Question

1) The planet Jupiter has a satellite, Io, which travels in an orbit of radius 4.220×108 m with a period of 1.77 days. Calculate the mass of Jupiter from this information.

2) The masses and coordinates of three spheres are as follows: 14 kg, x = 0.75 m, y = 1.50 m; 36 kg, x = -1.25 m, y = -3.00 m; 62 kg, x = 0.00 m, y= -0.75 m. What is the magnitude of the gravitational force on a 18 kg sphere located at the origin due to the other spheres?

Answer 1

(1) Here, the equations 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 = 152928 s

And r = 4.22 x 10^8 m

Now put these values in the above expression -

M = (2*3.14*4.22x10^8/152928)^2 x (4.22x10^8 / 6.7x10^-11) = 3.0x10^8 x 6.30x10^18 = 1.89 x 10^27 kg.

(2) Here, the radius from the origin to the first mass is:

R1 = sqrt(0.75^2 + 1.50^2) = 1.68 m

The gravitational force of the first mass on the origin is:

F1 = G*m0*m2/R^2 = (6.673 × 10^-11)*18*14/1.68^2 = 5.96*10^-9 N

This has x and y components:

F1x = F1*.75/1.68 = 2.66*10^-9 N

F1y = F1*1.50/1.68 = 5.32*10^-9N

The Radius to mass number 2 is:

R2 = sqrt(1.25^2 + 3.0^2) = 3.25 m

The gravitational force caused by the second mass is:

F2 = G*m0*m2/R2^2 = (6.673 × 10^-11)*18*36/3.25^2 = 4.09*10^-9N

This has the following components:

F2x = F2*(-1.25/3.25) = -1.57*10^-9 N

F2y = F2*(-3.0/3.25) = -3.77*10^-9N

The Radius to mass number 3 is: 0.75m

The gravitational force due to mass number 3 is:

F3 = G*m0*m3/R3^2 = (6.673 × 10^-11)*18*62/.75^2 = 132.40*10^-9 N

The x component of F3 is zero.

The y component of F3 is F3 directed along the negative y axis:

F3y = -132.40*10^-9N

Sum up the components:

Fx = F1x + F2x + F3x = 2.66*10^-9 N -1.57*10^-9 N = 1.09 x 10^-9 N

Fy = F1y + F2y + F3y = 5.32*10^-9N - 3.77*10^-9N - 132.40*10^-9N = - 130.85 x 10^-9 N

|F| = sqrt(Fx^2 + Fy^2) = 130.85 x 10^-9 N