Communications technology
46 - Buildings and structures
47 - Energy technology
48 - Biophysics
49 - Matter and Energy, Space and Time
50 - Energy Transformations
51 - Nanotechnology
52 - Website Videos
53 - ExtraStuff
55 - Modeling Motion, Friction, and Center of Mass
56 - Work and the Conservation of Energy

3B: Inclined plane and the conservation of energy

How does energy conservation govern motion down an inclined plane?

If you have ever skied down a mountain, biked down a hill, or ridden in a roller coaster you know going downhill causes your speed to increase. The higher the hill, the faster you can go (to a point). This investigation uses an interactive simulated hill to explore how gravitational potential energy and kinetic energy explain the changes in motion. For example, if you want to design a roller coaster to reach 30 mph, how high must it be at the start?

Part 1: Changes in energy for motion down an inclined plane

How to use the inclined plane interactive simulation

Set the initial parameters for the interactive simulation of an inclined plane: θ = 15°; h₀ = 100 m; x₀ = v₀ = μ = 0; and m = 20 kg.
Press play to watch the block slide down the ramp.

Graph displacement and speed. What are the shapes of the graphs? Why?

What is the speed at the bottom of the ramp? Change the mass to a few different values

Graph kinetic energy and potential energy. What are the shapes of the graphs? Why?

What are the values of the potential and kinetic energy at the beginning when the block is released? At the bottom? How are their changes related to each other?

What is the speed at the bottom of the ramp? Repeat for a few different values of the mass of the block. How does the speed change in each case? Why?

Change the initial speed of the block to −10 and +10 m/s. What is the change in kinetic energy at the bottom of the ramp? Explain why.

In this interactive simulation you will investigate motion of a block down a (frictionless) inclined plane. The slope of the plane, the initial height of the block, and its initial speed can all be varied. Up to two variables can be plotted at one time, allowing you to investigate distance and speed, or kinetic and potential energy, on the same graph.

Part 2: Designing a roller coaster that reaches 30 mph

Convert 30 mph into units of m/s. Your goal is to make this the final speed of the block when it reaches the bottom of the ramp.
Set the mass of the roller coaster to m = 2000 kg. Use the simulation to determine the speed of the roller coaster at the bottom of the ramp.
Vary the simulation parameters—such as the the vertical height h₀ or inclination angle θ—in order to produce a speed of 30 mph at the bottom of the ramp.
The fastest roller coaster in the United States reaches a maximum speed of 128 mph. Use the interactive simulation to estimate the height of this roller coaster.

Converting 30 mph into m/s

Since the interactive simulation uses meters per second, first we must convert 30 mph into m/s. Start by converting 30 miles into meters:

30 miles \times (\frac{5280 feet}{1 mile}) (\frac{12 inches}{1 foot}) (\frac{2 .54 cm}{1 inch}) (\frac{1 m}{100 cm}) = 48, 280 m

Then convert 1 hour into seconds:

1 hour \times (\frac{60 minutes}{1 hour}) (\frac{60 seconds}{1 minute}) = 3600 s

Finally, divide meters by seconds:

\frac{48, 280 m}{3600 s} = 13.4 {m s}^{- 1}

A speed of 30 mph is therefore equivalent to a speed of 13.4 m/s.
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