ASVAB: Mechanical Comprehension Review

Kinematics
To begin, we will look at the basics of physics. At its heart, physics is just a set of explanations for the ways in which matter and energy behave. There are three key concepts used to describe how matter moves:
1.  Displacement
2.  Velocity
3.  Acceleration

Displacement
Concept: where and how far an object has gone
Calculation: final position – initial position
When something changes its location from one place to another, it is said to have undergone displacement.
If a golf ball is hit across a sloped green into the hole, the displacement only takes into account the final and initial locations, not the path of the ball.


Displacement along a straight line is a very simple example of a vector quantity: that is, it has both a magnitude and a direction. Direction is as important as magnitude in many measurements. If we can determine the original and final position of the object, then we can determine the total displacement with this simple equation:


The hole (final position) is at the Cartesian coordinate location (2, 0) and the ball is hit from the location (1, 0). The displacement is:




The displacement has a magnitude of 1 and a direction of the positive x direction.

Velocity
Concept: the rate of moving from one position to another
Calculation: change in position / change in time
Velocity answers the question, 'How quickly is an object moving?' For example, if a car and a plane travel between two cities which are a hundred miles apart, but the car takes two hours and the plane takes one hour, the car has the same displacement as the plane, but a smaller velocity.


In order to solve some of the problems on the exam, you may need to assess the velocity of an object. If we want to calculate the average velocity of an object, we must know two things.
First, we must know its displacement. Second, we must know the time it took to cover this distance. The formula for average velocity is quite simple:


Or


To complete the example, the velocity of the plane is calculated to be:


The velocity of the car is less:


Often, people confuse the words speed and velocity.  There is a significant difference. The average velocity is based on the amount of displacement, a vector. Alternately, the average speed is based on the distance covered or the path length. The equation for speed is:


Notice that we used total distance and not change in position, because speed is path-dependent.
If the plane traveling between cities had needed to fly around a storm on its way, making the distance traveled 50 miles greater than the distance the car traveled, the plane would still have the same total displacement as the car.


The calculation for the speed: For this reason, average speed can be calculated:





Acceleration
Concept: how quickly something changes from one velocity to another velocity / change in time
Acceleration is the rate of change of the velocity of an object. If a car accelerates from zero velocity to 60 miles per hour (88 feet per second) in two seconds, the car has an impressive acceleration. But if a car performs the same change in velocity in eight seconds, the acceleration is much lower and not as impressive.


To calculate average acceleration, we may use the equation:


The acceleration of the cars is found to be:




Acceleration will be expressed in units of distance divided by time squared; for instance, meters per second squared or feet per second squared.

Projectile Motion
A specific application of the study of motion is projectile motion. Simple projectile motion occurs when an object is in the air and experiencing only the force of gravity. We will disregard drag for this topic. Some common examples of projectile motion are thrown balls, flying bullets, and falling rocks. The characteristics of projectile motion are: horizontal component of velocity doesn't change
2.  The vertical acceleration due to gravity affects the vertical component of velocity
Because gravity only acts downwards, objects in projectile motion only experience acceleration in the y direction (vertical). The horizontal component of the object's velocity does not change in flight. This means that if a rock is thrown out off a cliff, the horizontal velocity, (think the shadow if the sun is directly overhead) will not change until the ball hits the ground.


The velocity in the vertical direction is affected by gravity. Gravity imposes an acceleration of

 downward on projectiles. The vertical component of velocity at any point is equal to:


When these characteristics are combined, there are three points of particular interest in a projectile's flight. At the beginning of a flight, the object has a horizontal component and a vertical component giving it a large speed. At the top of a projectile's flight, the vertical velocity equals zero, making the top the slowest part of travel. When the object passes the same height as the launch, the vertical velocity is opposite of the initial vertical velocity making the speed equal to the initial speed.


If the object continues falling below the initial height from which it was launched (e.g., it was launched from the edge of a cliff), it will have an even greater velocity than it did initially from that point until it hits the ground.

Rotational Kinematics
Concept: increasing the radius increases the linear speed
Calculation:


Another interesting application of the study of motion is rotation. In practice, simple rotation is when an object rotates around a point at a constant speed. Most questions covering rotational kinematics will provide the distance from a rotating object to the center of rotation (radius) and ask about the linear speed of the object. A point will have a greater linear speed when it is farther from the center of rotation.
If a potter is spinning his wheel at a constant speed of one revolution per second, the clay six inches away from the center will be going faster than the clay three inches from the center. The clay directly in the center of the wheel will not have any linear velocity.


To find the linear speed of rotating objects using radians, we use the equation:


Using degrees, the equation is:


To find the speed of the pieces of clay we use the known values (rotational speed of 1 revolution per second, radii of 0 inches, 3 inches, and 6 inches) and the knowledge that one revolution = 2 pi.








Cams
In the study of motion, a final application often tested is the cam. A cam and follower system allow mechanical systems to have timed, specified, and repeating motion. Although cams come in varied forms, tests focus on rotary cams. In engines, a cam shaft coordinates the valves for intake and exhaust. Cams are often used to convert rotational motion into repeating linear motion.
Cams rotate around one point.
The follower sits on the edge of the cam and moves along with the edge. To understand simple cams, count the number of bumps on the cam. Each bump will cause the follower to move outwards.


Another way to consider cams is to unravel the cam profile into a straight object. The follower will then follow the top of the profile.






Kinetics

Newton's Three Laws of Mechanics
The questions on the exam may require you to demonstrate familiarity with the concepts expressed in Newton's three laws of motion which relate to the concept of force.
Newton's first law – A body at rest will tend to remain at rest, while a body in motion will tend to remain in motion, unless acted upon by an external force.
Newton's second law – The acceleration of an object is directly proportional to the force being exerted on it and inversely proportional to its mass.
Newton's third law – For every force, there is an equal and opposite force.

First Law
Concept: Unless something interferes, an object won't start or stop moving
Although intuition supports the idea that objects do not start moving until a force acts on them, the idea of an object continuing forever without any forces can seem odd. Before Newton formulated his laws of mechanics, general thought held that some force had to act on an object continuously in order for it to move at a constant velocity.
This seems to make sense: when an object is briefly pushed, it will eventually come to a stop. Newton, however, determined that unless some other force acted on the object (most notably friction or air resistance), it would continue in the direction it was pushed at the same velocity forever.



Second Law
Concept: Acceleration increases linearly with force.
Although Newton's second law can be conceptually understood as a series of relationships describing how an increase in one factor will decrease another factor, the law can be understood best in equation format:


Or




Each of the forms of the equation allows for a different look at the same relationships. To examine the relationships, change one factor and observe the result. If a steel ball, with a diameter of 6.3 cm, has a mass of 1 kg and an acceleration of 1 m/s2, then the net force on the ball will be 1 Newton.



Third Law
Concept: Nothing can push or pull without being pushed or pulled in return.
When any object exerts a force on another object, the other object exerts the opposite force back on the original object. To observe this, consider two spring-based fruit scales, both tipped on their sides as shown with the weighing surfaces facing each other. If fruit scale #1 is pressing fruit scale #2 into the wall, it exerts a force on fruit scale #2, measurable by the reading on scale #2. However, because fruit scale #1 is exerting a force on scale #2, scale #2 is exerting a force on scale
#1 with an opposite direction, but the same magnitude.



Force
Concept: a push or pull on an object
Calculation:


A force is a vector which causes acceleration of a body. Force has both magnitude and direction.
Furthermore, multiple forces acting on one object combine in vector addition.
This can be demonstrated by considering an object placed at the origin of the coordinate plane. If it is pushed along the positive direction of the x-axis, it will move in this direction; if the force acting on it is in the positive direction of the y-axis, it will move in that direction. However, if both forces are applied at the same time, then the object will move at an angle to both the x and y axes, an angle determined by the relative amount of force exerted in each direction. In this way, we may see that the resulting force is a vector sum; that is, a net force that has both magnitude and direction.



Mass
Concept: the amount of matter
Mass can be defined as the quantity of matter in an object. If we apply the same force to two objects of different mass, we will find that the resulting acceleration is different.
Newton's Second Law of Motion describes the relationship between mass, force, and acceleration in the equation: Force = mass x acceleration.
In other words, the acceleration of an object is directly proportional to the force being exerted on it and inversely proportional to its mass.



Gravity
Gravity is a force which exists between all objects with matter. Gravity is a pulling force between objects meaning that the forces on the objects point toward the opposite object. When Newton's third law is applied to gravity, the force pairs from gravity are shown to be equal in magnitude and opposite in direction.



Weight
Weight is sometimes confused with mass. While mass is the amount of matter, weight is the force exerted by the earth on an object with matter by gravity. The earth pulls every object of mass toward its center while every object of mass pulls the earth toward its center. The object's pull on the earth is equal in magnitude to the pull which the earth exerts, but, because the mass of the earth is very large in comparison (5.97 × 1024 kg), only the object appears to be affected by the force.


The gravity of earth causes a constant acceleration due to gravity (g) at a specific altitude. For most earthbound applications the acceleration due to gravity is 32.2 ft/s2 or 9.8 m/s2 in a downward direction. The equation for the force of gravity (weight) on an object is the equation from Newton's Second Law with the constant acceleration due to gravity (g).




The SI (International Standard of Units) unit for weight is the Newton

. The English
Engineering unit system uses the pound, or lb, as the unit for weight and force

. Thus, a 2 kg object under the influence of gravity would have a weight of:





Normal Force
Concept: the force perpendicular to a contact surface
The word 'normal' is used in mathematics to mean perpendicular, and so the force known as normal force should be remembered as the perpendicular force exerted on an object that is resting on some other surface. For instance, if a box is resting on a horizontal surface, we may say that the normal force is directed upwards through the box
(the opposite, downward force is the weight of the box). If the box is resting on a wedge, the normal force from the wedge is not vertical but is perpendicular to the wedge edge.



Tension
Concept: the pulling force from a cord
Another force that may come into play on the exam is called tension. Anytime a cord is attached to a body and pulled so that it is taut, we may say that the cord is under tension. The cord in tension applies a pulling tension force on the connected objects. This force is pointed away from the body and along the cord at the point of attachment. In simple considerations of tension, the cord is assumed to be both without mass and incapable of stretching. In other words, its only role is as the connector between two bodies. The cord is also assumed to pull on both ends with the same magnitude of tension force.



Friction
Concept: Friction is a resistance to motion between contacting surfaces
In order to illustrate the concept of friction, let us imagine a book resting on a table. As it sits, the force of its weight is equal to and opposite of the normal force. If, however, we were to exert a force on the book, attempting to push it to one side, a frictional force would arise, equal and opposite to our force. This kind of frictional force is known as static frictional force.


As we increase our force on the book, however, we will eventually cause it to accelerate in the direction of our force. At this point, the frictional force opposing us will be known as kinetic friction. For many combinations of surfaces, the magnitude of the kinetic frictional force is lower than that of the static frictional force, and consequently, the amount of force needed to maintain the movement of the book will be less than that needed to initiate the movement.



Rolling Friction
Occasionally, a question will ask you to consider the amount of friction generated by an object that is rolling. If a wheel is rolling at a constant speed, then the point at which it touches the ground will not slide, and there will be no friction between the ground and the wheel inhibiting movement. In fact, the friction at the point of contact between the wheel and the ground is static friction necessary to propulsion with wheels. When a vehicle accelerates, the static friction between the wheels and ground allows the vehicle to achieve acceleration. Without this friction, the vehicle would spin its wheels and go nowhere.
Although the static friction does not impede movement for the wheels, a combination of frictional forces can resist rolling motion. One such frictional force is bearing friction. Bearing friction is the kinetic friction between the wheel and an object it rotates around, such as a stationary axle.
Most questions will consider bearing friction the only force stopping a rotating wheel. There are many other factors that affect the efficiency of a rolling wheel such as deformation of the wheel, deformation of the surface, and force imbalances, but the net resulting friction can be modeled as a simple kinetic rolling friction. Rolling friction or rolling resistance is the catch-all friction for the combination of all the losses which impede wheels in real life.



Drag Force
Friction can also be generated when an object is moving through air or liquid. A drag force occurs when a body moves through some fluid (either liquid or gas) and experiences a force that opposes the motion of the body. The drag force is greater if the air or fluid is thicker or is moving in the direction opposite to the object.
Obviously, the higher the drag force, the greater amount of positive force required to keep the object moving forward.



Balanced Forces: An object is in equilibrium when the sum of all forces acting on the object is zero. When the forces on an object sum to zero, the object does not accelerate. Equilibrium can be obtained when forces in the y-direction sum to zero, forces in the x-direction sum to zero, or forces in both directions sum to zero.


In most cases, a problem will provide one or more forces acting on object and ask for a force to balance the system. The force will be the opposite of the current force or sum of current forces.



Rotational Kinetics
Many equations and concepts in linear kinematics and kinetics transfer to rotation. For example, angular position is an angle. 
Angular velocity, like linear velocity, is the change in the position (angle) divided by the time. Angular acceleration is the change in angular velocity divided by time. Although most tests will not require you to perform angular calculations, they will expect you to understand the angular version of force: torque.
Concept: Torque is a twisting force on an object
Calculation:


Torque, like force, is a vector and has magnitude and direction. As with force, the sum of torques on an object will affect the angular acceleration of that object. The key to solving problems with torque is understanding the lever arm. A better description of the torque equation is:




Because torque is directly proportional to the radius, or lever arm, a greater lever arm will result in a greater torque with the same amount of force. The wrench on the right has twice the radius and, as a result, twice the torque.


Alternatively, a greater force also increases torque. The wrench on the right has twice the force and twice the torque.



Work/Energy

Work
Concept: Work is the transfer of energy from one object to another Calculation:


The equation for work in one dimension is fairly simple:




In the equation, the force and the displacement are the magnitude of the force exerted and the total change in position of the object on which the force is exerted, respectively.
If force and displacement have the same direction, then the work is positive.
If they are in opposite directions, however, the work is negative.


For two-dimensional work, the equation is a bit more complex:




The angle in the equation is the angle between the direction of the force and the direction of the displacement. Thus, the work done when a box is pulled at a 20-degree angle with a force of 100 lb for 20 ft will be less than the work done when a differently weighted box is pulled horizontally with a force of 100 lb for 20 ft.







The unit

 is the unit for both work and energy.

Energy
Concept: the ability of a body to do work on another object
Energy is a word that has found a million different uses in the English language, but in physics it refers to the measure of a body's ability to do work. In physics, energy may not have a million meanings, but it does have many forms. Each of these forms, such as chemical, electric, and nuclear, is the capability of an object to perform work. However, for the purpose of most tests, mechanical energy and mechanical work are the only forms of energy worth understanding in depth.
Mechanical energy is the sum of an object's kinetic and potential energies.

Although they will be introduced in greater detail, these are the forms of mechanical energy:

Kinetic Energy – Energy an Object Has by Virtue of Its Motion
Gravitational Potential Energy – energy by virtue of an object's height
Elastic Potential Energy – energy stored in compression or tension

Neglecting frictional forces, mechanical energy is conserved. As an example, imagine a ball moving perpendicular to the surface of the earth, with its weight the only force acting on it. As the ball rises, the weight will be doing work on the ball, decreasing its speed and its kinetic energy, and slowing it down until it momentarily stops. During this ascent, the potential energy of the ball will be rising. Once the ball begins to fall back down, it will lose potential energy as it gains kinetic energy. Mechanical energy is conserved throughout; the potential energy of the ball at its highest point is equal to the kinetic energy of the ball at its lowest point prior to impact.



In systems where friction and air resistance are not negligible, we observe a different sort of result. For example, imagine a block sliding across the floor until it comes to a stop due to friction. Unlike a compressed spring or a ball flung into the air, there is no way for this block to regain its energy with a return trip. Therefore, we cannot say that the lost kinetic energy is being stored as potential energy.

Instead, it has been dissipated and cannot be recovered. The total mechanical energy of the block-floor system has been not conserved in this case but rather reduced. The total energy of the system has not decreased, since the kinetic energy has been converted into thermal energy, but that energy is no longer useful for work.



Energy, though it may change form, will be neither created nor destroyed during physical processes. However, if we construct a system and some external force performs work on it, the result may be slightly different. If the work is positive, then the overall store of energy is increased; if it is negative, however, we can say that the overall energy of the system has decreased.



Kinetic Energy
The kinetic energy of an object is the amount of energy it possesses by reason of being in motion. Kinetic energy cannot be negative.
Changes in kinetic energy will occur when a force does work on an object, such that the motion of the object is altered. This change in kinetic energy is equal to the amount of work that is done. This relationship is commonly referred to as the work-energy theorem.

One interesting application of the work-energy theorem is that of objects in a free fall. To begin with, let us assert that the force acting on such an object is its weight, equal to its mass times g (the force of gravity). The work done by this force will be positive, as the force is exerted in the direction in which the object is traveling. Kinetic energy will, therefore, increase, according to the work-kinetic energy theorem.
If the object is dropped from a great enough height, it eventually reaches its terminal velocity, where the drag force is equal to the weight, so the object is no longer accelerating and its kinetic energy remains constant.



Gravitational Potential Energy
Gravitational potential energy is simply the potential for a certain amount of work to be done by one object on another using gravity. For objects on earth, the gravitational potential energy is equal to the amount of work which the earth can act on the object. The work which gravity performs on objects moving entirely or partially in the vertical direction is equal to the force exerted by the earth (weight) times the distance traveled in the direction of the force (height above the ground or reference point):


Thus, the gravitational potential energy is the same as the potential work.





Elastic Potential Energy
Elastic potential energy is the potential for a certain amount of work to be done by one object on another using elastic compression or tension. The most common example is the spring. A spring will resist any compression or tension away from its equilibrium position (natural position). A small buggy is pressed into a large spring. The spring contains a large amount of elastic potential energy. If the buggy and spring are released, the spring will push exert a force on the buggy for a distance. This work will put kinetic energy into the buggy. The energy can be imagined as a liquid poured from one container into another. The spring pours its elastic energy into the buggy, which receives the energy as kinetic energy.



Power
Concept: the rate of work
Calculation:



On occasion, you may need to demonstrate an understanding of power, as it is defined in applied physics. Power is the rate at which work is done. Power, like work and energy, is a scalar quantity. Power can be calculated by dividing the amount of work performed by the amount of time in which the work was performed.



If more work is performed in a shorter amount of time, more power has been exerted. Power can be expressed in a variety of units. The preferred metric expression is one of watts or joules per seconds. For engine power, it is often expressed in horsepower.


Machines

Simple Machines
Concept: Tools which transform forces to make tasks easier. A. their job is to transform forces, simple machines have an input force and an output force or forces.

Simple machines transform forces in two ways: direction and magnitude. A machine can change the direction of a force, with respect to the input force, like a single stationary pulley which only changes the direction of the output force. A machine can also change the magnitude of the force like a lever.



Simple machines include the inclined plane, the wedge, the screw, the pulley, the lever, and the wheel.



Mechanical Advantage change a simple machine provides to the magnitude of a force

Calculation: output force/input force
Mechanical advantage is the measure of the output force divided by the input force. Thus, mechanical advantage measures the change performed by a machine. Machines cannot create energy, only transform it. Thus, in frictionless, ideal machines, the input work equals the output work.




This means that a simple machine can increase the force of the output by decreasing the distance which the output travels or it can increase the distance of the output by decreasing the force at the output. B. moving parts of the equation for work, we can arrive at the equation for mechanical advantage.



If the mechanical advantage is greater than one, the output force is greater than the input force and the input distance is greater than the output distance. Conversely, if the mechanical advantage is less than one, the input force is greater than the output force and the output distance is greater than the input distance. In equation form this is:

If Mechanical Advantage >
1:


If Mechanical Advantage <



Inclined Plane
The inclined plane is perhaps the most common of the simple machines. It is simply a flat surface that elevates as you move from one end to the other; a ramp is an easy example of an inclined plane. Consider how much easier it is for an elderly person to walk up a long ramp than to climb a shorter but steeper flight of stairs; this is because the force required is diminished as the distance increases. Indeed, the longer the ramp, the easier it is to ascend.


On the exam, this simple fact will most often be applied to moving heavy objects. For instance, if you have to move a heavy box onto the back of a truck, it is much easier to push it up a ramp than to lift it directly onto the truck bed. The longer the ramp, the greater the mechanical advantage, and the easier it will be to move the box.
The mechanical advantage of an inclined plane is equal to the slant length divided by the rise of the plane.



A. you solve this kind of problem, however, remember that the same amount of work is being performed whether the box is lifted directly or pushed up a twenty-foot ramp; a simple machine only changes the force and the distance.

Wedge
A wedge is a variation on the inclined plane, in which the wedge moves between objects or parts and forces them apart. The unique characteristic of a wedge is that, unlike an inclined plane, it is designed to move. Perhaps the most familiar use of the wedge is in splitting wood. A wedge is driven into the wood by hitting the flat back end.
The thin end of a wedge is easier to drive into the wood since it has less surface area and, therefore, transmits more force per area. As the wedge is driven in, the increased width helps to split the wood.



The exam may require you to select the wedge that has the highest mechanical advantage. This should be easy: the longer and thinner the wedge, the greater the mechanical advantage.

The equation for mechanical advantage is:



Screw
A screw is simply an inclined plane that has been wound around a cylinder so that it forms a sort of spiral.


When it is placed into some medium, as for instance wood, the screw will move either forward or backward when it is rotated. The principle of the screw is used in a number of different objects, from jar lids to flashlights. On the exam, you are unlikely to see many questions regarding screws, though you may be presented with a given screw rotation and asked in which direction the screw will move. However, for consistency's sake, the equation for the mechanical advantage is a modification of the inclined plane's equation. Again, the formula for an inclined plane is:



Because the rise of the inclined plane is the length along a screw, length between rotations = rise.

Also, the slant length will equal the circumference of one rotation = 2πr.





Lever
The lever is the most common kind of simple machine. See-saws, shovels, and baseball bats are all examples of levers. There are three classes of levers which are differentiated by the relative orientations of the fulcrum, resistance, and effort. The fulcrum is the point at which the lever rotates; the effort is the point on the lever where force is applied; the resistance is the part of the lever that acts in response to the effort.


The mechanical advantage of a lever depends on the distances of the effort and resistance from the fulcrum.
Mechanical advantage equals:



For each class of lever, the location of the important distance changes:







In a first-class lever, the fulcrum is between the effort and the resistance. A seesaw is a good example of a first-class lever: when effort is applied to force one end up, the other end goes down, and vice versa.
The shorter the distance between the fulcrum and the resistance, the easier it will be to move the resistance. As an example, consider whether it is easier to lift another person on a see-saw when they are sitting close to the middle or all the way at the end. A little practice will show you that it is much more difficult to lift a person the farther away he or she is on the see-saw.

In a second-class lever, the resistance is in-between the fulcrum and the effort. While a first-class lever is able to increase force and distance through mechanical advantage, a second-class lever is only able to increase force. A common example of a second-class lever is the wheelbarrow: the force exerted by your hand at one end of the wheelbarrow is magnified at the load. Basically, with a second-class lever you are trading distance for force; by moving your end of the wheelbarrow a bit farther, you produce greater force at the load.



Third class levers are used to produce greater distance. In a third-class lever, the force is applied in between the fulcrum and the resistance. A baseball bat is a classic example of a third-class lever; the bottom of the bat, below where you grip it, is considered the fulcrum. The end of the bat, where the ball is struck, is the resistance. By exerting effort at the base of the bat, close to the fulcrum, you are able to make the end of the bat fly quickly through the air. The closer your hands are to the base of the bat, the faster you will be able to make the other end of the bat travel.

Pulley
The pulley is a simple machine in which a rope is carried by the rotation of a wheel. Another name for a pulley is a block. Pulleys are typically used to allow the force to be directed from a convenient location. For instance, imagine you are given the task of lifting a heavy and tall bookcase. Rather than tying a rope to the bookcase and trying to lift it up, it would make sense to tie a pulley system to a rafter above the bookcase and run the rope through it, so that you could pull down on the rope and lift the bookcase. Pulling down allows you to incorporate your weight (normal force) into the act of lifting, thereby making it easier.



If there is just one pulley above the bookcase, you have created a first-class lever which will not diminish the amount of force that needs to be applied to lift the bookcase. There is another way to use a pulley, however, that can make the job of lifting a heavy object considerably easier. First, tie the rope directly to the rafter. Then, attach a pulley to the top of the bookcase and run the rope through it. If you can then stand so that you are above the bookcase, you will have a much easier time lifting this heavy object. Why? Because the weight of the bookcase is now being distributed: half of it is acting on the rafter, and half of it is acting on you. In other words, this arrangement allows you to lift an object with half the force. This simple pulley system, therefore, has a mechanical advantage of 2. Note that in this arrangement, the unfixed pulley is acting like a second-class lever. The price you pay for your mechanical advantage is that whatever distance you raise your end of the rope, the bookcase will only be lifted half as much.



Of course, it might be difficult for you to find a place high enough to enact this system. If this is the case, you can always tie another pulley to the rafter and run the rope through it and back down to the floor. Since this second pulley is fixed, the mechanical advantage will remain the same.



There are other, slightly more complex ways to obtain an even greater mechanical advantage with a system of pulleys. On the exam, you may be required to determine the pulley and tackle (rope) arrangement that creates the greatest mechanical advantage. The easiest way to determine the answer is to count the number of ropes that are going to and from the unfixed pulley; the more ropes coming and going, the greater the mechanical advantage.

Wheel and Axle
Another basic arrangement that makes use of simple machines is called the wheel and axle. When most people think of a wheel and axle, they immediately envision an automobile tire. The steering wheel of the car, however, operates on the same mechanical principle, namely that the force required to move the center of a circle is much greater than the force require to move the outer rim of a circle. When you turn the steering wheel, you are essentially using a second-class lever by increasing the output force by increasing the input distance. The force required to turn the wheel from the outer rim is much less than would be required to turn the wheel from its center. Just imagine how difficult it would be to drive a car if the steering wheel was the size of a saucer!

Conceptually, the mechanical advantage of a wheel is easy to understand. For instance, all other things being equal, the mechanical advantage created by a system will increase along with the radius. In other words, a steering wheel with a radius of 12 inches has a greater mechanical advantage than a steering wheel with a radius of ten inches; the same amount of force exerted on the rim of each wheel will produce greater force at the axis of the larger wheel.


The equation for the mechanical advantage of a wheel and axle is:


Thus, the mechanical advantage of the steering wheel with a larger radius will be:



Gears
The exam may ask you questions involving some slightly more complex mechanisms. It is very common, for instance, for there to be a couple of questions concerning gears. Gears are a system of interlocking wheels that can create immense mechanical advantages. The amount of mechanical advantage, however, will depend on the gear ratio; that is, on the relation in size between the gears.
When a small gear is driving a big gear, the speed of the big gear is relatively slow; when a big gear is driving a small gear, the speed of the small gear is relatively fast.

mechanical advantage is:


Note that mechanical advantage is greater than 1 when the output gear is larger.  In these cases, the output velocity (ω) will be lower.  The equation for the relative speed of a gear system is:





Uses of Gears
Gears are used to change direction of output torque, change location of output torque, change amount of output torque, and change angular velocity of output.



Gear Ratios
A gear ratio is a measure of how much the speed and torque are changing in a gear system. It is the ratio of output speed to input speed.
Because the number of teeth is directly proportional to the speed in meshing gears, a gear ratio can also be calculated using the number of teeth on the gears. When the driving gear has 30 teeth and the driven gear has 10 teeth, the gear ratio is 3:1.



This means that the smaller, driven gear rotates 3 times for every 1 rotation of the driving gear.

The Hydraulic Jack
The hydraulic jack is a simple machine using two tanks and two pistons to change the amount of an output force.


Since fluids are effectively incompressible, when you apply pressure to one part of a contained fluid, that pressure will have to be relieved in equal measure elsewhere in the container.  Supposed the input piston has half the surface area of the output piston (10 in2 compared to 20 in2), and it is being pushed downward with 50 pounds of force.  The pressure being applied to the fluid is

 or 5 psi.  When that 5 psi of pressure is applied to the output piston, it pushes that piston upward with a force of

.

The hydraulic jack functions similarly to a first-class lever, but with the important factor being the area of the pistons rather than the length of the lever arms. Note that the mechanical advantage is based on the relative areas, not the relative radii, of the pistons.  The radii must be squared to compute the relative areas.



Pulleys and Belts
Another system involves two pulleys connected by a drive belt (a looped band that goes around both pulleys).
The operation of this system is similar to that of gears, with the exception that the pulleys will rotate in the same direction, while interlocking gears will rotate in opposite directions. A smaller pulley will always spin faster than a larger pulley, though the larger pulley will generate more torque.



The speed ratio between the pulleys can be determined by comparing their radii; for instance, a 4-inch pulley and a 12-inch pulley will have a speed ratio of 3:1.


Momentum/Impulse

Linear Momentum
Concept: how much a body will resist stopping
Calculation:



In physics, linear momentum can be found by multiplying the mass and velocity of an object:


Momentum and velocity will always be in the same direction.

Newton's second law describes momentum, stating that the rate of change of momentum is proportional to the force exerted and is in the direction of the force. If we assume a closed and isolated system (one in which no objects leave or enter, and upon which the sum of external forces is zero), then we can assume that the momentum of the system will neither increase nor decrease. That is, we will find that the momentum is a constant. The law of conservation of linear momentum applies universally in physics, even in situations of extremely high velocity or with subatomic particles.

Collisions
This concept of momentum takes on new importance when we consider collisions. A collision is an isolated event in which a strong force acts between each of two or more colliding bodies for a brief period of time.
However, a collision is more intuitively defined as one or more objects hitting each other.

When two bodies collide, each object exerts a force on the opposite member. These equal and opposite forces change the linear momentum of the objects. However, when both bodies are considered, the net momentum in collisions is conserved.

There are two types of collisions: elastic and inelastic. The difference between the two lies in whether kinetic energy is conserved. If the total kinetic energy of the system is conserved, the collision is elastic. Visually, elastic collisions are collisions in which objects bounce perfectly. If some of the kinetic energy is transformed into heat or another form of energy, the collision is inelastic.
Visually, inelastic collisions are collisions in which the objects do not bounce perfectly or even stick to each other.


If the two bodies involved in an elastic collision have the same mass, then the body that was moving will stop completely, and the body that was at rest will begin moving at the same velocity as the projectile was moving before the collision.


Fluids
Concept: liquids and gasses
A few of the questions on the exam will probably require you to consider the behavior of fluids. It sounds obvious, perhaps, but fluids can best be defined as substances that flow. A fluid will conform, slowly or quickly, to any container in which it is placed.

Both liquids and gasses are considered to be fluids. Fluids are essentially those substances in which the atoms are not arranged in any permanent, rigid way. In ice, for instance, atoms are all lined up in what is known as a crystalline lattice, while in water and steam the only intermolecular arrangements are haphazard connections between neighboring molecules.



Flow Rates
When liquids flow in and out of containers with certain rates, the change in volume is the volumetric flow in minus the volumetric flow out. Volumetric flow is essentially the amount of volume moved past some point divided by the time it took for the volume to pass.


If the flow into a container is greater than the flow out, the container will fill with the fluid. However, if the flow out of a container is greater than the flow into a container, the container will drain of the fluid.



Density
Concept: how much mass is in a specific volume of a substance
Calculation:


Density is essentially how much stuff there is in a volume or space. The density of a fluid is generally expressed with the symbol ρ (the Greek letter rho.) The density may be found with the simple equation:


Density is a scalar property, meaning that it has no direction component.

Pressure
Concept: The amount of force applied per area
Calculation:


Pressure, like fluid density, is a scalar and does not have a direction. The equation for pressure is concerned only with the magnitude of that force, not with the direction in which it is pointing. The SI unit of pressure is the Newton per square meter, or Pascal.

A. every deep-sea diver knows, the pressure of water becomes greater the deeper you go below the surface; conversely, experienced mountain climbers know that air pressure decreases as they gain a higher altitude. These pressures are typically referred to as hydrostatic pressures because they involve fluids at rest.



Pascal's Principle
The exam may also require you to demonstrate some knowledge of how fluids move. Anytime you squeeze a tube of toothpaste, you are demonstrating the idea known as Pascal's principle. This principle states that a change in the pressure applied to an enclosed fluid is transmitted undiminished to every portion of the fluid as well as to the walls of the containing vessel.



Buoyant Force
If an object is submerged in water, it will have a buoyant force exerted on it in the upward direction.
Often, of course, this buoyant force is much too small to keep an object from sinking to the bottom. Buoyancy is summarized in Archimedes' principle; a body wholly or partially submerged in a fluid will be buoyed up by a force equal to the weight of the fluid that the body displaces.


If the buoyant force is greater than the weight of an object, the object will go upward. If the weight of the object is greater than the buoyant force, the object will sink. When an object is floating on the surface, the buoyant force has the same magnitude as the weight.


Even though the weight of a floating object is precisely balanced by a buoyant force, these forces will not necessarily act at the same point. The weight will act from the center of mass of the object, while the buoyancy will act from the center of mass of the hole in the water made by the object (known as the center of buoyancy). If the floating object is tilted, then the center of buoyancy will shift and the object may be unstable. In order to remain in equilibrium, the center of buoyancy must always shift in such a way that the buoyant force and weight provide a restoring torque, one that will restore the body to its upright position. This concept is, of course, crucial to the construction of boats which must always be made to encourage restoring torque.



Ideal Fluids
Because the motion of actual fluids is extremely complex, the exam usually assumes ideal fluids when they set up their problems. Using ideal fluids in fluid dynamics problems is like discounting friction in other problems. Therefore, when we deal with ideal fluids, we are making four assumptions. It is important to keep these in mind when considering the behavior of fluids on the exam. First, we are assuming that the flow is steady; in other words, the velocity of every part of the fluid is the same. Second, we assume that fluids are incompressible, and, therefore, have a consistent density. Third, we assume that fluids are nonviscous, meaning that they flow easily and without resistance. Fourth, we assume that the flow of ideal fluids is irrotational: that is, particles in the fluid will not rotate around a center of mass.

Bernoulli's Principle
When fluids move, they do not create or destroy energy; this modification of Newton's second law for fluid behavior is called Bernoulli's principle. It is essentially just a reformulation of the law of conservation of mechanical energy for fluid mechanics.
The most common application of Bernoulli's principle is that pressure and speed are inversely related, assuming constant altitude.  Thus, if the elevation of the fluid remains constant and the speed of a fluid particle increases as it travels along a streamline, the pressure will decrease. If the fluid slows down, the pressure will increase.




Heat Transfer
Heat is a type of energy.
Heat transfers from the hot object to the cold object through the three forms of heat transfer: conduction, convection, and radiation.


Conduction is the transfer of heat by physical contact. When you touch a hot pot, the pot transfers heat to your hand by conduction.
Convection is the transfer of heat by the movement of fluids. When you put your hand in steam, the steam transfers heat to your hand by convection.
Radiation is the transfer of heat by electromagnetic waves.
When you put your hand near a campfire, the fire heats your hand by radiation.

Phase Changes
Materials exist in four phases or states: solid, liquid, gas, and plasma. However, as most tests will not cover plasma, we will focus on solids, liquids, and gases. The solid state is the densest in almost all cases (water is the most notable exception), followed by liquid, and then gas.



The impetus for phase change (changing from one phase to another) is heat. When a solid is heated, it will change into a liquid. The same process of heating will change a liquid into a gas.





Optics

Optics
Lenses change the way light travels. Lenses are able to achieve this by the way in which light travels at different speeds in different mediums. The essentials to optics with lenses deal with concave and convex lenses. Concave lenses make objects appear smaller, while convex lenses make objects appear larger.