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Reference: Beginning Physics I

CHAPTER 7: ENERGY, POWER AND SIMPLE MACHINES

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KEY WORD LIST

Thermal Energy, Friction and Thermal Energy, Law of Conservation of Energy, Power, Simple Machine, Mechanical Advantage, Efficiency

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GLOSSARY

For details on the following concepts, please consult CHAPTER 7.

THERMAL ENERGY
When the motion of particles is of random nature (describable, in fact, only by statistical means) we call the associated energy thermal energy. Such energy manifests itself macroscopically in various ways, most notably as a rise in temperature.

FRICTION AND THERMAL ENERGY
Since friction always does negative work, the system that supplies the force of friction should always gain energy. The source of friction is the interaction between the surface layers of the two objects that are moving past each other. As a result, the random jiggling of the vast number of particles in the surface increases. This is the increase in the thermal energy of the surfaces.

LAW OF CONSERVATION OF ENERGY
If we include in our considerations thermal energy, as well as other forms of energy such as electromagnetic radiation (light) and more subtle form of mechanical energy such as sound, the law of conservation of energy still holds. Energy can be transformed from one type to another within a given system, and it can be transferred from one system to another system, but the total amount of energy remains the same.

POWER
Power is the rate at which work is done; that is, how much work is done per second by a force. The SI unit for power is the watt (W), where 1 W = 1 joule/second.

The instantaneous power is,

SIMPLE MACHINE
A simple machine is any device that allows a small force to move an object against a larger resisting force, or a force in one direction to move an object against a resisting force in another direction. Many simple machines do both. Examples of simple machines are lever, inclined plane and a pulley system.

MECHANICAL ADVANTAGE
The mechanical advantage of a machine is the ratio of the load to the applied force. The bigger the mechanical advantage the smaller is the applied force necessary to accomplish the task.

EFFICIENCY
The efficiency (e) of a simple machine is defined as the ratio,

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Reference: Beginning Physics I

CHAPTER 6: WORK AND MECHANICAL ENERGY

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KEY WORD LIST

Work, Spring Force, Kinetic Energy, Work-Kinetic Energy Theorem, Gravitational Potential Energy, Work-Energy Theorem, Total Mechanical Energy, Conservation of Mechanical Energy, Energy Transfer, Conservative Force, Gravitational Potential away from Earth, Escape Velocity

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GLOSSARY

For details on the following concepts, please consult CHAPTER 6.

WORK
The work W_F  due to a constant force F acting on an object while it moves through a displacement s is defined as the component of F along the s direction multiplied by the magnitude of s.

Even though the work involves two vector quantities F and s, it itself has no direction and is thus a scalar. The units of work are those of force times distance. The work is defined so that it can be positive, negative or zero, depending on whether the component of F along s is positive, negative or zero. Total work done is just the work by the resultant force.

SPRING FORCE
A stretched spring exerts a force whose magnitude is proportional to the length of the stretch. The proportionality constant k is called the spring constant:

F_sp = – kx;        F = kx

The work done by F in stretching the spring by a displacement x is

W_F = ½ kx²

KINETIC ENERGY
The expression ½ mv² is called the kinetic energy E_k of the mass m at velocity v. The kinetic energy has the units of work, and the SI units are Joules.

WORK-KINETIC ENERGY THEOREM
The work-kinetic energy theorem is expressed as follows.

Where W_T is total work done; E_k is kinetic energy; and E_k is the change in kinetic energy in going from the initial to the final position.

It can be shown, using the calculus, that the work-kinetic energy theorem is still true for the most general possible situation. No matter how complicated the path of motion, and no matter how complicated and numerous the forces are acting on the object, the total work done on the object in any interval equals the final minus the initial kinetic energy for that interval.

GRAVITATIONAL POTENTIAL ENERGY
The expression mgy is called the gravitational potential energy Ep of the mass m at height y.

It can be shown that this equation is true for any path of an object near the earth’s surface. More generally,

WORK-ENERGY THEOREM
The work done by all forces other than gravity on an object equals the sum of the changes in the gravitational potential energy and kinetic energy of the object.

TOTAL MECHANICAL ENERGY
The sum of the potential and kinetic energies at any point is called the total mechanical energy (E_T) at that point.

CONSERVATION OF MECHANICAL ENERGY
The total mechanical energy of an object stays constant (“is conserved”) throughout its motion if no forces other than gravity do work.

ENERGY TRANSFER
We can think of the work done by one system on another system as the mechanical transfer of energy between the systems.

CONSERVATIVE FORCE
Conservative force is any force that has the property that the work done by the force depends only on the starting and ending points, and not on what happened in between. The force of gravity near Earth’s surface is clearly such a force. The name “conservative” comes from the fact that if an object moves in a path that returns to the starting point, the total work done by such a force must be zero. We can define a potential energy for the conservative force. The spring force is also a conservative force.

NOTE: A conservative force of gravity is more like a force field in space that generates the same acceleration at all points in space.

GRAVITATIONAL POTENTIAL AWAY FROM EARTH
Gravitational force far from Earth’s surface is no longer constant, but it can be shown to be conservative. It thus has a potential energy. The gravitational potential energy is determined as,

ESCAPE VELOCITY
The escape velocity is the smallest burnout velocity for the rocket for no return. It is equal to

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Reference: Beginning Physics I

CHAPTER 5: NEWTON’S SECOND LAW

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KEY WORD LIST

Resultant Force and Acceleration, Newton’s Second Law, Mass, Inertia, Weight, Centripetal Force, Banking Equation, Newton’s Law of Gravitation

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GLOSSARY

For details on the following concepts, please consult CHAPTER 5.

RESULTANT FORCE AND ACCELERATION
The unbalanced force on an object causes its acceleration until it comes into balance. The acceleration continues with velocity increasing if the force is non-zero. When the unbalanced force vanishes, so does the acceleration, but the higher velocity continues at a constant rate unless some other resistance or counter force comes into play.

NEWTON’S SECOND LAW
When a nonzero resultant force F acts on a given object, the consequent acceleration a always points in the direction of F. Also, for a given magnitude of F, the magnitude of a is the same no matter what the direction of the force. On the other hand, if the magnitude of F doubles, the magnitude of a doubles; if the magnitude of F triples, the magnitude of a triples; etc. Thus, the magnitude of a is proportional to the magnitude of F. The proportionality constant is called the mass m of the object. This is expressed as the equation,

F = ma.

This equation is the mathematical statement of Newton’s second law.

MASS
The mass controls the response of the object to a given magnitude force. A small mass means a large acceleration, a large mass means a small acceleration. In a sense, the mass is a measure of the resistance of an object to having its velocity changed. This resistance is referred to as the inertia of the object. The relative magnitude of different masses can easily be established by applying the same magnitude force to different objects and measuring their accelerations. Then

The mass is an indestructible and unchanging property of any object that stays with the object even when it is combined into larger units. In the same way, when an object is broken into smaller parts, the sum of masses of the parts equals the original mass.

Units of mass: Kilogram; 1 lbm = 0.45359 kg; 1 slug = 32.2 lbm = 14.7 kg

INERTIA
Inertia is the resistance of an object to having its velocity changed. The inertia is exhibited while the velocity is changing. It vanishes when the velocity settles back to a higher constant value. At higher velocity, the mass of the object reduces (per the theory of relativity) by the amount of inertia overcome by the force. But this reduction in mass is infinitesimal and ignored in mathematical calculations at the level of matter. The mass reduces to almost zero when the speed of light is reached.

WEIGHT
The pull of gravity on an object is commonly called its weight. Weight and mass are proportional at a given point on earth’s surface.

CENTRIPETAL FORCE
When an object moves in uniform circular motion, it happens because it is continually being drawn toward the center of that path. The force drawing the object toward the center is called the centripetal force.

F = mv²/r

BANKING EQUATION
The banking equation gives the general relation among  (the banking angle), v (the velocity) and r (the radius of the curved path) that must hold in order to go around the curve, without the need for any frictional force. Note that the mass of vehicle does not enter the equation:

NEWTON’S LAW OF GRAVITATION

Newton’s Law of universal gravitation states that every particle of matter in the universe attracts every other particle of matter in the universe with a force directly proportional to the product of their masses and inversely proportional to the square of the distance between them. If we assume the proportional constant is G, the magnitude of this force is then

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