# Science Articles

The universe is a strange and fascinating place, from the subatomic all the way up to the astronomical. Study and conduct experiments in any and every branch of science that calls to you.

## Articles From Science

### Filter Results

Article / Updated 03-20-2024

Chemists aren’t satisfied with measuring length, mass, temperature, and time alone. On the contrary, chemistry often deals in calculated quantities. These kinds of quantities are expressed with derived units, which are built from combinations of base units. Here are some examples: Area (for example, catalytic surface). and area has units of length squared (square meters, or m2, for example). Volume (of a reaction vessel, for example). You calculate volume by using the familiar formula Because length, width, and height are all length units, you end up with or a length cubed (for example, cubic meters, or m³). Density (of an unidentified substance). Density, arguably the most important derived unit to a chemist, is built by using the basic formula Density = Mass / Volume. Pressure (of gaseous reactants, for example): Pressure units are derived using the formula Pressure = Force / Area. The SI units for force and area are newtons (N) and square meters (m²), so the SI unit of pressure, the pascal (Pa), can be expressed as N/m². Let’s try an example. A physicist measures the density of a substance to be 20 kg/m³. His chemist colleague, appalled with the excessively large units, decides to change the units of the measurement to the more familiar grams per cubic centimeter. What is the new expression of the density? The answer is 0.02 g/cm³. A kilogram contains 1,000 (10³) grams, so 20 kg equals 20,000 g. Well, 100 cm = 1 m; therefore, (100 cm)³= (1 m)³. In other words, there are 100³ (or 106) cubic centimeters in 1 cubic meter. Doing the division gives you 0.02 g/cm³. You can write out the conversion as follows:

View ArticleArticle / Updated 03-20-2024

Physics is filled with equations and formulas that deal with angular motion, Carnot engines, fluids, forces, moments of inertia, linear motion, simple harmonic motion, thermodynamics, and work and energy. Here’s a list of some important physics formulas and equations to keep on hand — arranged by topic — so you don’t have to go searching to find them. Angular motion Equations of angular motion are relevant wherever you have rotational motions around an axis. When the object has rotated through an angle of θ with an angular velocity of ω and an angular acceleration of α, then you can use these equations to tie these values together. You must use radians to measure the angle. Also, if you know that the distance from the axis is r, then you can work out the linear distance traveled, s, velocity, v, centripetal acceleration, ac, and force, Fc. When an object with moment of inertia, I (the angular equivalent of mass), has an angular acceleration, α, then there is a net torque Στ. Carnot engines A heat engine takes heat, Qh, from a high temperature source at temperature Th and moves it to a low temperature sink (temperature Tc) at a rate Qc and, in the process, does mechanical work, W. (This process can be reversed such that work can be performed to move the heat in the opposite direction — a heat pump.) The amount of work performed in proportion to the amount of heat extracted from the heat source is the efficiency of the engine. A Carnot engine is reversible and has the maximum possible efficiency, given by the following equations. The equivalent of efficiency for a heat pump is the coefficient of performance. Fluids A volume, V, of fluid with mass, m, has density, ρ. A force, F, over an area, A, gives rise to a pressure, P. The pressure of a fluid at a depth of h depends on the density and the gravitational constant, g. Objects immersed in a fluid causing a mass of weight, Wwater displaced, give rise to an upward directed buoyancy force, Fbuoyancy. Because of the conservation of mass, the volume flow rate of a fluid moving with velocity, v, through a cross-sectional area, A, is constant. Bernoulli’s equation relates the pressure and speed of a fluid. Forces A mass, m, accelerates at a rate, a, due to a force, F, acting. Frictional forces, FF, are in proportion to the normal force between the materials, FN, with a coefficient of friction, μ. Two masses, m1 and m2, separated by a distance, r, attract each other with a gravitational force, given by the following equations, in proportion to the gravitational constant G: Moments of inertia The rotational equivalent of mass is inertia, I, which depends on how an object’s mass is distributed through space. The moments of inertia for various shapes are shown here: Disk rotating around its center: Hollow cylinder rotating around its center: I = mr2 Hollow sphere rotating an axis through its center: Hoop rotating around its center: I = mr2 Point mass rotating at radius r: I = mr2 Rectangle rotating around an axis along one edge where the other edge is of length r: Rectangle rotating around an axis parallel to one edge and passing through the center, where the length of the other edge is r: Rod rotating around an axis perpendicular to it and through its center: Rod rotating around an axis perpendicular to it and through one end: Solid cylinder, rotating around an axis along its center line: The kinetic energy of a rotating body, with moment of inertia, I, and angular velocity, ω: The angular momentum of a rotating body with moment of inertia, I, and angular velocity, ω: Linear motion When an object at position x moves with velocity, v, and acceleration, a, resulting in displacement, s, each of these components is related by the following equations: Simple harmonic motion Particular kinds of force result in periodic motion, where the object repeats its motion with a period, T, having an angular frequency, ω, and amplitude, A. One example of such a force is provided by a spring with spring constant, k. The position, x, velocity, v, and acceleration, a, of an object undergoing simple harmonic motion can be expressed as sines and cosines. Thermodynamics The random vibrational and rotational motions of the molecules that make up an object of substance have energy; this energy is called thermal energy. When thermal energy moves from one place to another, it’s called heat, Q. When an object receives an amount of heat, its temperature, T, rises. Kelvin (K), Celsius (C), and Fahrenheit (F) are temperature scales. You can use these formulas to convert from one temperature scale to another: The heat required to cause a change in temperature of a mass, m, increases with a constant of proportionality, c, called the specific heat capacity. In a bar of material with a cross-sectional area A, length L, and a temperature difference across the ends of ΔT, there is a heat flow over a time, t, given by these formulas: The pressure, P, and volume, V, of n moles of an ideal gas at temperature T is given by this formula, where R is the gas constant: In an ideal gas, the average energy of each molecule KEavg, is in proportion to the temperature, with the Boltzman constant k: Work and energy When a force, F, moves an object through a distance, s, which is at an angle of Θ,then work, W, is done. Momentum, p, is the product of mass, m, and velocity, v. The energy that an object has on account of its motion is called KE.

View ArticleArticle / Updated 03-20-2024

When an object moves in a circle, if you know the magnitude of the angular velocity, then you can use physics to calculate the tangential velocity of the object on the curve. At any point on a circle, you can pick two special directions: The direction that points directly away from the center of the circle (along the radius) is called the radial direction, and the direction that’s perpendicular to this is called the tangential direction. When an object moves in a circle, you can think of its instantaneous velocity (the velocity at a given point in time) at any particular point on the circle as an arrow drawn from that point and directed in the tangential direction. For this reason, this velocity is called the tangential velocity. The magnitude of the tangential velocity is the tangential speed, which is simply the speed of an object moving in a circle. Given an angular velocity of magnitude the tangential velocity at any radius is of magnitude The idea that the tangential velocity increases as the radius increases makes sense, because given a rotating wheel, you’d expect a point at radius r to be going faster than a point closer to the hub of the wheel. A ball in circular motion has angular speed around the circle. Take a look at the figure, which shows a ball tied to a string. The ball is whipping around with angular velocity of magnitude You can easily find the magnitude of the ball’s velocity, v, if you measure the angles in radians. A circle has the complete distance around a circle — its circumference — is where r is the circle’s radius. In general, therefore, you can connect an angle measured in radians with the distance you cover along the circle, s, like this: where r is the radius of the circle. Now, you can say that v = s/t, where v is magnitude of the velocity, s is the distance, and t is time. You can substitute for s to get In other words, Now you can find the magnitude of the velocity. For example, say that the wheels of a motorcycle are turning with an angular velocity of If you can find the tangential velocity of any point on the outside edges of the wheels, you can find the motorcycle’s speed. Now assume that the radius of one of your motorcycle’s wheels is 40 centimeters. You know that so just plug in the numbers: Converting 27 meters/second to miles/hour gives you about 60 mph.

View ArticleArticle / Updated 03-20-2024

In physics, you can apply Hooke’s law, along with the concept of simple harmonic motion, to find the angular frequency of a mass on a spring. And because you can relate angular frequency and the mass on the spring, you can find the displacement, velocity, and acceleration of the mass. Hooke’s law says that F = –kx where F is the force exerted by the spring, k is the spring constant, and x is displacement from equilibrium. Because of Isaac Newton, you know that force also equals mass times acceleration: F = ma These force equations are in terms of displacement and acceleration, which you see in simple harmonic motion in the following forms: Inserting these two equations into the force equations gives you the following: You can now find the angular frequency (angular velocity) of a mass on a spring, as it relates to the spring constant and the mass. You can also tie the angular frequency to the frequency and period of oscillation by using the following equation: With this equation and the angular-frequency formula, you can write the formulas for frequency and period in terms of k and m: Say that the spring in the figure has a spring constant, k, of 15 newtons per meter and that you attach a 45-gram ball to the spring. The direction of force exerted by a spring. What’s the period of oscillation? After you convert from grams to kilograms, all you have to do is plug in the numbers: The period of the oscillation is 0.34 seconds. How many bounces will you get per second? The number of bounces represents the frequency, which you find this way: You get nearly 3 oscillations per second. Because you can relate the angular frequency, to the spring constant and the mass on the end of the spring, you can predict the displacement, velocity, and acceleration of the mass, using the following equations for simple harmonic motion: Using the example of the spring in the figure — with a spring constant of 15 newtons per meter and a 45-gram ball attached — you know that the angular frequency is the following: You may like to check how the units work out. Remember that so the units you get from the equation for the angular velocity work out to be Say, for example, that you pull the ball 10.0 centimeters before releasing it (making the amplitude 10.0 centimeters). In this case, you find that

View ArticleArticle / Updated 03-20-2024

In physics, how much torque you exert on an object depends on two things: the force you exert, F; and the lever arm. Also called the moment arm, the lever arm is the perpendicular distance from the pivot point to the point at which you exert your force and is related to the distance from the axis, r, by is the angle between the force and a line from the axis to the point where the force is applied. The torque you exert on a door depends on where you push it. Assume that you’re trying to open a door, as in the various scenarios in the figure. You know that if you push on the hinge, as in diagram A, the door won’t open; if you push the middle of the door, as in diagram B, the door will open; but if you push the edge of the door, as in diagram C, the door will open more easily. In the figure, the lever arm, l, is distance r from the hinge to the point at which you exert your force. The torque is the product of the magnitude of the perpendicular force multiplied by the lever arm. It has a special symbol, the Greek letter tau: The units of torque are force units multiplied by distance units, which are newton-meters in the MKS (meter-kilogram-second) system and foot-pounds in the foot-pound-second system. For example, the lever arm in the figure is distance r (because this lever arm is perpendicular to the force), so If you push with a force of 200 newtons and r is 0.5 meters, what’s the torque you see in the figure? In diagram A, you push on the hinge, so your distance from the pivot point is zero, which means the lever arm is zero. Therefore, the magnitude of the torque is zero. In diagram B, you exert the 200 newtons of force at a distance of 0.5 meters perpendicular to the hinge, so The magnitude of the torque here is 100 newton-meters. But now take a look at diagram C. You push with 200 newtons of force at a distance of 2r perpendicular to the hinge, which makes the lever arm 2r or 1.0 meter, so you get this torque: Now you have 200 newton-meters of torque, because you push at a point twice as far away from the pivot point. In other words, you double the magnitude of your torque. But what would happen if, say, the door were partially open when you exerted your force? Well, you would calculate the torque easily, if you have lever-arm mastery.

View ArticleArticle / Updated 03-20-2024

In physics, you can examine how much potential and kinetic energy is stored in a spring when you compress or stretch it. The work you do compressing or stretching the spring must go into the energy stored in the spring. That energy is called elastic potential energy and is equal to the force, F, times the distance, s: W = Fs As you stretch or compress a spring, the force varies, but it varies in a linear way (because in Hooke’s law, force is proportional to the displacement). The distance (or displacement), s, is just the difference in position, xf – xi, and the average force is (1/2)(Ff + Fi). Therefore, you can rewrite the equation as follows: Hooke’s law says that F = –kx. Therefore, you can substitute –kxf and –kxi for Ff and Fi: Distributing and simplifying the equation gives you the equation for work in terms of the spring constant and position: The work done on the spring changes the potential energy stored in the spring. Here’s how you give that potential energy, or the elastic potential energy: For example, suppose a spring is elastic and has a spring constant, k, of and you compress the spring by 10.0 centimeters. You store the following amount of energy in it: You can also note that when you let the spring go with a mass on the end of it, the mechanical energy (the sum of potential and kinetic energy) is conserved: PE1 + KE1 = PE2 + KE2 When you compress the spring 10.0 centimeters, you know that you have of energy stored up. When the moving mass reaches the equilibrium point and no force from the spring is acting on the mass, you have maximum velocity and therefore maximum kinetic energy — at that point, the kinetic energy is by the conservation of mechanical energy.

View ArticleArticle / Updated 03-20-2024

Physics constants are physical quantities with fixed numerical values. The following list contains the most common physics constants, including Avogadro’s number, Boltzmann’s constant, the mass of electron, the mass of a proton, the speed of light, the gravitational constant, and the gas constant. Avogadro’s number: Boltzmann’s constant: Mass of electron: Mass of proton: Speed of light: Gravitational constant: Gas constant:

View ArticleCheat Sheet / Updated 03-14-2024

The path to understanding astrophysics is both thought-provoking and brain-stretching. How did the universe come into existence, when will it end, and what role do our familiar planets and stars play in the grand scheme of the cosmos? There are many more questions in astrophysics than there are answers. The goal of this book is to put you in a position where you’re able to better formulate those questions, and know where to go for answers. As you’re getting started on your journey, use this Cheat Sheet to answer some of the first questions that come to mind.

View Cheat SheetCheat Sheet / Updated 02-22-2024

Welcome to the world of Human Geography. It is a whole world that a shockingly large number of people do not even know exists. Human geography is an academic discipline regularly taught at the high school and university level that actually encompasses quite a few subdisciplines of geographic study. The traditional divisions of human geography study the following major fields. Populations and migration Urban geography Economic geography Cultural geography Political geography Within those fields, a slew of other research areas can be included under the umbrella of human geography. Areas like medical geography study the relationship between space and medical care (like the spread of infectious diseases or the impact that location has on quality of life). Political geography is a geopolitics field that tries to understand the geographic factors that influence how different countries interact. The content in this book will give you a working overview of the terms and concepts covered within the field of human geography. The materials are comparable to the content covered in a lower-level undergraduate college course or an upper-level high school course. This book is not an in-depth dive into any particular human geography topic. In fact, every topic included in this book could have entire books written about just that one idea. Many researchers have spent untold hours building the human geography field. The purpose of this book is to give you a taste of the breadth of human geography in easily digestible tidbits. Also, this is not a textbook. Instead, it is a starting place for where human geography can take you.

View Cheat SheetCheat Sheet / Updated 02-12-2024

Neurobiology has all kinds of real-world (and not so real-world) applications. From curing paralysis to the possibility of cyborgs, neurobiology has answers to many fascinating questions.

View Cheat Sheet