Physics Equations

Gravitational Potential Energy

Coulomb's Law of Electrostatics

*ε*_{0} = 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.

*μ*_{0} = 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*10^{8} ^{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

*G* = 6.67384*10^{-11} m^{3}kg^{-1}s^{-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.

*m _{e}* = 9.10938291*10

*m _{p}* = 1.972621777*10

*R*_{∞} = 1.09737316*10^{7} m is the **Rydberg constant**. It describes the energy levels of the Hydrogen atom.

*N _{A}* = 6.02214129*10

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

*k _{B}* = 1.3806488*10

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

**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.

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

The following equations can be used to describe motion.

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:

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:

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

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

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.

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.

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

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.

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?

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.

Ex. What' the centripetal acceleration of a car traveling on a curve with radius of 40 meters at 15 m/s?

A yo-yo is traveling with an angular velocity of 5 rads/s on a string of .5 m, what is the centripetal acceleration?

Work is the amount of energy required to move an object a distance, r, using a force, F.

Ex. How much work is required to lift a 1000kg payload 1500m?

Kinetic energy is the energy associated with a moving object.

Ex: How much energy is associated with a 250g baseball pitched at 30 m/s?

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.

Ex. If the same baseball was directed straight up, how high would it go?

*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*:

Where **k _{e}** is Coulomb's constant (9.0*10

**q _{1}** and

**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?

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:

Example Problem:

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

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.

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.

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.

*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)cos*θ*

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 *d***A**). 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 related energy and frequency of an electromagnetic wave through the Planck constant (h) It is given as but it is also given in quantum mechanics as

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

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:

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.

From this formula, all the others can be derived in a similar manner to the rod, length (*L*), rotated about its end, shown below.

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.

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.

^{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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