Question

Three different planet-star systems, which are far apart from one another, are shown above. The masses of the planets are much less than the masses of the stars.

In System A , Planet A of mass Mp orbits Star A of mass Ms in a circular orbit of radius R .

In System B , Planet B of mass 4Mp orbits Star B of mass Ms in a circular orbit of radius R .

In System C , Planet C of mass Mp orbits Star C of mass 4Ms in a circular orbit of radius R .
(a) The gravitational force exerted on Planet A by Star A has a magnitude of F0 . Determine the magnitudes of the gravitational forces exerted in System B and System C .

___ Magnitude of gravitational force exerted on Planet B by Star B

___ Magnitude of gravitational force exerted on Planet C by Star C
(b) How do the tangential speeds of planets B and C compare to that of Planet A ? In a clear, coherent paragraph-length response that may also contain equations and/or drawings, provide claims about

why the tangential speed of Planet B is either greater than, less than, or the same as that of Planet A , and
why the tangential speed of Planet C is either greater than, less than, or the same as that of Planet A .

Answer 1

a) 4F0

b) Speed of planet B is the same as speed of planet A

Speed of planet C is twice the speed of planet A

Step-by-step explanation:

a)

The magnitude of the gravitational force between two objects is given by the formula

`F=G(m_1 m_2)/(r^2)`

where

G is the gravitational constant

m1, m2 are the masses of the 2 objects

r is the separation between the objects

For the system planet A - Star A, we have:

`m_1=M_p\\m_2 = M_s\\r=R`

So the force is

`F_A=G(M_p M_s)/(R^2)=F_0`

For the system planet B - Star B, we have:

`m_1 = 4 M_p\\m_2 = M_s\\r=R`

So the force is

`F=G(4M_p M_s)/(R^2)=4F_0`

So, the magnitude of the gravitational force exerted on planet B by star B is 4F0.

For the system planet C - Star C, we have:

`m_1 = M_p\\m_2 = 4M_s\\r=R`

So the force is

`F=G(M_p (4M_s))/(R^2)=4F_0`

So, the magnitude of the gravitational force exerted on planet C by star C is 4F0.

b)

The gravitational force on the planet orbiting around the star is equal to the centripetal force, therefore we can write:

`G(mM)/(r^2)=m(v^2)/(r)`

where

m is the mass of the planet

M is the mass of the star

v is the tangential speed

We can re-arrange the equation solving for v, and we find an expression for the speed:

`v=\sqrt{(GM)/(r)}`

For System A,

`M=M_s\\r=R`

So the tangential speed is

`v_A=\sqrt{(GM_s)/(R)}`

For system B,

`M=M_s\\r=R`

So the tangential speed is

`v_B=\sqrt{(GM_s)/(R)}=v_A`

So, the speed of planet B is the same as planet A.

For system C,

`M=4M_s\\r=R`

So the tangential speed is

`v_C=\sqrt{(G(4M_s))/(R)}=2(\sqrt{(GM_s)/(R)})=2v_A`

So, the speed of planet C is twice the speed of planet A.