## Physics Formulae

↦ Equations of angular motion are relevant wherever you have rotational motions around an axis. When the object has rotated through an angle of

You must use radians to measure the angle. Also, if you know that the distance from the axis is

↦ ω = ΔΘ/Δt

↦ α = Δω/Δt

↦ Θ = ω

↦ ω

↦ s=rΘ

↦ v=rω

↦ a=rα

↦ a

↦ F

↦ ΣT=Iα

↦ I=Σmr

**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,**a**, and force,_{c}**F**. When an object with moment of inertia,_{c}**I**(the angular equivalent of mass), has an angular acceleration,**α**, then there is a net torque**ΣT**.↦ ω = ΔΘ/Δt

↦ α = Δω/Δt

↦ Θ = ω

_{i}(t_{t}-t_{i}) + 1/2α(t_{t}-t_{i})^{2}↦ ω

_{t}^{2}- ω_{i}^{2}= 2αΘ↦ s=rΘ

↦ v=rω

↦ a=rα

↦ a

_{c}= v^{2}/r↦ F

_{c}= mv^{2}/r↦ ΣT=Iα

↦ I=Σmr

^{2}↦ A heat engine takes heat, Q

↦ Efficiency = W/Q

↦ Q

↦ Efficiency = 1 - (Q

↦ Coefficient of performance = Q

↦ Coefficient of performance = 1/(1-Q

_{h}, from a high temperature source at temperature T_{h}and moves it to a low temperature sink (temperature T_{c}) at a rate Q_{c}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.↦ Efficiency = W/Q

_{h}↦ Q

_{c}/Q_{h}= T_{c}/T_{h}↦ Efficiency = 1 - (Q

_{c}/Q_{h}) = 1 - (T_{c}/T_{h})↦ Coefficient of performance = Q

_{h}/W↦ Coefficient of performance = 1/(1-Q

_{c}/Q_{h}) = 1/(1-T_{c}/T_{h})↦ A volume, V, of fluid with mass, m, has density, p. 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, W

↦ p=m/V

↦ ΔP = pgh

↦ F

↦ p

↦ P

_{water}displaced, give rise to an upward directed buoyancy force, F_{buoyancy}. 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.↦ p=m/V

↦ ΔP = pgh

↦ F

_{buoyancy}= W_{water displaced}↦ p

_{1}A_{1}v_{1}= p_{2}A_{2}v_{2}↦ P

_{1}+ 1/2 * pv_{1}^{2}+ pgy_{1}= P_{2}+ 1/2 * pv_{2}^{2}+ pgy_{2}↦ A mass, m, accelerates at a rate, a, due to a force, F, acting. Frictional forces, F

↦ ΣF=ma

↦ F

↦ F = Gm

_{F}, are in proportion to the normal force between the materials, F_{N}, with a coefficient of friction, µ. Two masses, m_{1}and m_{2}, separated by a distance, r, attract each other with a gravitational force, given by the following equations, in proportion to the gravitational constant G:↦ ΣF=ma

↦ F

_{F}= µF_{N}↦ F = Gm

_{1}m_{2}/r^{2}↦ 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:

1/2 * mr

↦ Hollow cylinder rotating around its center:

I = mr

↦ Hollow sphere rotating an axis through its center:

I = 2/3 * mr

↦ Hoop rotating around its center:

I = mr

↦ Point mass rotating at radius r:

I = mr

↦ Rectangle rotating around an axis along one edge where the other edge is of length r:

I = 1/3 * mr

↦ Rectangle rotating around an axis parallel to one edge and passing through the center, where the length of the other edge is r:

I = 1/12 * mr

↦ Rod rotating around an axis perpendicular to it and through its center:

I = 1/12 * mr

↦ Rod rotating around an axis perpendicular to it and through one end:

I = 1/3 * mr

↦ Solid cylinder, rotating around an axis along its center line:

I = 1/2 * mr

↦ The kinetic energy of a rotating body, with moment of inertia, I, and angular velocity, ω:

KE = 1/2 * Iω

↦ The angular momentum of a rotating body with moment of inertia, I, and angular velocity, ω:

L = Iω

↦ Disk rotating around its center:

1/2 * mr

^{2}↦ Hollow cylinder rotating around its center:

I = mr

^{2}↦ Hollow sphere rotating an axis through its center:

I = 2/3 * mr

^{2}↦ Hoop rotating around its center:

I = mr

^{2}↦ Point mass rotating at radius r:

I = mr

^{2}↦ Rectangle rotating around an axis along one edge where the other edge is of length r:

I = 1/3 * mr

^{2}↦ Rectangle rotating around an axis parallel to one edge and passing through the center, where the length of the other edge is r:

I = 1/12 * mr

^{2}↦ Rod rotating around an axis perpendicular to it and through its center:

I = 1/12 * mr

^{2}↦ Rod rotating around an axis perpendicular to it and through one end:

I = 1/3 * mr

^{2}↦ Solid cylinder, rotating around an axis along its center line:

I = 1/2 * mr

^{2}↦ The kinetic energy of a rotating body, with moment of inertia, I, and angular velocity, ω:

KE = 1/2 * Iω

^{2}↦ The angular momentum of a rotating body with moment of inertia, I, and angular velocity, ω:

L = Iω

↦ 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:

↦ v

= 2a(x

↦ v = Δx/Δt

↦ a = Δv/Δt

↦ s = v

↦ v

_{f}^{2}- v_{i}^{2}= 2as= 2a(x

_{f}-x_{i})↦ v = Δx/Δt

↦ a = Δv/Δt

↦ s = v

_{i}(t_{f}-t_{i}) + 1/2* (t_{f}-t_{i})^{2}↦
Particular kinds of force result in periodic motion, where the object repeats its motion with a period, T, having an angular frequency, ?, 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.

↦ x = Acosωt

↦ v

↦ a=-Aω

↦ F=-kx

↦ T = 2π/ω

↦ x = Acosωt

↦ v

_{X}=-Aωsinωt↦ a=-Aω

^{2}cosωt↦ F=-kx

↦ T = 2π/ω

↦ 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:

↦ C = 5/9 * (F -32)

↦ F = 9/5 *C +32

↦ K = C + 273.15

↦ The heat required to cause a change in temperature of a mass, m, increases with a constant of proportionality, c, called the

↦ Q = cmΔT

↦ Q = (kAΔTt)/L

↦ Q=eσAtT

↦ 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:

PV=nRT

↦ In an ideal gas, the average energy of each molecule KE

KE

Kelvin (K), Celsius (C), and Fahrenheit (F) are temperature scales. You can use these formulas to convert from one temperature scale to another:

↦ C = 5/9 * (F -32)

↦ F = 9/5 *C +32

↦ K = C + 273.15

↦ 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:↦ Q = cmΔT

↦ Q = (kAΔTt)/L

↦ Q=eσAtT

^{4}↦ 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:

PV=nRT

↦ In an ideal gas, the average energy of each molecule KE

_{avg}, is in proportion to the temperature, with the Boltzman constant k:KE

_{avg}=3/2*kT↦ 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.

↦ W = FscosΘ

↦ p = mv

↦ KE = 1/2mv

↦ t = FrsinΘ

↦ W = FscosΘ

↦ p = mv

↦ KE = 1/2mv

^{2}↦ t = FrsinΘ