Sector speed

Sector speed is the area (displayed in green) that the radius vector of a particle sweeps per unit of time as it moves along the curve (displayed in blue).

Sector velocity is a physical quantity that determines the rate of change in the area swept by the radius vector of a point as it moves along a curve. Sector speed is a vector quantity and is equal to half the vector product of the radius vector by the point velocity vector:

{\begin{aligned}{\vec {\sigma }}={\frac {1}{2}}[{\vec {r}}\times {\frac {d{\vec {r}}}{dt}}]\end{aligned}}

{\ displaystyle {\ begin {aligned} {\ vec {\ sigma}} = {\ frac {1} {2}} [{\ vec {r}} \ times {\ frac {d {\ vec {r}} } {dt}}] \ end {aligned}}}

{\ displaystyle {\ begin {aligned} {\ vec {\ sigma}} = {\ frac {1} {2}} [{\ vec {r}} \ times {\ frac {d {\ vec {r}} } {dt}}] \ end {aligned}}}

Illustration to Kepler’s second law. The planet moves faster near the Sun in such a way that its radius vector throws the same area per unit time, including at a distance from the Sun, where the planet moves more slowly.

Relation to angular momentum

The concept of sector speed is historically closely related to the concept of angular momentum. Kepler’s second law states that the sectorial velocity of the planet remains if the origin is in the focus of the ellipse where the Sun is located. Generally speaking, the concept of sectorial velocity plays an important role in the study of motion under the action of central forces, since in this motion the sectorial velocity remains constant. Isaac Newton was the first scientist to recognize the dynamic significance of Kepler’s second law. In 1684, that any planet that is attracted by a fixed center sweeps equal areas for equal time intervals (area theorem).

The time derivative of the sector velocity is called the sector acceleration of the point.

{\dot {\vec {\sigma }}}={\frac {1}{2}}[{\vec {r}}\times {\vec {\omega }}]

{\ displaystyle {\ dot {\ vec {\ sigma}}} = {\ frac {1} {2}} [{\ vec {r}} \ times {\ vec {\ omega}}]}

where:

{\vec {\omega }}

{\ displaystyle {\ vec {\ omega}}}

- acceleration point.

There is a relationship between the angular momentum and the sector speed:

2m{\dot {\vec {\sigma }}}={\vec {L}}

{\ displaystyle 2m {\ dot {\ vec {\ sigma}}} = {\ vec {L}}}

Sector velocity in a cylindrical coordinate system

If a point moves along a plane curve and its position is determined by the polar coordinates ρ and φ, then

\sigma _{z}={\frac {1}{2}}\rho ^{2}{\dot {\phi }}

{\ displaystyle \ sigma _ {z} = {\ frac {1} {2}} \ rho ^ {2} {\ dot {\ phi}}}

\omega _{\phi }={\frac {2}{\rho }}{\frac {d\sigma _{z}}{dt}}

{\ displaystyle \ omega _ {\ phi} = {\ frac {2} {\ rho}} {\ frac {d \ sigma _ {z}} {dt}}}

Literature

Olkhovsky, I.I. The course of theoretical mechanics for physicists. - 4th ed. - Doe, 2009. - ISBN 978-5-8114-0857-3 .

Sector speed

Relation to angular momentum

Sector velocity in a cylindrical coordinate system

Literature

See also

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