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Reference: Beginning Physics I

 CHAPTER 9: RIGID BODIES I: EQUILIBRIUM & CENTER OF GRAVITY

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KEY WORD LIST

Equilibrium, Torque, Moment, Laws of Equilibrium (Rigid Bodies), Equivalent Sets of Coplanar Forces, Center of Gravity, Couple

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GLOSSARY

For details on the following concepts, please consult CHAPTER 9.

EQUILIBRIUM
For introduction to translational and rotational equilibrium. See Chapter 4.

For a situation in which every slight tilt from equilibrium gives rise to a couple that restores equilibrium, we say that the equilibrium is stable.

Whenever a slight tilt of an object away from equilibrium gives rise to a couple that continues the motion away from equilibrium, we say the equilibrium is unstable.

TORQUE
The words torque and moment are synonymous. They are used interchangeably in the present context.

MOMENT
The moment of a force is a tendency to produce motion about an axis. It is the product of a force and its directed distance from an axis. The axis is selected as a point in the plane of the force, and rigidly linked to the body. A moment arm is defined as the perpendicular distance from this point to the line of action of the force. The moment of the force is defined as the product of the magnitude of the force and length of the moment arm.

The plus sign indicates that the force would tend to rotate the body counterclockwise about the point chosen as the axis. The negative sign indicates clockwise rotation.

LAWS OF EQUILIBRIUM (RIGID BODIES)
The following are the necessary and sufficient conditions for translational and rotational equilibrium for a rigid body acted on by any number of coplanar forces:

1. The vector sum of the forces must vanish.
2. The algebraic sum of the torques about a given point must vanish.

Mathematically,

EQUIVALENT SETS OF COPLANAR FORCES
One can always replace one set of forces acting on a rigid body by any other set of forces having the same vector sum and the same resultant torque (about any chosen point) to get the same effect on the motion of the body.

CENTER OF GRAVITY
The center of gravity is the point in the body where the total weight can be assumed to be acting. The center of gravity and the center of mass are one and the same point.

COUPLE
A couple is a pair of equal and opposite forces giving rise to a torque.

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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 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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