Newtons Laws EX 35

Io has an orbit of radius $r=4.22\times 10^{8}m$ and the period $T=1.77days=1.53\times 10^{5}s$. From the period and the radius, we can find the speed of Io:

$v=\frac{2\pi r}{T}=\frac{2\pi (4.22\times 10^8)}{1.53\times 10^5}=1.73\times 10^{4}m/s$

Mass of Jupiter

Then we can use the Newton's second law to find the mass of Jupiter:

$\frac{G{M}_J{m}_I}{r^2}=m_I\frac{v^2}{r}$

Cancel out $r$ and $m_I$ and substitute known values into the equation.

$\frac{(6.67\times 10^{-11})M_J}{4.22\times 10^8}=(1.73\times 10^4{)}^2$

From this equation we compute $M_J=1.9\times 10^{27}kg$.

Radius of Europa's orbit

The period of Europa is $T=3.55days=3.07\times 10^{5}s$. We substitute the formula for $v=\frac{2\pi r}{T}$ into the Newton's law

$\frac{G{M}_J{m}_E}{r^2}=m_E\frac{(\frac{2\pi r}{T}{)}^2}{r}$

and then cancel out the mass of Europa ($m_E$) and substitute the known values of other constants to obtain:

$\frac{(6.67\times 10^{-11})(1.9\times 10^{27})}{r^3}=\frac{4{\pi }^2}{(3.07\times 10^5{)}^2}$

We solve to find $r=6.71\times 10^{8}m$.