NWHS Physics

Equations Page This is a listing of all equations that we are using (updated as we go) from the text and class discussions.  Feel free to print out a copy and update as we go in class.  It will also help to have if equations are permitted on tests! Equation What it is and when to use it... This is simply how we define a duration of time.  The quantities t1 and t2 represent two events (with 1 being first).  The difference in the two time measurements represents a duration of time.  Typically, this is measured in seconds, but always in units of time. This is simply how we define a displacement in the x-direction.  The quantities x1 and x2 represent two positions (with 1 being the starting location, and 2 being the ending location).  The difference in the two position measurements (measured from some common reference point - usually the origin point, or zero) represents a change in position.  Typically, this is measured in meters, but always in units of distance.  The sign of the value designates a direction (positive or negative x). This is just a generic version of the above equation, using the variable d to represent some displacement in normal, three-dimensional space.  This is also measured in units of distance.  The sign of this number simply denotes whether the displacement was away from (positive) or toward (negative) the origin of measurement. Average velocity, measured in units of distance per unit time (typically, meters per second), is the average distance traveled during some time interval.  If the object moves with a constant velocity, it will have the same average velocity during all time durations. When examining an object's displacement-time graph, the slope of a line is equal to the average velocity of the object.  If the object's displacement-time graph is a straight line itself, then the object is traveling with a constant velocity.  If the graph is not a straight line (i.e., a curve) then the slope of that curve's tangent line at some specific time is equal to the object's instantaneous velocity. This is just an equation relating the three main ways average acceleration is expressed in equations.  Remember that if the object has a constant acceleration, its average acceleration is the exact same value. Average acceleration, measured in units of distance per time-squared (typically, meters per second per second), is the average rate at which an object's velocity changes over a given time interval.  This tells us how quickly the object speeds up, slows down, or changes direction only.  This equation is both the definition of average acceleration and the fact that it is the slope of a velocity-time graph.  Like velocity, if the graph is not a straight line then the acceleration is not constant. This is a simple re-write of the definition of acceleration.  It is useful when solving for the final velocity of an object with a known initial velocity and constant acceleration over some time interval. If an object goes from an initial velocity to a final velocity, undergoing constant acceleration, you can simply "average" the two velocities this way.  This is particularly helpful and easy to use if you know that it starts with zero velocity (just divide the final velocity in half). This is a simple re-write of the old distance-equals-rate-times-time formula with average velocity defined as above. This is a very important formula for later use.  It can be used to calculate an object's displacement using initial velocity, constant acceleration, and time.  This is often times used to calculate how far an object moves vertically under the influence of gravity (agravity = g = 9.81 m/s2). Though a bit more complex looking, this equation is really an excellent way to find final velocity knowing only initial velocity, average acceleration, and displacement.  Don't forget to take the square-root to finish solving for vf. This equation is the definition of a vector (in this case, the vector A) through its vertical and horizontal components.  Recall that x is horizontal and y is vertical. This equation relates the lengths of the vector and its components.  It is taken directly from the Pythagorean theorem relating the side lengths of a right triangle. The length of a vector's horizontal component can be found by knowing the length of the vector and the angle it makes with the positive-x axis (in this case, the Greek letter theta). The length of a vector's vertical component can be found by knowing the length of the vector and the angle it makes with the positive-x axis (in this case, the Greek letter theta). Because the components of a vector are perpendicular to each other, and they form a right triangle with the vector as the hypotenuse, the tangent of the vector's angle with the positive-x axis is equal to the ratio of the vertical component length to the horizontal component length.  This is useful for calculating the angle that a vector is pointed when only the components are known. This is Newton's Second Law, written as a definition of the term "force".  Simply put, a force is what is required to cause a mass to accelerate.  Forces are measured in Newtons (N), which are defined in terms of kilograms (kg) of mass and meters per second-squared (m/s2) of acceleration. This is simply a reworking of Newton's 2nd Law to state that the "weight" of an object is really the force that gravity (see our old friend g = -9.81 m/s2) pulls it down with.  Since 'g' is already a negative value, we don't have to mess around with putting a negative to show direction (down is negative in our x-y reference frame). Through experimentation, physicists came to learn that the frictional force between two surfaces depends on two things: the type of material that the surfaces are made of; and how strong a force it is acting perpendicularly between them.  These two factors are seen here in this equation: the Greek letter 'mu' is the coefficient of friction (always positive); and the normal force (normal literally means perpendicular).  Since both are positive, we must include a negative to account for friction's oppositional nature (always goes against motion). Another way to interpret Newton's 2nd Law is to say that the net (sum total) force on an object is what causes its acceleration.  Hence, there may be any number of forces acting on an object, but it is the resultant of all of them that actually causes any acceleration.  Remember, however, that these are force vectors, not just numbers.  We must add them just as we would add vectors. A simple if-then statement that holds true due to Newton's 2nd Law.  If the mass is not accelerated (meaning: sped up, slowed down, or changed direction), then there must be no net force acting on it.  This is not to say that there is no force acting on it, just that the sum total of all the forces acting on it is equal to zero -- all the forces "cancel out". Since force is a vector, I can simply focus on its components when I wish.  So, if I have a series of forces acting on a mass, the sum of their x-components must be equal to the x-component of the net force on the mass.  And, by Newton's 2nd Law, this must be equal to the mass times the x-component of the acceleration (since mass has no direction, and acceleration is also a vector). Similarly as above, if I have a series of forces acting on a mass, the sum of their y-components must be equal to the y-component of the net force on the mass.  And, by Newton's 2nd Law, this must be equal to the mass times the y-component of the acceleration (since mass has no direction, and acceleration is also a vector). If we calculate (or just know) the x- and y-components of the net force acting on an object, it is a snap to find the total net force.  As with any vector, it is merely the sum of its components (added together like a right triangle, of course).  This equation becomes ridiculously easy to use if either one of the components is zero. The definition of momentum is simply mass times velocity.  Take note that an object can have different velocities measured from different reference frames. Newton's 2nd Law re-written as an expression of momentum change.  This is actually how Newton first thought of his law. The Impulse-Momentum Theorem is just an algebraic manipulation of Newton's 2nd Law.  It allows us to think of momentum change as "impulse" (force over some time), and apply the law in a much simpler fashion. In a closed, isolated system, the total momentum of all the objects does not change.  Since "closed" means nothing coming in or going out, we can imagine all our applications talking about a fixed set of objects.  Since "isolated" means no interactions with anything outside the system, we must imagine all our applications involve nothing but those objects and forces that we consider.  These are tough prices to pay, but the result is an INCREDIBLY powerful tool -- total momentum before an interaction is equal to total momentum afterward. In two dimensions, the law still holds -- we just pay attention to the components of the total momentum.  Here, a' refers to object a after the collision. This equation shows the relationship between arclength (s), radius (r), and angle (theta - measured in radians).  It is useful for finding the distance around any circular path (or portion thereof) at a given radial distance. This equation shows the relationship between the period of a pendulum and its length.  It was first discovered by Galileo that the arc of a pendulums swing and the mass at the end of a pendulum do not factor noticeably into the amount of time each swing takes.  Only the length of the pendulum matters. The tangential velocity of an object in uniform (unchanging) circular motion is how fast it is moving tangent to the circle.  Literally the distance around the circle divided by the period of rotation (time for one full rotation). The centripetal acceleration of an object in uniform circular motion is how much its velocity (because of direction, not speed) changes toward the center of the circle in order for it to continue moving in a circle. The force that is required to keep an object moving in a circular path is the centripetal force acting on the object.  This force, directed towards the center of the circle, is really just a derivative of Newton's 2nd Law using centripetal acceleration. The work done on an object is found by multiplying force and distance, but there is a catch.  The force and distance must be parallel to each other.  Only the component of the force in the same direction as the distance traveled does any work.  Hence, if a force applied is perpendicular to the distance traveled, no work is done.  The equation becomes force times distance times the cosine of the angle between them.  Work is measured in units of newtons times meters, or joules (J). Power is a physical quantity equal to the rate at which work is done.  The more time it takes to do the same work, the smaller the power generated, and vice-versa.  Power is measured in units of joules per second, or watts (W). Kinetic energy is simply the energy of motion - the more something is moving (or the more there is to that something), the more kinetic energy it possesses.  Kinetic energy, like all forms of energy, is measured in units of joules (J). Since work and energy have the same units, it stands to reason that they are related.  Well, they are.  Energy is really defined as the ability to do mechanical work.  Therefore, if positive work is done on an object, that object gains kinetic energy (it gets moved). This is just a different version of the above equation.  It is commonly referred to as the Work-Energy Theorem. Gravity is a constant force - always there and always the same.  Since this is the case, we can say that as an object gains height (near the surface of the Earth), it gains some potential to do work (when it eventually falls).  This potential energy is stored energy that can be turned into kinetic later. The total mechanical (motion-related) energy of an object is found by adding the kinetic plus the potential energies for that object - energy due to how fast it is currently going and due to how fast it could go because of its position. This is a simplified mathematical re-statement of the law of energy conservation.  If we have a closed and isolated system, the total mechanical energy does not change. Another way of expressing mechanical energy conservation, this formula says that the total before energy (PE + KE) must equal the total after energy (PE + KE). This is the definition that links the relationship between frequency (measured in Hz -- or cycles per second) and period (measured in units of time -- seconds per cycle).  The inverse relationship between the two is important in relating wave speed with wavelength. The speed of a wave is due to only two features, the frequency of the wave pattern and the wavelength (how far apart the waves are in space).  It is important to note that there is no dependence on the amplitude of the wave for calculating the frequency.  This equation is derived from the simple, constant speed equation -- distance = rate x time. The energy carried by a wave is proportional to the square of the amplitude of the wave (and has nothing to do with wave speed).  Therefore, if I were to double the amplitude of a wave (like doubling the intensity of a sound) I am actually quadrupling the energy that it carries. This equation shows the relationship between three variables of a string attached at two ends and the velocity of a transverse wave that would travel between them.  The variable F is the tension force in the string; the variable m is the mass of the string; and the variable L is the length of the string.  Therefore, in order to make a wave travel faster in a string (like a guitar string), I can do any one of three things while keeping the others constant: increase the tension, decrease the mass of the string, or increase the length of the string.  The denominator (m/L) is sometimes written as the Greek letter mu, and referred to as the "linear density" of the string. Sound waves (or any other form of three dimensional emanation) can be ranked by their intensity -- an objective measure of the amount of energy they carry.  At some distance, r, from a point source of sound with power output, P, the intensity can be calculated in Watts per square-meter.  This is a much more objective view of "loudness" than is measured by the decibel scale, in which the frequencies of the sound matter due to limitations on the human range of hearing (20 Hz to 20 kHz). The Doppler Effect can be detected whenever a wave source and observer are in relative motion.  If they are moving towards each other, then the frequency is observed to be higher than what is actually emitted, and vice versa.  In this equation, the top sign (+ in the numerator, - in the denominator) is used if the source (s) and observer (-) are moving towards each other.  Otherwise, the bottom sign is used in either case.  The entire factor in parentheses is actually a unit-less quantity that acts as a multiplier for the emitted frequency, f. For either an open-ended resonator or a sting attached at both ends, this equation allows you to calculate the frequency of a standing wave with the integer, n, number of antinodes (or loops).  You must know the length of the tube or string, the number of antinodes, and the velocity of the wave in the tube or along the string.  If n = 1, the resulting value will be the 1st resonating frequency (or fundamental harmonic). Incorporating the simple wave speed equation along with the previous equation, this allows us to calculate the wavelength of any resonating frequency knowing only the number of antinodes (therefore, the harmonic number) and the length of the open tube or string.  With it, you could predict the fundamental frequency that would be played by a string of any length (how frets are placed on a guitar). For either a closed-ended resonator (like blowing across the top of a pop bottle), this equation allows you to calculate the frequency of a standing wave with the integer, n, number of antinodes (or loops).  You must know the length of the tube, the number of antinodes, and the velocity of the wave in the tube.  If n = 1, the resulting value will be the 1st resonating frequency (or fundamental harmonic).  Important to note that closed-resonators are able to achieve the same resonant frequency, but at one-half the length. Incorporating the simple wave speed equation along with the previous equation, this allows us to calculate the wavelength of any resonating frequency knowing only the number of antinodes (therefore, the harmonic number) and the length of the closed tube.  With it, you could predict the fundamental frequency that would be played by a pop bottle with any level of water within it (therefore, of any length).