Steven Holzner

Article / 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.

Article / 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.

Article / 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

Article / 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.

Article / 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.

Article / 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:

Article / Updated 07-31-2023

In physics, it’s important to know the difference between conservative and nonconservative forces. The work a conservative force does on an object is path-independent; the actual path taken by the object makes no difference. Fifty meters up in the air has the same gravitational potential energy whether you get there by taking the steps or by hopping on a Ferris wheel. That’s different from the force of friction, which dissipates kinetic energy as heat. When friction is involved, the path you take matters — a longer path will dissipate more kinetic energy than a short one. For that reason, friction is a nonconservative force. For example, suppose you and some buddies arrive at Mt. Newton, a majestic peak that rises h meters into the air. You can take two ways up — the quick way or the scenic route. Your friends drive up the quick route, and you drive up the scenic way, taking time out to have a picnic and to solve a few physics problems. They greet you at the top by saying, “Guess what — our potential energy compared to before is mgh greater.” “Mine, too,” you say, looking out over the view. You pull out this equation: ΔPE = mg(hf - hi) This equation basically states that the actual path you take when going vertically from hi to hf doesn’t matter. All that matters is your beginning height compared to your ending height. Because the path taken by the object against gravity doesn’t matter, gravity is a conservative force. Here’s another way of looking at conservative and nonconservative forces. Say you’re vacationing in the Alps and your hotel is at the top of Mt. Newton. You spend the whole day driving around — down to a lake one minute, to the top of a higher peak the next. At the end of the day, you end up back at the same location: your hotel on top of Mt. Newton. What’s the change in your gravitational potential energy? In other words, how much net work did gravity perform on you during the day? Gravity is a conservative force, so the change in your gravitational potential energy is 0. Because you’ve experienced no net change in your gravitational potential energy, gravity did no net work on you during the day. The road exerted a normal force on your car as you drove around, but that force was always perpendicular to the road (meaning no force parallel to your motion), so it didn’t do any work, either. Conservative forces are easier to work with in physics because they don’t “leak” energy as you move around a path — if you end up in the same place, you have the same amount of energy. If you have to deal with nonconservative forces such as friction, including air friction, the situation is different. If you’re dragging something over a field carpeted with sandpaper, for example, the force of friction does different amounts of work on you depending on your path. A path that’s twice as long will involve twice as much work to overcome friction. What’s really not being conserved around a track with friction is the total potential and kinetic energy, which taken together is mechanical energy. When friction is involved, the loss in mechanical energy goes into heat energy. You can say that the total amount of energy doesn’t change if you include that heat energy. However, the heat energy dissipates into the environment quickly, so it isn’t recoverable or convertible. For that and other reasons, physicists often work in terms of mechanical energy.

Article / Updated 06-28-2023

Thanks to the principle of conservation of mechanical energy, you can use physics to determine the final height of a moving object. At this very moment, for example, suppose Tarzan is swinging on a vine over a crocodile-infested river at a speed of 13.0 meters/second. He needs to reach the opposite river bank 9.0 meters above his present position in order to be safe. Can he swing it? The principle of conservation of mechanical energy gives you the answer: At Tarzan’s maximum height at the end of the swing, his speed, v2, will be 0 meters/second, and assuming h1 = 0 meters — meaning that he started swinging from the same height as the tree branch he's swinging to — you can relate h2 to v1 like this: Solving for h2, this means that Tarzan will come up 0.4 meters short of the 9.0 meters he needs to be safe, so he needs some help.

Article / Updated 12-23-2022

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.”

Article / Updated 10-06-2022

At some point, your quantum physics instructor may want you to find the x, y, and z equations for three-dimensional free particle problems. Take a look at the x equation for the free particle, You can write its general solution as where Ay and Az are constants. Because you get this for where A= Ax Ay Az. The part in the parentheses in the exponent is the dot product of the vectors That is, if the vector a = (ax, ay, az) in terms of components and the vector b = (bx, by, bz), then the dot product of a and b is So here’s how you can rewrite the