A software development company has three jobs to do. Two of the jobs require three programmers, and the other requires four. If the company employs ten programmers, how many different ways are there to assign them to the jobs?
Chapter One Useful Unit Conversions 1 in = 2.54 cm 1 ft = 0.305 m 1 mile = 1.609 km 1 mph = 0.447 m/s 1 m = 39.37 in 1 km = 0.621 mile 1 m/s = 2.24 mph ● the number of significant figures after multiplication or division is equal to the number of significant figures of the least accurately known quantity ● the number of decimal places after addition or subtraction is equal to the smallest number of decimal places Chapter Three ● dimensions refer to the quality ([L]], [M], [T], etc…) ● units refer to the quantity ● don’t say “the length of the table is 6 ft” ● instead say “the measure of the length of the table is 6 ft” ● scalar: has magnitude (number with units) ● vector: quantity with magnitude and direction ● vectors are written in boldface or with an arrow on top ● displacement has to do with your initial and final position ● moving along the xaxis can be represented with x or i ● moving along the yaxis can be represented with y or j ● moving along the zaxis can be represented with z or k ● A =B only A and B have the same magnitude and direction ● we can move vectors around as long as you don’t change orientation or length ● it is common practice to measure the angle from the positive xaxis and to measure it in a counterclockwise (ccw) direction ● adding vectors geographically: place the tail of the second vector at the tip of the first (tip to tail method) ○ you can do this with as many vectors as you want ● the negative of a vector has the same magnitude but is pointing in the other direction (flip it 180 degrees on its tip) ● Adding vectors using components (quantitative method) ○ Find the x and y components of each vector to be added ○ Add all the x and all the y components separately ○ Find the magnitude of the resultant vector Chapter Two ● mechanics is the study of how objects move, how they respond to external forces, and how other factors such as size, mass, and mass distribution affect their motion ● kinematics is the description of motion ● dynamics is the study of the cause of motion ● we must set up a coordinate system (origin and positive direction) in order to define motion ● if an object is in the air, its acceleration is a constant 9.81 m/s^2 ○ this is the only acceleration acting on it ● average speed is defined as the distance traveled divided by the time the trip took ● average velocity is displacement divided by time ● speed is always positive as it is a scalar quantity ● velocity can be positive or negative based on the direction of displacement because it is a vector quantity ● a speedometer measures speed, not velocity ● in a position vs. time graph, the slope is the average velocity ○ it is a motion diagram put as a graph ● in a velocity vs. time graph, the slope is the average acceleration ● in a position vs. time graph, the line tangent to a point is the instantaneous velocity at that point ● between any two times on a position vs. time graph, draw a line between the two points and that slope is the average velocity ● uniform velocity is constant velocity motion ● when velocity is constant, the average velocity over any interval is equal to the instantaneous velocity at any point ● the magnitude of the average velocity is only equal to speed when it is a straight line ● acceleration and deceleration should not be confused with the direction of velocity and acceleration ● if you are slowing down, your acceleration and velocity will be in opposite directions ● freefall is the motion of an object subject only to the influence of gravity ● once you release something from your hand, it is in freefall ● when you throw something up, at its peak the velocity is zero but the acceleration is still a constant g ● air resistance only has a big effect if the object is light like a feather or a piece of paper ● mass doesn’t cause the effect of air resistance (surface area does) ● on the earth’s surface, the acceleration on the yaxis is never zero ● when you throw something, its trajectory has a symmetric shape and velocity (parabola shaped) ● g is always 9.8 but your acceleration can be g (you will never have g = 9.8) Three Main Equations of Motion Chapter Four ● two dimensional motion is when an object is moving in two dimensions at the same time (x and y) ● make sure you break these vectors down into x and y components ● x component = dcos(theta) ● y component = dsin(theta) ● motion in the x and y directions should be solved separately ● ignore air resistance ● gravity always acts downward ● if the y axis points up, acceleration in the x direction is zero and acceleration in the y direction is 9.81 m/s^2 ● solve x and y independently; the only thing they have in common is time ● acceleration is independent of direction ● in projectile motion, x velocity is constant and x acceleration is zero ● launch angle: direction of initial velocity with respect to the horizontal ● horizontal launch is a zero launch angle; there is only an x component of initial velocity ● if your launch angle is not zero, you need to look at both the x and y components ● range: horizontal distance an object travels ● if the final and initial elevation are the same, you can use R = (V0g)sin(2theta) ● range is a maximum when theta = 45 degrees ● if you have two complementary angles (add up to 90 degrees) they have the same range ● if final and initial elevation are the same and you’re solving for range, you don’t have to break it into x and y components ● the least speed is found at the highest point in the trajectory ● if all projectiles reach the same height, they are in the air the same amount of time (time in the air is determined by height reached, not the distance traveled!) Physics Test Two Chapter Five ● Newton’s Laws of Motion ○ An object at rest will stay at rest and an object in motion will stay in motion unless acted on by an outside force ○ Force is equal to the mass of an object times its acceleration (F = ma) ○ For each force, there is an equal and opposite force. ● if two object have the same force acting upon them, the smaller object will accelerate more (e.g. hammer and nail) ● dynamics is the study of the cause of motion ● force is a vector because it has magnitude and direction ○ sometimes force must be broken up into its x and y components ● mass is a measure of how hard it is to change an object’s velocity Newton’s First Law ● inertia means an object wants to stay where it is ○ an inertial reference frame is one in which the first law of motion is true ○ if you stop pushing an object, it doesn’t stop moving unless there are external forces acting on it such as friction ○ you require a force to start the motion of an object, but it does not need force to keep moving (an object can be in motion without force) ○ law of inertia: in the absence of any net external force, an object will keep moving at a constant speed in a straight line, or remain at rest ● two equal weights exert twice the force of one and this can be used for the calibration of a spring Newton’s Second Law ● acceleration is inversely proportional to mass ● the direction of acceleration is in the same direction as the resulting force ● Net Force = F1 + F2 + F3….. ● 1 newton = 1 (kg x m)/s^2 ● force can be measured in newtons ● contact force: agent of force that is actually making contact with the surface ● there are also long range forces such as gravity ● forces cannot act by themselves which means there will always be an agent or external action ● weight: the gravitational pull of the earth on an object or near the surface of the earth ● tension is an equilibrium force such as something hanging by a string ● spring force: contact force that can push when compressed or pull when stretched ● drag: resistive force that opposes the direction of motion ○ e.g. friction and air resistance ○ ignore air resistance unless told to consider it ● thrust: occurs when a jet or rocket engine expels gas molecules at high speed (this is a contact force) ● normal force: due to the atomic structure of solids, if an object is placed on some material that material will exert an upward spring force ○ always perpendicular to the surface ● kinetic friction: occurs when an object is already moving and it opposed the motion and its vector points in the opposite direction ● static friction: occurs when the object is at rest ○ it keeps it at rest so it is a force opposite the direction ○ the object would move if there was more force ● if something is not moving or has constant velocity, you can treat it basically the same because acceleration is zero ● Free Body Diagram Steps ○ identify all forces ○ draw a coordinate system ○ represent the object as a dot at the origin of the coordinate axes ○ draw vector representing each of the forces ○ draw and label the net force vector Newton’s Third Law ● every force occurs as one member of an action/reaction pair ● members of the action/reaction pair act on different objects ● the forces are equal in magnitude and opposite in direction ● if an object is moving at constant speed, the net force is zero ● apparent weight: your perception of your weight is based on the contact forces between your body and your surroundings ○ e.g. elevator problems ○ if your surroundings are accelerating, your apparent weight may be more or less than your actual weight ○ if you’re stationary or moving with constant speed, your apparent weight is equal to your actual weight ○ if you’re accelerating upward, your apparent weight is greater than your actual weight ○ if you’re accelerating downward, your apparent weight is less than your actual weight ● weightlessness: apparent weight is zero ○ force is gone so g = a and you appear to have no weight ○ this is the case of astronauts in space ● even if you have a tension force and you’re moving upward, if you speed is constant then it’s like you’re at rest Chapter Six ● friction has its basis in surfaces that are not completely smooth ● ● kinetic friction is equal to the coefficient of kinetic friction times the normal force ● static friction is equal to the coefficient of static friction times the normal force ● kinetic friction is the friction experienced by objects sliding against one another ● kinetic and static friction are independent of the relative speed of the surfaces and of their area of contact force ● static friction is experienced by objects at rest ● static friction can range from zero to its maximum value ○ the maximum value is just before the object starts to move and right before it switches to kinetic friction ● whenever you pull a string, it becomes taut so there is tension in the spring ● the tension in an actual rope depends on the length and mass of the rope, but for this class we will assume there is no mass to the rope ● with a pulley, there are two tension forces that both point toward the suspension force ● Hooke’s Law: the force in a spring increases with the amount the spring in compressed or stretched ○ the force on a spring is equal to kx where k is the spring constant ● an object is in equilibrium when the force acting on it is zero ● this happens when an object is not moving or moving at a constant speed (a = 0) ● when forces are exerted on connected objects, they will have the same acceleration ● if there are two boxes connected by a string, we treat each box as a separate system ● NORMAL FORCE DOES NOT ALWAYS EQUAL WEIGHT ● Uniform Circular Motion is when speed is constant but direction is constantly changing so the acceleration is nonzero ○ if you take any two vectors, their difference will point to the center of the circle ○ centripetal acceleration always points to the center of the circle ○ the resulting force of this acceleration is the centripetal force ○ centripetal acceleration = ○ there must be a force acting on the objects because if there wasn’t they would be moving in a straight line and not a circle (this is the centripetal force) ○ centripetal force = ○ centripetal force can be produced by friction, tension, gravity,normal force, etc…. ○ when a call is swinging in a circle on a string, the centripetal force is the tension in the circle ○ in a vertical circle, tension and weight can or cannot be in the same direction ○ a car going around a loop is kept in the circle by friction pointing to the middle of the circle ○ if there’s no friction, the car might be on a bank (inclined) and you’ll have to use the xcomponent of the normal force Physics Test Three Chapter Seven ● The basic definition of energy is work ● Work is force times displacement, or the change in kinetic energy ● Work is a scalar quantity that comes from two vectors ● You want the component of the force that is parallel to the displacement, which may not simply be the force ● If the displacement is perpendicular to the force, there is no work ● Work may be positive, negative, or zero ● If the angle of the force to displacement is acute, the work is positive ● If the angle of the force to displacement is obtuse, the work is negative ● If you have multiple forces on an object, you can find the work done by the net force ○ Find the components of the net force that’s parallel to the displacement and multiple it by the displacement ● Energy transformations within the system do not change the overall energy of the system ● Work Energy Theorem: Work is the change in kinetic energy ● If force is constant, we can calculate work graphically ○ Graph force on the yaxis and displacement on the xaxis ○ The area under the curve is work ● If force is changing but has several successive constant forces, you can calculate it graphically as well in the same way ● We can also calculate a continuously varying force using integration (this is calculus and more than what we’ll do in this class) ● Work done by the spring force: ● Power is the rate at which work is done ○ Power = work/time ○ Power is measured in watts or horsepower (1 hp = 746 W) ○ In cars, use horsepower ● Two situations may have the same final kinetic energy, but their power is different because they took different times Chapter Eight ● Conservative force: work done by this force is stored as usable energy (like putting money from your checking account into your savings account) ● Gravity is an example of a conservative force (by something falling due to gravity, kinetic energy is created so gravity is a force that has potential to be used) ● Nonconservative force: cannot be stored as usable energy (like taking money from checking account and spending it) ● Friction is an example of a nonconservative force (friction creates kinetic energy when you rub your hands together but when you stop you can’t get that heat back) ● When moving an object around a closed path, work done by a conservative force is zero ○ This is not true for a nonconservative force ● Work done by friction is negative (decreasing kinetic energy) because friction is a force against motion ● U = potential energy ○ Work = initial U final U ● Work you do by conservative forces is stored as potential energy ● You can choose your zero potential energy based on where you decide to start ● Spring force is also a conservative force ● Mechanical energy = U + K ○ Using this formula, and only using conservative forces, initial energy equals final energy ○ If there are only conservative forces, mechanical energy is constant ● Law of Conservation of Energy: the sum of potential and kinetic energy is equal before and after ● Gravitational potential energy = mgh (mass x gravity x height) Chapter Nine ● Linear momentum: p = mv ● Unless an outside force acts on it, an object in motion stays in motion and this natural tendency is called momentum ● Impetus: the ability for an object to move in that straight line path ○ Momentum is the measure of impetus ● Momentum is like inertia in how it resists motion ● Momentum is a vector whose direction is the same as the velocity ● Newton’s Second Law of Motion is only true for objects of constant (so most everything we work with) ● When mass is changing, force = change in momentum/change in time ● Impulse: momentary forward force ○ E.g. kicking a soccer ball ○ Huge amount of force in a very short time interval ○ Impulse = force x time = change in momentum ○ Impulse is a vector in the same direction as the force ○ Impulse is the area under a force x time graph ● Change in momentum can be a small force acting for a long time or a large force acting for a short time ● Conservation of Momentum Concept ○ Conservation of Linear Momentum: if the net force acting on an object is zero, its momentum is conserved ○ Collision: two objects striking each other and the time of collision is short enough that external forces can be ignored ○ Inelastic collision: the momentum of a system is conserved, but kinetic energy is not conserved ■ Two objects colliding and moving away from each other ■ Completely inelastic collision: two things stick together after the collision ■ Perfectly inelastic: two objects move together at a common final velocity after collision ○ Elastic collision: momentum and kinetic energy are conserved ■ One dimensional elastic collision: moving object collides with stationary object then stops moving while the stationary ball starts moving (billiard ball example) ■ With elastic collisions, we have an equation for kinetic energy and an equation for momentum so we can work with both of those to make these: ■ You can put these on your equation sheet Physics Test Four Chapter Eleven ● Static Equilibrium ○ The resultant external force is zero ○ The external torque around any axis is zero ● First condition implies that a = 0 ● Second condition implies that alpha = 0 ● We can choose our axis of rotation to be wherever we want ● When you have a fulcrum you have to consider the position of the fulcrum ● Angular momentum = L = Iw = rmv = rp ● Torque is the change in angular momentum Chapter Twelve ● G = Gravitational constant = 6.67 x 10^11 ● Law of universal gravitation says F = force acting between any two objects = Gm1m2/r^2 ○ R is the distance between the two centers of mass ● Gravitational force is a vector and a force exists between any two massive particles ● Inverse square law = force is inversely proportional to radius ● So as the distance between two particles increase, the gravitational force will decrease ● Since G is really small, the objects have to be really massive for the gravitational force to not be negligible ● Principle of superposition: net force is the vector sum of the individual forces ● Force between any object on the surface of the earth = F = mg = m (GMearth/(r of earth)^2 ● Kepler’s three laws of planetary motion ○ Each planet moves in an elliptical orbit with the sun at one focus ○ The line from the sun to any planet sweeps out equal areas at equal time intervals ○ For each planet, the square of the period is proportional to the cube of the average radius ● K = 2.9 x 10^19 ○ This is for all planets ○ This does not involve the orbiting object ● Anything moving in a circle in constant velocity has centripetal force ● Geosynchronous satellite: orbit around the earth is 24 hours ● GPS satellites are 12 hours, twice the speed of geosynchronous satellites