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Review 29:008 Exam 2

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Ch. 6 Energy & Oscillations

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Kinetic Energy, and the Work-Energy Principle Apply a force to accelerate a bus: the work done here is We define the kinetic energy:

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6-3 Kinetic Energy, and the Work-Energy Principle This means that the work done is equal to the change in the kinetic energy: If the net work is positive, the kinetic energy increases. If the net work is negative, the kinetic energy decreases. (6-4)

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6-3 Kinetic Energy, and the Work-Energy Principle Because work and kinetic energy can be equated, they must have the same units: kinetic energy is measured in joules.

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6-4 Potential Energy An object can have potential energy by virtue of its surroundings. Familiar examples of potential energy: A wound-up spring A stretched elastic band An object at some height above the ground

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Potential Energy In raising a mass m to a height h, the work done by the external force is We therefore define the gravitational potential energy: (6-5a) (6-6)

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Potential Energy This potential energy can become kinetic energy if the object is dropped. If, where do we measure y from? It turns out not to matter, as long as we are consistent about where we choose y = 0. Only changes in potential energy can be measured.

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Potential Energy Potential energy can also be stored in a spring when it is compressed; the figure below shows potential energy yielding kinetic energy.

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6-4 Potential Energy The force required to compress or stretch a spring is: where k is called the spring constant, and needs to be measured for each spring.

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Potential Energy The force increases as the spring is stretched or compressed further. We find that the potential energy of the compressed or stretched spring, measured from its equilibrium position, can be written:

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Conservative and Nonconservative Forces If friction is present, the work done depends not only on the starting and ending points, but also on the path taken. Friction is called a nonconservative force.

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Conservative and Nonconservative Forces Potential energy can only be defined for conservative forces.

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Conservation of Energy. total energy: Total energy conserved for a system that has no non-conservative forces acting on it

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Problem Solving Using Conservation of Mechanical Energy

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If there is no friction, the speed of a roller coaster will depend only on its height compared to its starting height.

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Problem Solving Using Conservation of Mechanical Energy For an elastic force, conservation of energy tells us:

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Power Power is the rate at which work is done – (6-17) In the SI system, the units of power are watts:

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Ch. 7 Momentum & Impulse

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Momentum and Its Relation to Force Momentum is a vector symbolized by the symbol p, and is defined as The rate of change of momentum is equal to the net force:

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Conservation of Momentum During a collision, total momentum does not change:

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Conservation of Momentum The law of conservation of momentum states: The total momentum of an isolated system of objects remains constant.

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Conservation of Momentum Momentum conservation works for a rocket as long as we consider the rocket and its fuel to be one system, and account for the mass loss of the rocket.

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Collisions and Impulse Impulse = change in momentum = average force X time

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Conservation of Energy and Momentum in Collisions Momentum is conserved in all collisions. Collisions in which kinetic energy is conserved as well are called elastic collisions, and those in which it is not are called inelastic.

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Center of Gravity The center of gravity can be found experimentally by suspending an object from different points. The CM need not be within the actual object – a doughnuts CM is in the center of the hole.

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Ch. 8 Rotational Motion

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Angular Quantities Angular displacement: The average rotational velocity is defined as the total angular displacement divided by time:

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Angular Quantities The rotational acceleration is the rate at which the angular velocity changes with time:

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Angular Quantities Correspondence between linear and rotational quantities:

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Constant Angular Acceleration The equations of motion for constant angular acceleration are like those for linear motion:

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Torque The perpendicular distance from the axis of rotation to the line along which the force acts is called the lever arm. Torque = Perpendicular Force X Lever Arm

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Rotational Inertia The quantity is called the rotational inertia of an object. The distribution of mass matters here – these two objects have the same mass, but the one on the left has a greater rotational inertia, as so much of its mass is far from the axis of rotation.

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Rotational Inertia The rotational inertia of an object depends : Mass Shape Location of axis

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Angular Momentum and Its Conservation In analogy with linear momentum, we can define angular momentum L: Conservation: If the net torque on an object is zero, the total angular momentum is constant.

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Angular Momentum and Its Conservation Therefore, systems that can change their rotational inertia through internal forces will also change their rate of rotation:

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Vector Nature of Angular Quantities Vectors for rotational velocity & Angular momentum: point along the axis of rotation direction is found using a right hand rule:

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Ch. 9 Fluids

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Density The density ρ of an object is its mass per unit volume:

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Pressure in Fluids Pressure is defined as the force per unit area. the units are pascals: 1 Pa = 1 N/m 2 Pressure is the same in every direction in a fluid at a given depth; if it were not, the fluid would flow.

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Pressure in Fluids Also for a fluid at rest, there is no component of force parallel to any solid surface – once again, if there were the fluid would flow.

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Pascals Principle If an external pressure is applied to a confined fluid, the pressure at every point within the fluid increases by that amount. This principle is used in hydraulic lifts

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Buoyancy and Archimedes Principle For a floating object, the portion of the object that is submerged displaces a mass of water equal to the mass of the entire object.

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Continuity of Flow

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Bernoullis Principle Lift on an airplane wing is due to the different air speeds and pressures on the two surfaces of the wing.

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Ch. 10 Temperature & Heat

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Heat As Energy Transfer Heat is a form of energy. 1 cal is the amount of heat necessary to raise the temperature of 1 g of water by 1 Celsius degree.

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Definition of heat Heat is energy transferred from one object to another because of a difference in temperature.

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Internal Energy The total of all the energy of all the molecules in a substance is its internal energy. Temperature: measures average kinetic energy of molecules Internal energy: total energy of all molecules

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Specific Heat The amount of heat required to change the temperature of a material is proportional to: mass temperature change specific heat, c (a material propert)

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1Latent Heat Energy is required for a material to change phase, even though its temperature is not changing.

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Latent Heat Heat of fusion, L F : heat required to change 1.0 kg of material from solid to liquid Heat of vaporization, L V : heat required to change 1.0 kg of material from liquid to vapor

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Latent Heat The total heat required for a phase change depends on the total mass and the latent heat:

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Heat Transfer: Convection Convection occurs when heat flows by the mass movement of molecules from one place to another. It may be natural or forced; both these examples are natural convection.

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Heat Transfer: Radiation The most familiar example of radiation is our own Sun, which radiates at a temperature of almost 6000 K.

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Summary p. 1 In an isolated system, heat gained by one part of the system must be lost by another. Calorimetry measures heat exchange quantitatively. Phase changes require energy even though the temperature does not change. Heat of fusion: amount of energy required to melt 1 kg of material. Heat of vaporization: amount of energy required to change 1 kg of material from liquid to vapor.

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Summary p. 2 Heat transfer takes place by conduction, convection, and radiation. In conduction, energy is transferred through the collisions of molecules in the substance. In convection, bulk quantities of the substance flow to areas of different temperature. Radiation is the transfer of energy by electromagnetic waves.

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Ch. 11 Thermodynamics

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The First Law of Thermodynamics The change in internal energy of a closed system will be equal to the energy added to the system minus the work done by the system on its surroundings. This is the law of conservation of energy, written in a form useful to systems involving heat transfer.

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Isothermal process In order for an isothermal process to take place, heat flows from a hot body (heat reservoir) to a cold body. An isothermal process is one where the temperature does not change.

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Adiabatic Processes An adiabatic process is one where there is no heat flow into or out of the system.

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The Second Law of Thermodynamics – Introduction The process above doesnt happen. This tells us that conservation of energy (First Law) is not the whole story. If it were, movies run backwards would look perfectly normal to us!

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The Second Law of Thermodynamics – Introduction The second law of thermodynamics is a statement about which processes occur and which do not. There are many ways to state the second law; here is one: Heat can flow spontaneously from a hot object to a cold object; it will not flow spontaneously from a cold object to a hot object.

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15-5 Heat Engines It is easy to produce thermal energy using work, but how does one produce work using thermal energy? This is a heat engine; mechanical energy can be obtained from thermal energy only when heat can flow from a higher temperature to a lower temperature.

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Heat Engines We will discuss only engines that run in a repeating cycle; the change in internal energy over a cycle is zero, as the system returns to its initial state. The high temperature reservoir transfers an amount of heat Q H to the engine: part of it is transformed into work W the rest, Q L, is exhausted to the lower temperature reservoir. Note that all three of these quantities are positive.

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Heat Engines A steam engine is one type of heat engine.

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Heat Engines The efficiency of the heat engine is the ratio of the work done to the heat input:

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Heat Engines The Carnot engine is idealized, as it has no friction. Each leg of its cycle is reversible. The Carnot cycle consists of: Isothermal expansion Adiabatic expansion Isothermal compression Adiabatic compression

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Heat Engines

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Carnot engine efficiency is: (15-5) From this we see that 100% efficiency can be achieved only if the cold reservoir is at absolute zero, which is impossible. Real engines have some frictional losses; the best achieve 60-80% of the Carnot value of efficiency.

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Refrigerators, Air Conditioners, and Heat Pumps These appliances can be thought of as heat engines operating in reverse. By doing work, heat is extracted from the cold reservoir and exhausted to the hot reservoir.

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Refrigerators, Air Conditioners, and Heat Pumps

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A heat pump is like an air conditioner turned around, so that the coils dump heat inside the house:

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Entropy and the Second Law of Thermodynamics Another statement of the second law of thermodynamics: The total entropy of an isolated system never decreases.

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Order to Disorder Entropy is a measure of the disorder of a system. This gives us yet another statement of the second law: Natural processes tend to move toward a state of greater disorder. Example: If you put milk and sugar in your coffee and stir it, you wind up with coffee that is uniformly milky and sweet. No amount of stirring will get the milk and sugar to come back out of solution.

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