Convert rotation to second of arc
Learn how to convert
1
rotation to
second of arc
step by step.
Calculation Breakdown
Set up the equation
\(1.0\left(rotation\right)={\color{rgb(20,165,174)} x}\left(second \text{ } of \text{ } arc\right)\)
Define the base values of the selected units in relation to the SI unit \(\left(radian\right)\)
\(\text{Left side: 1.0 } \left(rotation\right) = {\color{rgb(89,182,91)} 2.0 \times π\left(radian\right)} = {\color{rgb(89,182,91)} 2.0 \times π\left(rad\right)}\)
\(\text{Right side: 1.0 } \left(second \text{ } of \text{ } arc\right) = {\color{rgb(125,164,120)} \dfrac{π}{6.48 \times 10^{5}}\left(radian\right)} = {\color{rgb(125,164,120)} \dfrac{π}{6.48 \times 10^{5}}\left(rad\right)}\)
Insert known values into the conversion equation to determine \({\color{rgb(20,165,174)} x}\)
\(1.0\left(rotation\right)={\color{rgb(20,165,174)} x}\left(second \text{ } of \text{ } arc\right)\)
\(\text{Insert known values } =>\)
\(1.0 \times {\color{rgb(89,182,91)} 2.0 \times π} \times {\color{rgb(89,182,91)} \left(radian\right)} = {\color{rgb(20,165,174)} x} \times {\color{rgb(125,164,120)} {\color{rgb(125,164,120)} \dfrac{π}{6.48 \times 10^{5}}}} \times {\color{rgb(125,164,120)} \left(radian\right)}\)
\(\text{Or}\)
\(1.0 \cdot {\color{rgb(89,182,91)} 2.0 \times π} \cdot {\color{rgb(89,182,91)} \left(rad\right)} = {\color{rgb(20,165,174)} x} \cdot {\color{rgb(125,164,120)} \dfrac{π}{6.48 \times 10^{5}}} \cdot {\color{rgb(125,164,120)} \left(rad\right)}\)
\(\text{Cancel SI units}\)
\(1.0 \times {\color{rgb(89,182,91)} 2.0 \times π} \cdot {\color{rgb(89,182,91)} \cancel{\left(rad\right)}} = {\color{rgb(20,165,174)} x} \times {\color{rgb(125,164,120)} \dfrac{π}{6.48 \times 10^{5}}} \times {\color{rgb(125,164,120)} \cancel{\left(rad\right)}}\)
\(\text{Conversion Equation}\)
\(2.0 \times π = {\color{rgb(20,165,174)} x} \times \dfrac{π}{6.48 \times 10^{5}}\)
Cancel factors on both sides
\(\text{Cancel factors}\)
\({\color{rgb(255,204,153)} \cancel{π}} \times 2.0 = {\color{rgb(20,165,174)} x} \times \dfrac{{\color{rgb(255,204,153)} \cancel{π}}}{6.48 \times 10^{5}}\)
\(\text{Simplify}\)
\(2.0 = {\color{rgb(20,165,174)} x} \times \dfrac{1.0}{6.48 \times 10^{5}}\)
Switch sides
\({\color{rgb(20,165,174)} x} \times \dfrac{1.0}{6.48 \times 10^{5}} = 2.0\)
Isolate \({\color{rgb(20,165,174)} x}\)
Multiply both sides by \(\left(\dfrac{6.48 \times 10^{5}}{1.0}\right)\)
\({\color{rgb(20,165,174)} x} \times \dfrac{1.0}{6.48 \times 10^{5}} \times \dfrac{6.48 \times 10^{5}}{1.0} = 2.0 \times \dfrac{6.48 \times 10^{5}}{1.0}\)
\(\text{Cancel}\)
\({\color{rgb(20,165,174)} x} \times \dfrac{{\color{rgb(255,204,153)} \cancel{1.0}} \times {\color{rgb(99,194,222)} \cancel{6.48}} \times {\color{rgb(166,218,227)} \cancel{10^{5}}}}{{\color{rgb(99,194,222)} \cancel{6.48}} \times {\color{rgb(166,218,227)} \cancel{10^{5}}} \times {\color{rgb(255,204,153)} \cancel{1.0}}} = 2.0 \times \dfrac{6.48 \times 10^{5}}{1.0}\)
\(\text{Simplify}\)
\({\color{rgb(20,165,174)} x} = 2.0 \times 6.48 \times 10^{5}\)
Solve \({\color{rgb(20,165,174)} x}\)
\({\color{rgb(20,165,174)} x} = 1296000 = 1.296 \times 10^{6}\)
\(\text{Conversion Equation}\)
\(1.0\left(rotation\right) = {\color{rgb(20,165,174)} 1.296 \times 10^{6}}\left(second \text{ } of \text{ } arc\right)\)
