joule second

SI coherent derived unit whose name and symbol includes an SI coherent derived unit with a special name and symbol

Name

Symbol

Quantity

Expressed in terms of SI base units

joule second

J s

action,
angular momentum

kg m² s⁻¹

Definition

The joule second, symbol J s, is the SI coherent derived unit of action, and of angular momentum.

The joule second is also the unit used to express the Planck constant, one of the seven SI defining constants.

Planck constant

In quantum mechanics, the Planck–Einstein relation states that the energy, E, of a photon is directly proportional to the frequency, ν, of its associated wave:

$E \propto \nu$

When expressed in the form of an equation, the Planck constant, h, is the proportionality constant:

Using SI coherent units,

$E = h \nu$

where:

E is energy in joules, symbol J,
ν is frequency in hertz, symbol Hz,
h is the Planck constant, in kg m² s⁻¹.

Inverting this relation gives an expression for the Planck constant in terms of energy and frequency:

$h = \dfrac{E}{\nu}$

Substituting frequency, ν, with period of oscillation, T, gives an expression for the Planck constant in terms of energy and time:

$h = E \mspace{4mu} T$

where:

h is the Planck constant, in joule seconds, symbol J s,
m is mass in kilograms, symbol kg,
T is period of oscillation in seconds, symbol s.

SI defining constant

The fixed numerical value of the Planck constant, h, is defined as 6.626 070 15 × 10^‑34 when expressed in the unit J s.

$h \mspace{6mu} = 6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34} \mspace{6mu} \text{J} \mspace{6mu} \text{s}$

Inverting this relation gives an exact expression for the joule second in terms of the SI defining constant h :

$1 \mspace{6mu} \text{J} \mspace{6mu} \text{s} \mspace{6mu} = \dfrac{h}{6.626 \mspace{4mu} 070 \mspace{4mu} 15 \times 10^{-34}}$

The effect of this definition is that one joule second is exactly ¹⁄_{6.626 070 15 × 10⁻³⁴} of the Planck constant.

Angular momentum

When expressed in terms of SI base units, the joule second is equal to the unit kg m² s⁻¹, which is the SI coherent derived unit of angular momentum.

Angular momentum is a vector quantity.

The angular momentum of an object is defined as the product of its moment of inertia, I, and its angular velocity, ω.

$L = I \times \omega$

where:

L is angular momentum, in kg m² s⁻¹,
I is moment of inertia, in kg m²,
ω is angular velocity, in radians per second, symbol rad s⁻¹.

Angular momentum – comparison with linear momentum

In a closed system,

angular momentum remains constant in both magnitude and direction.
linear momentum remains constant in both magnitude and direction.

The angular momentum, L, of an object is the product of its moment of inertia, I, and its angular velocity, ω.

$L = I \times \omega$

The linear momentum, p, of an object is the product of its mass, m, and its velocity, v.

$p = m \times v$