AFIT Kneeboard: STEM Review Materials

AFIT Kneeboard: STEM Review Materials
Physics Equations

Table of Contents

Fundamental Constants

Kinematics

Newton's Second Law

Gravitational Force

Coefficient of Friction

Centripetal Acceleration

Work

Kinetic Energy

Gravitational Potential Energy

Coulomb's Law of Electrostatics

Electric Field Force

Wave Formula

Maxwell's Equations

Planck's Relation

Moment of Inertia

 

Fundamental Constants

ε0 = 8.85418781*10-12 F m-1 is the electric constant, also known as the permittivity of free space or the vacuum permittivity. This constant describes the amount of electric field that is created by a certain quantity of charge.

μ0 = 4π*10-7 N A-2 is the magnetic constant, also known as the permeability of free space or the vacuum permeability. The magnetic constant describes the amount of magnetic field that is created by a certain quantity of current.

c = 2.9979245*108 m/s is the speed of light in a vacuum. It represents the "speed limit" for all objects in the universe. It is also related to the electric and magnetic constants because light is an electomagnetic wave. By manipulating Maxwell's Equations, one can show that the speed of light is equal to Speed o' Light

G = 6.67384*10-11 m3kg-1s-2 is the Newtonian constant of gravitation. It describes the how strongly objects of certain masses attract one another.

h = 6.62606957*10-34 J s is Plank's constant. It describes the relationship between a photon's frequency and its energy. Plank's constant is also used to describe black-body radiation, the photoelectric effect, and the uncertainty principle.

e = 1.602176565*10-19 C is the elementary charge. It is the magnitude of the charge of an electron or proton.

me = 9.10938291*10-31 kg is the mass of the electron.

mp = 1.972621777*10-27 kg is the mass of the proton.

R = 1.09737316*107 m is the Rydberg constant. It describes the energy levels of the Hydrogen atom.

NA = 6.02214129*1023 mol-1 is Avogadro's constant, sometimes called Avagadro's number. It is the number of particles in a mole of substance. It is useful in many chemistry problems and for using the ideal gas law.

R = 8.3144621 J mol-1K-2 is the Molar Gas constant. It is used to relate the quantities of the ideal gas law when it takes the form: Aero Ideal Gas Law (usually used in Aeronautical Engineering problems.

kB = 1.3806488*10-23 J K-1 is the Boltzmann constant. It is also used in the ideal gas law, but usually in physics problems where the ideal gas law is written in the form: Physics Ideal Gas Law The Boltzmann constant relates to the molar gas constant and Avagadro's number in the following way: kB = R/NA. The Boltzmann constant also relates the average kinetic energy of a gas to its temperature.

Non-SI Units

The electron volt (eV) is a unit of energy that is useful in describing the relatively small amounts of energy that often appear in quantum physics. The electron volt is the amount of energy required to move an electron across a potential difference of 1 volt which is equal to 1.602176565*1-19 J.

The unified atomic mass unit is a unit of mass that is useful in describing the small amounts of mass associated with single atoms, molecules, and subatomic particles. The atomic mass unit is equal to 1.660538921*10-27 kg. It is defined as u definition

Kinematics

Position is a vector (geometric object with magnitude and direction that can be added with other vectors) which describes an object's location in relation to a reference point.

Velocity is a vector and therefore contains two components: speed and direction. It measures how quickly the position of an object changes with respect to time.
velocity

Acceleration is another vector which measures the change in velocity with respect to time.

 

Kinematic Equations14

The following equations can be used to describe motion.
Position

Velocity

Ex. A car is moving down the road with a velocity of 10 m/s. It accelerates at 5m/s/s for 5 seconds. The final velocity can be found with:
velocity
velocity example

Ex. Now you want to know how far the car traveled. You noticed that the car passed mile marker zero with a velocity of 10m/s, when it accelerated for 5 seconds. The distance the car went can be found with:
Position
velocity example

Newton's Second Law15

Newton's second law mathematically describes the phenomena called force, which is the amount of 'push/pull' felt by an object. It is measured in units of Newton's or kg*m/s^2.
2nd Law

Ex. Let's say you have a ball of mass .5kg and throw it with an acceleration of 5m/s/s. The amount of force that you applied to the ball is Example of force
Now you apply a force of 10 N to a ball with a mass of .5 kg and want to know how much it accelerates. second example of force

Gravitational Force16

All objects with mass attract each other with a force called gravity. The equation below models the force and contains a constant, G, that has been experimentally measured.
Gravity Constant
Gravity

Ex. What force does the earth exert on the moon to keep it in orbit? (assume a circular orbit)

Coefficient of Friction17

Friction is a non-conservative force (conservative forces have the property that the work done to move between two points is the same regardless of the path taken) that resists motion and is created when two surfaces come in contact with one another. The coefficient of friction is directly proportional to the magnitude of the force and can be caluclated using the equaiton below. There are two basic types of friction: static friction and kinetic friction. A force must overcome static friction to put an object in motion, while kinetic friction resists the motion after the initial movement.
Gravity

Ex. A 5 kg box sits on a table with a coefficient of static friction of 0.8. What is the force required to overcome friction?
Example of coefficient of friction

Centripetal acceleration18

Since velocity contains both magnitude and direction, centripetal acceleration occurs when there is a change in the direction of an object's velocity. It can be calculated using either an object's linear or angular velocity and the radius of curvature of the path traveled.

Centripetal Accel

Ex. What' the centripetal acceleration of a car traveling on a curve with radius of 40 meters at 15 m/s?
 ((15m))²/40m=5.625 m/s²

A yo-yo is traveling with an angular velocity of 5 rads/s on a string of .5 m, what is the centripetal acceleration?
 (5rads/s)²*.5m=12.5 m/s²

Work19

Work is the amount of energy required to move an object a distance, r, using a force, F.
Gravity
Ex. How much work is required to lift a 1000kg payload 1500m?
 F=1000kg*9.81/s² =9810 N, so W=9810 N*1500 m = 14715000 Joules or J
work

Kinetic Energy20

Kinetic energy is the energy associated with a moving object.
kinetic energy

Ex: How much energy is associated with a 250g baseball pitched at 30 m/s?
 K= .25kg*(30m/s)²/2 = 112.5 J
kinetic energy

Gravitational Potential Energy21

Gravitational potential energy is the energy associated with objects elevated in a gravitational field. Since force is in the same direction as the direction of movement the dot product equals the product of the magnitude of the two vectors.
Gravity
Ex. If the same baseball was directed straight up, how high would it go?
 K=U
potential

Coulomb's Law of Electrostatics22

Charge is a property of matter that causes it to experience an electrical force. The electrical force causes like charges to repel one another and opposite charges to attract one another. The magnitude of the force felt by interacting charges is described by Coulomb's Law of Electrostatics:

Coulumbs Law

Where ke is Coulomb's constant (9.0*109 N*m2 *C-2).

q1 and q2 are the magnitudes of the two charges.

r is the distance separating the two charges.

 

Example Problem:

A mystery particle is causing a 2C particle to experience a 2N force from a distance of 3m. What is the magnitude of the charge of the mystery particle?

Coulomb Problem
Coulomb Workout

Electric Field Force23

In the previous section, we discussed the interaction of two charged, stationary particles, but what if we have only one particle? Coulomb's law tells us that if we have one charged particle, it changes the space around it so that if we drop a test charge in the vicinity of the first charge, that test charge will experience a force. We use the words electric field to describe the change that the first charge creates in space around it. Electric fields are measured in N/C or V/m:

N/C to V/m

Example Problem:

A 3C test charge is placed in a 4V/m electric field. What force will the test charge experience?

Electric Picture
Electric force calculation

Wave Formula24

The wave formula describes the relationship between velocity, wavelength, and frequency. It is commonly used with the speed of light (c) as velocity to find either frequency or wavelength.
wave formula

Ex. Find the frequency of a wave given a wavelength of 1 mm and the fact that it is traveling at the speed of light in a vacuum.
wave example

Maxwell's Equations25

In 1864, James Clerk Maxwell made one of the most significant contributions in the history of physics by discovering that electricity and magnetism are the two closely related components of the electromagetic field. His findings are described mathematically with Maxwell's four equations:




Let's look at the equations individually.



Gauss's Law for Electrical Charge


Stationary charges create electric fields.



Gauss's Law for Magnetism


Magnetic field lines go in loops. Every magnet has a positive and negative (or north and south) end. There are no magnetic monopoles.



Faraday's Law


Changing magnetic fields create electric fields. If we move magnets, bring more magnets, take away magnets, or do anything else to change a magnetic field, an electric field is created.



Ampere's Law with Maxwell's Modification


Changing electric fields create magnetic fields. If we move charges (i.e. in a current), bring more charges or take away charges (i.e. on capacitor plates), or do anything else to change the electric field, a magnetic field is created.

Flux26

Flux is a concept that comes up frequently when working with Maxwell's Equations. Flux is the dot product of an area vector and a field vector. Area vectors are orthogonal to the surface that they represent:

When we calculate flux, if the area vector is parallel to the field, then flux is simply the product of the magnitudes of the area and the field:

If we want to calculate flux where the area vector and the field are not parallel, then we calcute flux with the formula: Φ = Field(Area)cosθ

Now that we know how to calculate flux, we can focus on the flux through a special type of surface, a Gaussian Surface. If we refer back the first of Maxwell's Equations (also known as Gauss's Law) , we notice a circle around the integral sign and that the integral is evaluated with respect to Area (hence the dA). This means that we are concerned with a surface that can be any shape, but that surface must be closed (i.e. a spherical shell is closed, while a sheet of paper is not). Using Gauss's Law, we can find the electric field at a Gaussian surface based on the amount of charge that surface encloses.

Example Problem: A large sphere has a charge density of 1 Coulomb per cubic meter. What is the magnitude of the electric field 2 meters from the sphere's center.


Example Problem 2: A copper wire carries 3A of current. What is the magnetic field 0.5m from the wire?


Planck relation28

Planck related energy and frequency of an electromagnetic wave through the Planck constant (h) It is given as Planck relation but it is also given in quantum mechanics as Planck pi formula

Ex. What is the energy of an electromagnetic wave that has a frequency of 256MHz
planck example

Moment of Inertia29

The moment of inertia (I) is the same for rotational motion as mass is for linear motion. Because of this, the moment of inertia must be specified with respect to a rotational axis. It describes the resistance to change in rotation an object has about its axis due to Newton's law. In most cases, we do not deal with a point mass rotating about an axis, but, rather, some other shape with a mass distribution. With this in mind, it is important to understand how the moment of inertia can be calculated. The moment of inertia of a single point mass can be derived from the definitions of angular momentum and angular velocity:
MOI point mass graph about L

The general formula is based on this definition. Since everything else can be constructed from multiple point masses, the general formula is calculated by integrating over the entire mass distribution.
MoI general
From this formula, all the others can be derived in a similar manner to the rod, length (L), rotated about its end, shown below.
Rod Derivation
There are several other scenarios in which the moments of inertia are common enough that the formulas have been worked out and are commonly known. Several of these have been listed below.
MoI Equations
Ex. Two spheres of the same mass and radius are released at the top of a ramp. One is a hollow shell while the other is solid. Which will reach the bottom of the ramp first? (Assume no slipping)

The solid sphere will reach the bottom first since it has a lower moment of inertia. That means that it will resist the change in motion less than the hollow shell.
MoI Example

 

Citations

14 Miller, Franklin Jr, College Physics. 4th ed. New York: Harcourt Brace Jovanovich, 1977. 26-48.
15 Ibid. 59.
16 Ibid. 68-70.
17 Ibid. 87-92.
18 Ibid. 177-178.
19 Ibid. 118-120.
20 Ibid. 122-124.
21 Ibid. 122-124.
22 Ibid. 387-389.
23 Ibid. 400-404.
24 Ibid. 234-235.
25 Edminister, Joseph A. Schaum's Easy Outlines: Electromagnetics. New York: McGraw-Hill, 2003. 69-83.
26 Ibid. 25-29.
27 Miller, Franklin Jr, College Physics. 4th ed. New York: Harcourt Brace Jovanovich, 1977. 338-339.
28 Singh, Jasprit. Modern Physics for Engineers. New York: John Wiley & Sons, Inc., 1999. 31-35.
29 Miller, Franklin Jr, College Physics. 4th ed. New York: Harcourt Brace Jovanovich, 1977. 181-183.
30 Anderson, John D. Fundamentals of Aerodynamics. New York: McGraw-Hill, 2011. 53-54.
31 Ibid. 209.

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