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 .
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
Explanation:
a)
The magnitude of the gravitational force between two objects is given by the formula
[tex]F=G\frac{m_1 m_2}{r^2}[/tex]
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:
[tex]m_1=M_p\\m_2 = M_s\\r=R[/tex]
So the force is
[tex]F_A=G\frac{M_p M_s}{R^2}=F_0[/tex]
For the system planet B – Star B, we have:
[tex]m_1 = 4 M_p\\m_2 = M_s\\r=R[/tex]
So the force is
[tex]F=G\frac{4M_p M_s}{R^2}=4F_0[/tex]
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:
[tex]m_1 = M_p\\m_2 = 4M_s\\r=R[/tex]
So the force is
[tex]F=G\frac{M_p (4M_s)}{R^2}=4F_0[/tex]
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:
[tex]G\frac{mM}{r^2}=m\frac{v^2}{r}[/tex]
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:
[tex]v=\sqrt{\frac{GM}{r}}[/tex]
For System A,
[tex]M=M_s\\r=R[/tex]
So the tangential speed is
[tex]v_A=\sqrt{\frac{GM_s}{R}}[/tex]
For system B,
[tex]M=M_s\\r=R[/tex]
So the tangential speed is
[tex]v_B=\sqrt{\frac{GM_s}{R}}=v_A[/tex]
So, the speed of planet B is the same as planet A.
For system C,
[tex]M=4M_s\\r=R[/tex]
So the tangential speed is
[tex]v_C=\sqrt{\frac{G(4M_s)}{R}}=2(\sqrt{\frac{GM_s}{R}})=2v_A[/tex]
So, the speed of planet C is twice the speed of planet A.