Physics Problem: Determining the mass of Jupiter?

Question

Io, a satellite of Jupiter has an orbital period of 1.77 days and an orbital radius of 4.22 x 10^5 km. From these data, determine the mass of Jupiter.

I don’t understand this problem in my book. Please help me by explaining how to solve and what equations used to solve them. Thank You.

Answer 1

The equations you need to determine the mass of Jupiter are:

v = (GM / r)^(1/2)

T = 2πr / v

Use those to solve for the mass of Jupiter:

2πr / T = (GM / r)^(1/2)

M = (2πr / T)² x (r / G)

First, you must convert days into seconds and kilometers into meters:

1.77 days x 24 hours x 60 minutes x 60 seconds = 152,928 s

(4.22 x 10^5 km) x 1,000 m = 4.22 x 10^8 m

I got an answer of 1.90 x 10^27 kg using that method.

Answer 2

divide 10 into 5 then that anwser x 4.22

Answer 3

You can find the mass of a planet if you know certain quantities about

a satellite in orbit around it.

The formula is   T   =   2*pi*SQR( R^3/G*M)

where:-

  T   =  the time it takes the satellite to make 1 orbit  -  in Seconds

      =   1.77 x 24 x 3600   =   1.529 E5  sec.

  R   =  radius of satellite orbit in metres

       =  4.22 E5 x 1000   =   4.22 E8 metres

  G  =  Gravitational constant  

      =   6.7 E-11

  M  =  mass of Jupiter, in Kg

It is better to rearrange the formula first.

  T^2   =   4*pi^2 * R^3 / G*M

  So   M   =   4 * pi^2 * R^3 / G * T^2

Substituting values:-

     M  =    4 * 9.87 *  (4.22 E8)^3 / 6.7 E-11 * (1.529 E5)^2

          =   1.89 E27 Kg.