The figure above (which is not to scale!) shows two point masses in the middle o

Question

The figure above (which is not to scale!) shows two point masses in the middle of space. Particle A has mass, m, and particle B has mass, 3m. The separation between the two masses is D, and because of the force of gravity, they rotate about their center of mass.

a) Using the definition of the center of mass, find the radius, a, of the orbit of particle A in terms of D.

b) What is the radius, b, of the orbit of particle B?

c) What is the tangential velocity, va, of particle A?

d) What is the tangential velocity, vb, of particle B?

Explanation / Answer

the center of mass is at distance a from A

Xcm = (mA*xA + mB*xB)/(mA+mB)

a = (0 + 3mD)/(m+3m)

a = (3D/4)

___________

b = D-a = D - (3/4)D = (D/4)

___________________

force of gravity = Fg = G*mA*mB/D^2 = G*3m^2/D^2

mA = m

mB = 3m

in rotatory motion

force on A = Fc = mA*va^2/a = m*va^2/(3/4)*D

Fc = Fg

m*va^2/(3D/4) = G*3m^2/D^2

va = sqrt(9*G*m/4D)

_________

for B

3m*Vb^2/b = G*3m^2/D^2

vb^2 = (Gmb/D^2)

b = D/4

Vb = sqrt(GM/4D)

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