Physics I For Dummies, 3rd Edition book cover

Physics I For Dummies, 3rd Edition

By: Steven Holzner Published: 03-28-2022

Physics classes are recommended or required courses for a wide variety of majors, and they can be quite challenging. Physics I For Dummies, 3rd Edition tracks specifically to an introductory course and, keeping with the traditionally easy-to-follow Dummies style, teaches you the basic principles and formulas in a clear and concise manner, proving that you don't have to be Einstein to understand physics!

Articles From Physics I For Dummies, 3rd Edition

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131 results
131 results
Physics I For Dummies Cheat Sheet

Cheat Sheet / Updated 01-24-2022

Physics involves a lot of calculations and problem solving. Having on hand the most frequently used physics equations and formulas helps you perform these tasks more efficiently and accurately. This Cheat Sheet also includes a list physics constants that you’ll find useful in a broad range of physics problems.

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How to Calculate a Spring Constant Using Hooke's Law

Article / Updated 10-29-2021

Any physicist knows that if an object applies a force to a spring, then the spring applies an equal and opposite force to the object. Hooke’s law gives the force a spring exerts on an object attached to it with the following equation: F = –kx The minus sign shows that this force is in the opposite direction of the force that’s stretching or compressing the spring. The variables of the equation are F, which represents force, k, which is called the spring constant and measures how stiff and strong the spring is, and x, the distance the spring is stretched or compressed away from its equilibrium or rest position. The force exerted by a spring is called a restoring force; it always acts to restore the spring toward equilibrium. In Hooke’s law, the negative sign on the spring’s force means that the force exerted by the spring opposes the spring’s displacement. Understanding springs and their direction of force The direction of force exerted by a spring The preceding figure shows a ball attached to a spring. You can see that if the spring isn’t stretched or compressed, it exerts no force on the ball. If you push the spring, however, it pushes back, and if you pull the spring, it pulls back. Hooke’s law is valid as long as the elastic material you’re dealing with stays elastic — that is, it stays within its elastic limit. If you pull a spring too far, it loses its stretchy ability. As long as a spring stays within its elastic limit, you can say that F = –kx. When a spring stays within its elastic limit and obeys Hooke’s law, the spring is called an ideal spring. How to find the spring constant (example problem) Suppose that a group of car designers knocks on your door and asks whether you can help design a suspension system. “Sure,” you say. They inform you that the car will have a mass of 1,000 kilograms, and you have four shock absorbers, each 0.5 meters long, to work with. How strong do the springs have to be? Assuming these shock absorbers use springs, each one has to support a mass of at least 250 kilograms, which weighs the following: F = mg = (250 kg)(9.8 m/s2) = 2,450 N where F equals force, m equals the mass of the object, and g equals the acceleration due to gravity, 9.8 meters per second2. The spring in the shock absorber will, at a minimum, have to give you 2,450 newtons of force at the maximum compression of 0.5 meters. What does this mean the spring constant should be? In order to figure out how to calculate the spring constant, we must remember what Hooke’s law says: F = –kx Now, we need to rework the equation so that we are calculating for the missing metric, which is the spring constant, or k. Looking only at the magnitudes and therefore omitting the negative sign, you get Time to plug in the numbers: The springs used in the shock absorbers must have spring constants of at least 4,900 newtons per meter. The car designers rush out, ecstatic, but you call after them, “Don’t forget, you need to at least double that if you actually want your car to be able to handle potholes.”

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How to Calculate Force Based on Pressure

Article / Updated 10-07-2021

Pressure and force are related, and so you can calculate one if you know the other by using the physics equation, P = F/A. Because pressure is force divided by area, its meter-kilogram-second (MKS) units are newtons per square meter, or N/m2. In the foot-pound-second (FPS) system, the units are pounds per square inch, or psi. The unit newtons per square meter is so common in physics that it has a special name: the pascal, which equals 1 newton per square meter. The pascal is abbreviated as Pa. You don’t have to be underwater to experience pressure from a fluid. Air exerts pressure, too, due to the weight of the air above you. Here’s how much pressure the air exerts on you at sea level: The air pressure at sea level is a standard pressure that people refer to as 1 atmosphere (abbreviated atm): If you convert an atmosphere to pounds per square inch, it’s about 14.7 psi. That means that 14.7 pounds of force are pressing in on every square inch of your body at sea level. Your body pushes back with 14.7 psi, so you don’t feel any pressure on you at all. But if you suddenly got transported to outer space, the inward pressure of the air pushing on you would be gone, and all that would remain would be the 14.7 pounds per square inch your body exerted outward. You wouldn’t explode, but your lungs could burst if you tried to hold your breath. The change in pressure could also cause the nitrogen in your blood to form bubbles and give you the bends! Here’s a pressure example problem using water pressure. Say you’re in your neighbor’s pool, waiting near the bottom until your neighbors give up trying to chase you off and go back into the house. You’re near the deep end of the pool, and using the handy pressure gauge you always carry, you measure the pressure on the back of your hand as What force does the water exert on the back of your hand? The back of your hand has an area of about You reason that if P = F/A, then the following is true: F = PA Plugging in the numbers and solving gives you the answer: Yikes. A thousand newtons! You whip out your underwater calculator to find that’s about 230 pounds. Forces add up quickly when you’re underwater because water is a heavy liquid. The force you feel is the weight of the water above you.

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How to Calculate Time and Distance from Acceleration and Velocity

Article / Updated 09-09-2021

In a physics equation, given a constant acceleration and the change in velocity of an object, you can figure out both the time involved and the distance traveled. For instance, imagine you’re a drag racer. Your acceleration is 26.6 meters per second2, and your final speed is 146.3 meters per second. Now find the total distance traveled. Got you, huh? “Not at all,” you say, supremely confident. “Just let me get my calculator.” You know the acceleration and the final speed, and you want to know the total distance required to get to that speed. This problem looks like a puzzler, but if you need the time, you can always solve for it. You know the final speed, vf, and the initial speed, vi (which is zero), and you know the acceleration, a. Because vf – vi = at, you know that Now you have the time. You still need the distance, and you can get it this way: The second term drops out because vi = 0, so all you have to do is plug in the numbers: In other words, the total distance traveled is 402 meters, or a quarter mile. Must be a quarter-mile racetrack.

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How to Calculate Angular Momentum

Article / Updated 05-01-2017

Picture a small child on a spinning playground ride, such as a merry-go-round, and she’s yelling that she wants to get off. You have to stop the spinning ride, but it’s going to take some effort. Why? Because it has angular momentum. In physics, you can calculate angular momentum in the same way that you calculate linear momentum — just substitute moment of inertia for mass, and angular velocity for velocity. What is angular momentum? Angular momentum is the quantity of rotation of a body, which is the product of its moment of inertia and its angular velocity. Linear momentum, p, is defined as the product of mass and velocity: p = mv This is a quantity that is conserved when there are no external forces acting. The more massive and faster moving an object, the greater the magnitude of momentum. The angular momentum equation Physics also features angular momentum, L. The equation for angular momentum looks like this: The angular momentum equation features three variables: L = angular momentum / = the moment of inertia W = the angular velocity Note that angular momentum is a vector quantity, meaning it has a magnitude and a direction. the thumb of your right hand points when you wrap your fingers around in the direction the object is turning). in the MKS (meter-kilogram-second) system. The important idea about angular momentum, much as with linear momentum, is that it’s conserved. The principle of conservation of angular momentum states that angular momentum is conserved if no net torques are involved. This principle comes in handy in all sorts of problems, such as when two ice skaters start off holding each other close while spinning but then end up at arm’s length. Given their initial angular velocity, you can find their final angular velocity, because angular momentum is conserved: If you can find the initial moment of inertia and the final moment of inertia, you’re set. But you also come across less obvious cases where the principle of conservation of angular momentum helps out. For example, satellites don’t have to travel in circular orbits; they can travel in ellipses. And when they do, the math can get a lot more complicated. Lucky for you, the principle of conservation of angular momentum can make the problems simple. Angular momentum example problem Say that NASA planned to put a satellite into a circular orbit around Pluto for studies, but the situation got a little out of hand and the satellite ended up with an elliptical orbit. At its nearest point to Pluto, the satellite zips along at 9,000 meters per second. at that point? The answer is tough to figure out unless you can come up with an angle here, and that angle is angular momentum. Angular momentum is conserved because there are no external torques the satellite must deal with (gravity always acts parallel to the orbital radius). Because angular momentum is conserved, you can say that Because the satellite is so small compared to the radius of its orbit at any location, you can consider the satellite a point mass. Therefore, the moment of inertia, I, equals mr2. The magnitude of the angular velocity equals v/r, so you can express the conservation of angular momentum in terms of the velocity like so: You can put v2 on one side of the equation by dividing by mr2: You have your solution; no fancy math involved at all, because you can rely on the principle of conservation of angular momentum to do the work for you. All you need to do is plug in the numbers: At its closest point to Pluto, the satellite will be screaming along at 9,000 meters per second, and at its farthest point, it will be moving at 2,700 meters per second. Easy enough to figure out, as long as you have the principle of conservation of angular momentum under your belt.

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How to Calculate Displacement in a Physics Problem

Article / Updated 05-01-2017

Displacement is the distance between an object’s initial position and its final position and is usually measured or defined along a straight line. Since this is a calculation that measures distance, the standard unit is the meter (m). How to find displacement In physics, you find displacement by calculating the distance between an object’s initial position and its final position. In physics terms, you often see displacement referred to as the variable s. The official displacement formula is as follows: s = sf – si s = displacement si = initial position sf = final position Calculating displacement example Say, for example, that you have a fine new golf ball that’s prone to rolling around. This particular golf ball likes to roll around on top of a large measuring stick and you want to know how to calculate displacement when the ball moves. You place the golf ball at the 0 position on the measuring stick, as shown in the below figure, diagram A. The golf ball rolls over to a new point, 3 meters to the right, as you see in the figure, diagram B. The golf ball has moved, so displacement has taken place. In this case, the displacement is just 3 meters to the right. Its initial position was 0 meters, and its final position is at +3 meters. The displacement is 3 meters. Scientists, being who they are, like to go into even more detail. You often see the term si, which describes initial position, (the i stands for initial). And you may see the term sf used to describe final position. In these terms, moving from diagram A to diagram B in the figure, si is at the 0-meter mark and sf is at +3 meters. The displacement, s, equals the final position minus the initial position: Displacements don’t have to be positive; they can be zero or negative as well. If the positive direction is to the right, then a negative displacement means that the object has moved to the left. In diagram C, the restless golf ball has moved to a new location, which is measured as –4 meters on the measuring stick. The displacement is given by the difference between the initial and final position. If you want to know the displacement of the ball from its position in diagram B, take the initial position of the ball to be si = 3 meters; then the displacement is given by When working on physics problems, you can choose to place the origin of your position-measuring system wherever is convenient. The measurement of the position of an object depends on where you choose to place your origin; however, displacement from an initial position si to a final position sf does not depend on the position of the origin because the displacement depends only on the difference between the positions, not the positions themselves.

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How to Calculate Acceleration

Article / Updated 04-26-2017

In physics terms, acceleration, a, is the amount by which your velocity changes in a given amount of time. Given the initial and final velocities, vi and vf, and the initial and final times over which your speed changes, ti and tf, you can write the equation like this: In terms of units, the equation looks like this: Distance per time squared? Don’t let that throw you. You end up with time squared in the denominator because you divide velocity by time. In other words, acceleration is the rate at which your velocity changes, because rates have time in the denominator. For acceleration, you see units of meters per second2, centimeters per second2, miles per second2, feet per second2, or even kilometers per hour2. It may be easier, for a given problem, to use units such as mph/s (miles per hour per second). This would be useful if the velocity in question had a magnitude of something like several miles per hour that changed typically over a number of seconds. Say you become a drag racer in order to analyze your acceleration down the dragway. After a test race, you know the distance you went — 402 meters, or about 0.25 miles (the magnitude of your displacement) — and you know the time it took — 5.5 seconds. So what was your acceleration as you blasted down the track? Well, you can relate displacement, acceleration, and time as follows: and that’s what you want — you always work the algebra so that you end up relating all the quantities you know to the one quantity you don’t know. In this case, you have (Keep in mind that in this case, your initial velocity is 0 — you’re not allowed to take a running start at the drag race!) You can rearrange this equation with a little algebra to solve for acceleration; just divide both sides by t2 and multiply by 2 to get Great. Plugging in the numbers, you get the following: Okay, the acceleration is approximately 27 meters per second2. What’s that in more understandable terms? The acceleration due to gravity, g, is 9.8 meters per second2, so this is about 2.7 g’s — you’d feel yourself pushed back into your seat with a force about 2.7 times your own weight.

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A List of Physics Constants

Article / Updated 03-26-2016

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:

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Physics Equations and Formulas

Article / Updated 03-26-2016

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.

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Calculating Tangential Velocity on a Curve

Article / Updated 03-26-2016

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.

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