Root mean square speed

Root mean square speed DEFAULT

Calculate Root Mean Square Velocity of Gas Particles

This example problem demonstrates how to calculate the root mean square (RMS) velocity of particles in an ideal gas. This value is the square root of the average velocity-squared of molecules in a gas. While the value is an approximation, especially for real gases, it offers useful information when studying kinetic theory.

Root Mean Square Velocity Problem

What is the average velocity or root mean square velocity of a molecule in a sample of oxygen at 0 degrees Celsius?


Gases consist of atoms or molecules that move at different speeds in random directions. The root mean square velocity (RMS velocity) is a way to find a single velocity value for the particles. The average velocity of gas particles is found using the root mean square velocity formula:

μrms = (3RT/M)½
μrms = root mean square velocity in m/sec
R = ideal gas constant = 8.3145 (kg·m2/sec2)/K·mol
T = absolute temperature in Kelvin
M = mass of a mole of the gas in kilograms.

Really, the RMS calculation gives you root mean squarespeed, not velocity. This is because velocity is a vector quantity that has magnitude and direction. The RMS calculation only gives the magnitude or speed. The temperature must be converted to Kelvin and the molar mass must be found in kg to complete this problem.

Step 1

Find the absolute temperature using the Celsius to Kelvin conversion formula:

  • T = °C + 273
  • T = 0 + 273
  • T = 273 K

Step 2

Find molar mass in kg:
From the periodic table, the molar mass of oxygen = 16 g/mol.
Oxygen gas (O2) is comprised of two oxygen atoms bonded together. Therefore:

  • molar mass of O2 = 2 x 16
  • molar mass of O2 = 32 g/mol
  • Convert this to kg/mol:
  • molar mass of O2 = 32 g/mol x 1 kg/1000 g
  • molar mass of O2 = 3.2 x 10-2 kg/mol

Step 3

Find μrms:

  • μrms = (3RT/M)½
  • μrms = [3(8.3145 (kg·m2/sec2)/K·mol)(273 K)/3.2 x 10-2 kg/mol]½
  • μrms = (2.128 x 105 m2/sec2)½
  • μrms = 461 m/sec


The average velocity or root mean square velocity of a molecule in a sample of oxygen at 0 degrees Celcius is 461 m/sec.


9.15: Kinetic Theory of Gases- Molecular Speeds

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Other sections state that increasing the temperature increases the speeds at which molecules move. We are now in a position to find just how large that increase is for a gaseous substance. Combining the ideal gas law with Eq. (1) from The Total Molecular Kinetic Energy, we obtain

\(\begin{align} & PV=nRT=\tfrac{\text{1}}{\text{3}}Nm\text{(}u^{\text{2}}\text{)}_{\text{ave}}\\ & \text{or } \text{3}RT=\frac{Nm}{n}\text{(}u^{\text{2}}\text{)}_{\text{ave}} \label{1}\end{align}\)

Since N is the number of molecules and m is the mass of each molecule, Nm is the total mass of gas. Dividing total mass by amount of substance gives molar mass M: \[M=\frac{Nm}{n}\] Substituting in Eq. \(\ref{1}\), we have \(\begin{align} & \text{ 3}RT=M(u^{\text{2}})_{\text{ave}} \\ & \text{or }(u^{\text{2}})_{\text{ave}}=\frac{\text{3}RT}{M} \\ & \text{so that }u_{rms}=\sqrt{\text{(}u^{\text{2}}\text{)}_{\text{ave}}}=\sqrt{\frac{\text{3}RT}{M}}\text{ (2)} \end{align}\) The quantity urms is called the root-mean-square (rms) velocity because it is the square root of the mean square velocity.

The rms velocity is directly proportional to the square root of temperature and inversely proportional to the square root of molar mass. Thus quadrupling the temperature of a given gas doubles the rms velocity of the molecules. Doubling this average velocity doubles the number of collisions between gas molecules and the walls of a container. It also doubles the impulse of each collision. Thus the pressure quadruples. This is indicated graphically in Figure \(\PageIndex{1}\). Pressure is thus directly proportional to temperature, as required by Gay-Lussac’s law.


The inverse proportionality between root-mean-square velocity and the square root of molar mass means that the heavier a molecule is, the slower it moves, which is verified by the examples below

We can compare the rates of effusion or diffusion of a known gas with that of an unknown gas to determine the molar mass of the unknown gas. A convenient equation can be derived easily by considering the kinetic energy of individual molecules rather than moles of gas:

Knowing that kinetic energy is proportional to temperature, if the two gases are at the same temperature,

\(\text{K} \text{E}_{1} = \text{K} \text{E}_{2} \) where 1 and 2 denote the two gases. Since \(KE= \frac{1}{2} m v^{2}\),
\( \frac{1}{2} m_{1} ( u_{\text{rms, 1}} )^{2} = \frac{1}{2} m_{2} ( u_{\text{rms, 2}} )^{2}\) where m is the atomic weight in amu/average molecule, and urms is the velocity.


\[\frac{m_{1}}{m_{2}} = \frac{ ( u_{rms,2} )^{2} }{u_{rms,1} )^{2} } \]

Example \(\PageIndex{1}\) : Molar Mass

What is the molar mass of an unknown gas if the gas effuses through a pinhole into a vacuum at a rate of 2 mL/min, and H2 effuses at 11 mL/min. Assume that the rate of effusion is proportional to the gas molecule velocities.


\[\frac{m_{1}}{m_{2}} = \frac{ ( u_{rms,2} )^{2} }{u_{rms,1} )^{2} } \\ \frac{4}{m_{2}} = \frac{2^{2}}{11^{2}} \\ m_{2} = 121 \]

Example \(\PageIndex{2}\) : RMS Velocity

Find the rms velocity for (a) H2 and (b) O2 molecules at 27°C.

Solution This problem is much easier to solve if we use SI units. Thus we choose

R = 8.314 J mol–1 K–1 = 8.314 kg m2s–2 mol–1 K–1

a)For H2\(\begin{align}u_{\text{rms}}=\sqrt{\frac{\text{3}RT}{M}} & =\sqrt{\frac{\text{3 }\times \text{ 8}\text{.314 J mol}^{-\text{1}}\text{ K}^{-\text{1}}\text{ }\times \text{ 300 K}}{\text{2}\text{.016 g mol}^{-\text{1}}}}\\ & =\sqrt{\text{3}\text{.712 }\times \text{ 10}^{\text{3}}\text{ }\frac{\text{kg m}^{\text{2}}\text{s}^{-\text{2}}}{\text{g}}}\\ & =\sqrt{\text{3}\text{.712 }\times \text{ 10}^{\text{3}}\text{ }\times \text{ 10}^{\text{3}}\text{ }\frac{\text{g m}^{\text{2}}\text{s}^{-\text{2}}}{\text{g}}}\\ & =\sqrt{\text{3}\text{.712 }}\times \text{ 10}^{\text{3}}\text{ m s}^{-\text{1}}=\text{1}\text{.927 }\times \text{ 10}^{\text{3}}\text{ m s}^{-\text{1}}\end{align}\) b)For O2\[u_{\text{rms}}=\sqrt{\frac{\text{3 }\times \text{ 8}\text{.314 J mol}^{-\text{1}}\text{ K}^{-\text{1}}\text{ }\times \text{ 300 K}}{\text{32}\text{.00 g mol}^{-\text{1}}}}=\text{4}\text{.836 }\times \text{ 10}^{\text{2}}\text{ m s}^{-\text{1}}\]
The rms velocities 1927 m s–1 and 484 m s–1 correspond to about 4300 miles per hour and 1080 miles per hour, respectively. The O2 molecules in air at room temperature move about 50 percent faster than jet planes, and H2 molecules are nearly 4 times speedier yet. Of course an O2 molecule would take a lot longer to get from New York to Chicago than a jet would. Gas molecules never go far in a straight line before colliding with other molecules.

Now we can see the microscopic basis for Avogadro’s law. Most of the volume in H2, O2 or any gas is empty space, and that empty space is the same for a given amount of any gas at the same temperature and pressure. This happens because the total kinetic energy of the molecules is the same for H2 or O2 or any other gas. The more energy they have, the more room the molecules can make for themselves by expanding against a constant pressure. This is illustrated in Figure \(\PageIndex{2}\), where equal numbers of H2 and O2 molecules occupy separate containers at the same temperature and pressure.


The volumes are seen to be the same. Because O2 molecules are 16 times heavier than H2 molecules, the average speed of H2 molecules is 4 times faster. H2 molecules therefore make 4 times as many collisions with walls. Based on mass, each collision of an H2 molecule with the wall has one-sixteenth the effect of an O2 collision, but an H2 collision has 4 times the effect of an O2 collision when molecular velocity is considered. The net result is that each H2 collision is only one-fourth as effective as an O2 collision. But since there are four times as many collisions, each one-fourth as effective, the same pressure results. Thus the same number of O2 molecules as H2 molecules is required to occupy the same volume at the same temperature and pressure.

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Root mean square speed

Root mean square speed (vrms)

Root mean square speed (vrms) is defined as the square root of the mean of the square of speeds of all molecules. It is denoted by vrms = √v2

Equation (9.8) can be re-written as,

From the equation (9.18) we infer the following

(i) rms speed is directly proportional to square root of the temperature and inversely proportional to square root of mass of the molecule. At a given temperature the molecules of lighter mass move faster on an average than the molecules with heavier masses.

Example: Lighter molecules like hydrogen and helium have high ‘vrms’ than heavier molecules such as oxygen and nitrogen at the same temperature.

(ii) Increasing the temperature will increase the r.m.s speed of molecules

We can also write the vrms in terms of gas constant R. Equation (9.18) can be rewritten as follows

Where NA is Avogadro number.

Since NAk = R and NAm = M (molar mass)

The root mean square speed or r.m.s speed

The equation (9.6) can also be written in terms of rms speed


Impact of vrms in nature:

1. Moon has no atmosphere.

The escape speed of gases on the surface of Moon is much less than the root mean square speeds of gases due to low gravity. Due to this all the gases escape from the surface of the Moon.

2. No hydrogen in Earth’s atmosphere.

As the root mean square speed of hydrogen is much less than that of nitrogen, it easily escapes from the earth’s atmosphere.

In fact, the presence of nonreactive nitrogen instead of highly combustible hydrogen deters many disastrous consequences.


A room contains oxygen and hydrogen molecules in the ratio 3:1. The temperature of the room is 27°C. The molar mass of 02 is 32 g mol-1 and for H2 2 g mol-1. The value of gas constant R is 8.32 J mol-1K-1


(a) rms speed of oxygen and hydrogen molecule

(b) Average kinetic energy per oxygen molecule and per hydrogen molecule

(c) Ratio of average kinetic energy of oxygen molecules and hydrogen molecules


(a) Absolute Temperature

T=27°C =27+273=300 K.

Gas constant R=8.32 J mol-1k-1

For Oxygen molecule: Molar mass

M=32 gm=32 x 10-3 kg mol-1

Note that the rms speed is inversely proportional to √M and the molar mass of oxygen is 16 times higher than molar mass of hydrogen. It implies that the rms speed of hydrogen is 4 times greater than rms speed of oxygen at the same temperature.

1934/484 ≈ 4 .

(b) The  average  kinetic  energy  per molecule is 3/2 kT. It depends only on absolute temperature of the gas and is independent of the nature of molecules. Since both the gas molecules are at the same temperature, they have the same average kinetic energy per molecule. k is Boltzmaan constant.

(c) Average  kinetic  energy  of  total oxygen molecules = 3/2 N0kT where N0 - number of oxygen molecules in the room

Average  kinetic  energy  of  total hydrogen  molecules = 3/2 NHkT where NH - number of hydrogen molecules in the room.

It is given that the number of oxygen molecules is 3 times more than number of hydrogen molecules in the room. So the ratio of average kinetic energy of oxygen molecules with average kinetic energy of hydrogen molecules is 3:1

Root-mean-square Speed of Gas Particles


Learning Objective

  • Recall the mathematical formulation of the root-mean-square velocity for a gas.

Key Points

    • All gas particles move with random speed and direction.
    • Solving for the average velocity of gas particles gives us the average velocity of zero, assuming that all particles are moving equally in different directions.
    • You can acquire the average speed of gaseous particles by taking the root of the square of the average velocities.
    • The root-mean-square speed takes into account both molecular weight and temperature, two factors that directly affect a material’s kinetic energy.


  • velocitya vector quantity that denotes the rate of change of position, with respect to time or a speed with a directional component

Kinetic Molecular Theory and Root-Mean-Square Speed

According to Kinetic Molecular Theory, gaseous particles are in a state of constant random motion; individual particles move at different speeds, constantly colliding and changing directions. We use velocity to describe the movement of gas particles, thereby taking into account both speed and direction.

Although the velocity of gaseous particles is constantly changing, the distribution of velocities does not change. We cannot gauge the velocity of each individual particle, so we often reason in terms of the particles’ average behavior. Particles moving in opposite directions have velocities of opposite signs. Since a gas’ particles are in random motion, it is plausible that there will be about as many moving in one direction as in the opposite direction, meaning that the average velocity for a collection of gas particles equals zero; as this value is unhelpful, the average of velocities can be determined using an alternative method.

By squaring the velocities and taking the square root, we overcome the “directional” component of velocity and simultaneously acquire the particles’ average velocity. Since the value excludes the particles’ direction, we now refer to the value as the average speed. The root-mean-square speed is the measure of the speed of particles in a gas, defined as the square root of the average velocity-squared of the molecules in a gas.

It is represented by the equation: [latex]v_{rms}=\sqrt{\frac{3RT}{M}}[/latex], where vrms is the root-mean-square of the velocity, Mm is the molar mass of the gas in kilograms per mole, R is the molar gas constant, and T is the temperature in Kelvin.

The root-mean-square speed takes into account both molecular weight and temperature, two factors that directly affect the kinetic energy of a material.


  • What is the root-mean-square speed for a sample of oxygen gas at 298 K?




Speed root mean square

Root Square Mean Velocity Example Problem

Gases are made up of individual atoms or molecules freely moving in random directions with a wide variety of speeds. Kinetic molecular theory tries to explain the properties of gases by investigating the behavior of individual atoms or molecules making up the gas. This example problem shows how to find the average or root mean square velocity (rms) of particles in a gas sample for a given temperature.

Root Mean Square Problem

What is the root mean square velocity of the molecules in a sample of oxygen gas at 0 °C and 100 °C?


Root mean square velocity is the average velocity of the molecules that make up a gas. This value can be found using the formula:

vrms = [3RT/M]1/2

vrms = average velocity or root mean square velocity
R = ideal gas constant
T = absolute temperature
M = molar mass

The first step is to convert the temperatures to absolute temperatures. In other words, convert to the Kelvin temperature scale:

K = 273 + °C
T1 = 273 + 0 °C = 273 K
T2 = 273 + 100 °C = 373 K

The second step is to find the molecular mass of the gas molecules.

Use the gas constant 8.3145 J/mol·K to get the units we need. Remember 1 J = 1 kg·m2/s2. Substitute these units into the gas constant:

R = 8.3145 kg·m2/s2/K·mol

Oxygen gas is made up of two oxygen atoms bonded together. The molecular mass of a single oxygen atom is 16 g/mol. The molecular mass of O2 is 32 g/mol.

The units on R use kg, so the molar mass must also use kg.

32 g/mol x 1 kg/1000 g = 0.032 kg/mol

Use these values to find the vrms.

0 °C:
vrms = [3RT/M]1/2
vrms = [3(8.3145 kg·m2/s2/K·mol)(273 K)/(0.032 kg/mol)]1/2
vrms = [212799 m2/s2]1/2
vrms = 461.3 m/s

100 °C
vrms = [3RT/M]1/2
vrms = [3(8.3145 kg·m2/s2/K·mol)(373 K)/(0.032 kg/mol)]1/2
vrms = [290748 m2/s2]1/2
vrms = 539.2 m/s


The average or root mean square velocity of the oxygen gas molecules at 0 °C is 461.3 m/s and 539.2 m/s at 100 °C.

Watch Now: How to Calculate Velocity

Root Mean Square Speed

Root mean square

Square root of the mean square

In mathematics and its applications, the root mean square (RMS or RMS or rms) is defined as the square root of the mean square (the arithmetic mean of the squares of a set of numbers).[1] The RMS is also known as the quadratic mean[2][3] and is a particular case of the generalized mean with exponent 2. RMS can also be defined for a continuously varying function in terms of an integral of the squares of the instantaneous values during a cycle.

For alternating electric current, RMS is equal to the value of the constant direct current that would produce the same power dissipation in a resistive load.[1]

In estimation theory, the root-mean-square deviation of an estimator is a measure of the imperfection of the fit of the estimator to the data.


The RMS value of a set of values (or a continuous-timewaveform) is the square root of the arithmetic mean of the squares of the values, or the square of the function that defines the continuous waveform. In physics, the RMS current value can also be defined as the "value of the direct current that dissipates the same power in a resistor."

In the case of a set of n values \{x_{1},x_{2},\dots ,x_{n}\}, the RMS is

{\displaystyle x_{\text{RMS}}={\sqrt {{\frac {1}{n}}\left(x_{1}^{2}+x_{2}^{2}+\cdots +x_{n}^{2}\right)}}.}

The corresponding formula for a continuous function (or waveform) f(t) defined over the interval T_{1}\leq t\leq T_{2} is

{\displaystyle f_{\text{RMS}}={\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{[f(t)]}^{2}\,{\rm {d}}t}}},}

and the RMS for a function over all time is

{\displaystyle f_{\text{RMS}}=\lim _{T\rightarrow \infty }{\sqrt {{1 \over {2T}}{\int _{-T}^{T}{[f(t)]}^{2}\,{\rm {d}}t}}}.}

The RMS over all time of a periodic function is equal to the RMS of one period of the function. The RMS value of a continuous function or signal can be approximated by taking the RMS of a sample consisting of equally spaced observations. Additionally, the RMS value of various waveforms can also be determined without calculus, as shown by Cartwright.[4]

In the case of the RMS statistic of a random process, the expected value is used instead of the mean.

In common waveforms[edit]

Sine, square, triangle, and sawtoothwaveforms. In each, the centerline is at 0, the positive peak is at {\displaystyle y=A_{1}}and the negative peak is at {\displaystyle y=-A_{1}}
A rectangular pulse wave of duty cycle D, the ratio between the pulse duration (\tau ) and the period (T); illustrated here with a= 1.
Graph of a sine wave's voltage vs. time (in degrees), showing RMS, peak (PK), and peak-to-peak (PP) voltages.

If the waveform is a pure sine wave, the relationships between amplitudes (peak-to-peak, peak) and RMS are fixed and known, as they are for any continuous periodic wave. However, this is not true for an arbitrary waveform, which may not be periodic or continuous. For a zero-mean sine wave, the relationship between RMS and peak-to-peak amplitude is:

Peak-to-peak{\displaystyle =2{\sqrt {2}}\times {\text{RMS}}\approx 2.8\times {\text{RMS}}.}

For other waveforms, the relationships are not the same as they are for sine waves. For example, for either a triangular or sawtooth wave

Peak-to-peak{\displaystyle =2{\sqrt {3}}\times {\text{RMS}}\approx 3.5\times {\text{RMS}}.}
WaveformVariables and operatorsRMS
DC{\displaystyle y=A_{0}\,}{\displaystyle A_{0}\,}
Sine wave{\displaystyle y=A_{1}\sin(2\pi ft)\,}{\displaystyle {\frac {A_{1}}{\sqrt {2}}}}
Square wave{\displaystyle y={\begin{cases}A_{1}&\operatorname {frac} (ft)<0.5\\-A_{1}&\operatorname {frac} (ft)>0.5\end{cases}}}{\displaystyle A_{1}\,}
DC-shifted square wave {\displaystyle y=A_{0}+{\begin{cases}A_{1}&\operatorname {frac} (ft)<0.5\\-A_{1}&\operatorname {frac} (ft)>0.5\end{cases}}}{\displaystyle {\sqrt {A_{0}^{2}+A_{1}^{2}}}\,}
Modified sine wave{\displaystyle y={\begin{cases}0&\operatorname {frac} (ft)<0.25\\A_{1}&0.25<\operatorname {frac} (ft)<0.5\\0&0.5<\operatorname {frac} (ft)<0.75\\-A_{1}&\operatorname {frac} (ft)>0.75\end{cases}}}{\displaystyle {\frac {A_{1}}{\sqrt {2}}}}
Triangle wave{\displaystyle y=\left|2A_{1}\operatorname {frac} (ft)-A_{1}\right|}{\displaystyle A_{1} \over {\sqrt {3}}}
Sawtooth wave{\displaystyle y=2A_{1}\operatorname {frac} (ft)-A_{1}\,}{\displaystyle A_{1} \over {\sqrt {3}}}
Pulse wave{\displaystyle y={\begin{cases}A_{1}&\operatorname {frac} (ft)<D\\0&\operatorname {frac} (ft)>D\end{cases}}}{\displaystyle A_{1}{\sqrt {D}}}
Phase-to-phase voltage{\displaystyle y=A_{1}\sin(t)-A_{1}\sin \left(t-{\frac {2\pi }{3}}\right)\,}{\displaystyle A_{1}{\sqrt {\frac {3}{2}}}}
y is displacement,
t is time,
f is frequency,
Ai is amplitude (peak value),
D is the duty cycle or the proportion of the time period (1/f) spent high,
frac(r) is the fractional part of r.

In waveform combinations[edit]

Waveforms made by summing known simple waveforms have an RMS value that is the root of the sum of squares of the component RMS values, if the component waveforms are orthogonal (that is, if the average of the product of one simple waveform with another is zero for all pairs other than a waveform times itself).[5]

{\displaystyle {\text{RMS}}_{\text{Total}}={\sqrt {{\text{RMS}}_{1}^{2}+{\text{RMS}}_{2}^{2}+\cdots +{\text{RMS}}_{n}^{2}}}}

Alternatively, for waveforms that are perfectly positively correlated, or "in phase" with each other, their RMS values sum directly.


In electrical engineering[edit]


Further information: Root mean square AC voltage

A special case of RMS of waveform combinations is:[6]

{\displaystyle {\text{RMS}}_{\text{AC+DC}}={\sqrt {{\text{RMS}}_{\text{DC}}^{2}+{\text{RMS}}_{\text{AC}}^{2}}}}

where {\displaystyle {\text{RMS}}_{\text{DC}}} refers to the direct current (or average) component of the signal, and {\displaystyle {\text{RMS}}_{\text{AC}}} is the alternating current component of the signal.

Average electrical power[edit]

Further information: AC power

Electrical engineers often need to know the power, P, dissipated by an electrical resistance, R. It is easy to do the calculation when there is a constant current, I, through the resistance. For a load of R ohms, power is defined simply as:


However, if the current is a time-varying function, I(t), this formula must be extended to reflect the fact that the current (and thus the instantaneous power) is varying over time. If the function is periodic (such as household AC power), it is still meaningful to discuss the average power dissipated over time, which is calculated by taking the average power dissipation:

{\displaystyle {\begin{aligned}P_{av}&=\left(I(t)^{2}R\right)_{av}&&{\text{where }}\left(\cdots \right)_{av}{\text{ denotes the temporal mean of a function}}\\[3pt]&=\left(I(t)^{2}\right)_{av}R&&{\text{(as }}R{\text{ does not vary over time, it can be factored out)}}\\[3pt]&=I_{\text{RMS}}^{2}R&&{\text{by definition of root-mean-square}}\end{aligned}}}

So, the RMS value, IRMS, of the function I(t) is the constant current that yields the same power dissipation as the time-averaged power dissipation of the current I(t).

Average power can also be found using the same method that in the case of a time-varying voltage, V(t), with RMS value VRMS,

{\displaystyle P_{\text{Avg}}={V_{\text{RMS}}^{2} \over R}.}

This equation can be used for any periodic waveform, such as a sinusoidal or sawtooth waveform, allowing us to calculate the mean power delivered into a specified load.

By taking the square root of both these equations and multiplying them together, the power is found to be:

{\displaystyle P_{\text{Avg}}=V_{\text{RMS}}I_{\text{RMS}}.}

Both derivations depend on voltage and current being proportional (that is, the load, R, is purely resistive). Reactive loads (that is, loads capable of not just dissipating energy but also storing it) are discussed under the topic of AC power.

In the common case of alternating current when I(t) is a sinusoidal current, as is approximately true for mains power, the RMS value is easy to calculate from the continuous case equation above. If Ip is defined to be the peak current, then:

{\displaystyle I_{\text{RMS}}={\sqrt {{1 \over {T_{2}-T_{1}}}\int _{T_{1}}^{T_{2}}\left[I_{\text{p}}\sin(\omega t)\right]^{2}dt}},}

where t is time and ω is the angular frequency (ω = 2π/T, where T is the period of the wave).

Since Ip is a positive constant:

{\displaystyle I_{\text{RMS}}=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{\sin ^{2}(\omega t)}\,dt}}}.}

Using a trigonometric identity to eliminate squaring of trig function:

{\displaystyle {\begin{aligned}I_{\text{RMS}}&=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}{\int _{T_{1}}^{T_{2}}{1-\cos(2\omega t) \over 2}\,dt}}}\\[3pt]&=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}\left[{t \over 2}-{\sin(2\omega t) \over 4\omega }\right]_{T_{1}}^{T_{2}}}}\end{aligned}}}

but since the interval is a whole number of complete cycles (per definition of RMS), the sine terms will cancel out, leaving:

{\displaystyle I_{\text{RMS}}=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}\left[{t \over 2}\right]_{T_{1}}^{T_{2}}}}=I_{\text{p}}{\sqrt {{1 \over {T_{2}-T_{1}}}{{T_{2}-T_{1}} \over 2}}}={I_{\text{p}} \over {\sqrt {2}}}.}

A similar analysis leads to the analogous equation for sinusoidal voltage:

{\displaystyle V_{\text{RMS}}={V_{\text{p}} \over {\sqrt {2}}},}

where IP represents the peak current and VP represents the peak voltage.

Because of their usefulness in carrying out power calculations, listed voltages for power outlets (for example, 120 V in the USA, or 230 V in Europe) are almost always quoted in RMS values, and not peak values. Peak values can be calculated from RMS values from the above formula, which implies VP = VRMS × √2, assuming the source is a pure sine wave. Thus the peak value of the mains voltage in the USA is about 120 × √2, or about 170 volts. The peak-to-peak voltage, being double this, is about 340 volts. A similar calculation indicates that the peak mains voltage in Europe is about 325 volts, and the peak-to-peak mains voltage, about 650 volts.

RMS quantities such as electric current are usually calculated over one cycle. However, for some purposes the RMS current over a longer period is required when calculating transmission power losses. The same principle applies, and (for example) a current of 10 amps used for 12 hours each 24-hour day represents an average current of 5 amps, but an RMS current of 7.07 amps, in the long term.

The term RMS power is sometimes erroneously used in the audio industry as a synonym for mean power or average power (it is proportional to the square of the RMS voltage or RMS current in a resistive load). For a discussion of audio power measurements and their shortcomings, see Audio power.


Main article: Root-mean-square speed

In the physics of gas molecules, the root-mean-square speed is defined as the square root of the average squared-speed. The RMS speed of an ideal gas is calculated using the following equation:

{\displaystyle v_{\text{RMS}}={\sqrt {3RT \over M}}}

where R represents the gas constant, 8.314 J/(mol·K), T is the temperature of the gas in kelvins, and M is the molar mass of the gas in kilograms per mole. In physics, speed is defined as the scalar magnitude of velocity. For a stationary gas, the average speed of its molecules can be in the order of thousands of km/hr, even though the average velocity of its molecules is zero.


Main article: Root-mean-square deviation

When two data sets — one set from theoretical prediction and the other from actual measurement of some physical variable, for instance — are compared, the RMS of the pairwise differences of the two data sets can serve as a measure how far on average the error is from 0. The mean of the absolute values of the pairwise differences could be a useful measure of the variability of the differences. However, the RMS of the differences is usually the preferred measure, probably due to mathematical convention and compatibility with other formulae.

In frequency domain[edit]

The RMS can be computed in the frequency domain, using Parseval's theorem. For a sampled signal {\displaystyle x[n]=x(t=nT)}, where T is the sampling period,

{\displaystyle \sum _{n=1}^{N}{x^{2}[n]}={\frac {1}{N}}\sum _{m=1}^{N}\left|X[m]\right|^{2},}

where {\displaystyle X[m]=\operatorname {FFT} \{x[n]\}} and N is the sample size, that is, the number of observations in the sample and FFT coefficients.

In this case, the RMS computed in the time domain is the same as in the frequency domain:

{\displaystyle {\text{RMS}}\{x[n]\}={\sqrt {{\frac {1}{N}}\sum _{n}{x^{2}[n]}}}={\sqrt {{\frac {1}{N^{2}}}\sum _{m}{{\bigl |}X[m]{\bigr |}}^{2}}}={\sqrt {\sum _{m}{\left|{\frac {X[m]}{N}}\right|^{2}}}}.}

Relationship to other statistics[edit]

If {\bar {x}} is the arithmetic mean and \sigma _{x} is the standard deviation of a population or a waveform, then:[8]

{\displaystyle x_{\text{rms}}^{2}={\overline {x}}^{2}+\sigma _{x}^{2}={\overline {x^{2}}}.}

From this it is clear that the RMS value is always greater than or equal to the average, in that the RMS includes the "error" / square deviation as well.

Physical scientists often use the term root mean square as a synonym for standard deviation when it can be assumed the input signal has zero mean, that is, referring to the square root of the mean squared deviation of a signal from a given baseline or fit.[9][10] This is useful for electrical engineers in calculating the "AC only" RMS of a signal. Standard deviation being the RMS of a signal's variation about the mean, rather than about 0, the DC component is removed (that is, RMS(signal) = stdev(signal) if the mean signal is 0).

See also[edit]


  1. ^ ab"Root-mean-square value". A Dictionary of Physics (6 ed.). Oxford University Press. 2009. ISBN .
  2. ^Thompson, Sylvanus P. (1965). Calculus Made Easy. Macmillan International Higher Education. p. 185. ISBN . Retrieved 5 July 2020.
  3. ^Jones, Alan R. (2018). Probability, Statistics and Other Frightening Stuff. Routledge. p. 48. ISBN . Retrieved 5 July 2020.
  4. ^Cartwright, Kenneth V (Fall 2007). "Determining the Effective or RMS Voltage of Various Waveforms without Calculus"(PDF). Technology Interface. 8 (1): 20 pages.
  5. ^Nastase, Adrian S. "How to Derive the RMS Value of Pulse and Square Waveforms". Retrieved 21 January 2015.
  6. ^"Make Better AC RMS Measurements with your Digital Multimeter"(PDF). Keysight. Keysight. Retrieved 15 January 2019.
  7. ^If AC = a and BC = b. OC = AM of a and b, and radius r = QO = OG.
    Using Pythagoras' theorem, QC² = QO² + OC² ∴ QC = √QO² + OC² = QM.
    Using Pythagoras' theorem, OC² = OG² + GC² ∴ GC = √OC² − OG² = GM.
    Using similar triangles, HC/GC = GC/OC ∴ HC = GC²/OC = HM.
  8. ^Chris C. Bissell; David A. Chapman (1992). Digital signal transmission (2nd ed.). Cambridge University Press. p. 64. ISBN .
  9. ^Weisstein, Eric W. "Root-Mean-Square". MathWorld.
  10. ^"ROOT, TH1:GetRMS".

External links[edit]


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