Question

Two stars, each of mass M, orbit around their center of mass. The radius of their common orbit is r (their separation is 2r).

4. Two stars, each of mass M, orbit around their center of mass. The radius of their common orbit is r (their separation is 2r). A planetoid of mass m (<< M) happens to move along the axis of the system (the line perpendicular to the orbital plane which intersects the center of mass) as shown in the figure. b. c a. Calculate directly the force on the planetoid if it is displaced a distance z from the center of mass (you'll need to use vector components for this). Calculate the gravitational potential energy as a function of the displacement z and use it to verify the result of part a. (HINT: we know F =- do in this case, and we know U =- GMm) C. Find approximate expressions (using Taylor's theorem) for the potential energy and the force in the cases z>> rand z << r. d. Show that if the planetoid is displaced slightly from the center of mass, simple harmonic motion occurs. (Hint, consider the equation for the mechanical energy of the planetoid given your approximate potential energy from part c with z small).

A planetoid of mass m (<< M) happens to move along the axis of the system (the line perpendicular to the orbital plane which

intersects the center of mass) as shown in the figure.

a. Calculate directly the force on the planetoid if it is displaced a distance z from the

center of mass (you’ll need to use vector components for this).

b. Calculate the gravitational potential energy as a function of the displacement z

and use it to verify the result of part a. (HINT: we know F = - dU/dz in this case, and

we know U = − GMm/r)

c. Find approximate expressions (using Taylor’s theorem) for the potential energy

and the force in the cases z >> r and z << r.

d. Show that if the planetoid is displaced slightly from the center of mass, simple

harmonic motion occurs. (Hint, consider the equation for the mechanical energy

of the planetoid given your approximate potential energy from part c with z small)

Answer 1

Exp. No.......... fi m fz #2 Case z >> U - 2_GMM f coso f cose U - 2 - = i >> G Mm t z f = GMm 22 +22 f = GMm g2+2² fnet af sine fnet = 2+G Mmx2 (72+2)/2 (f, + f2) sino f loso = f loso So cancelout Eacho Huer sino = 2 Case r » z - u = -2G Mm No2+22 192,22 U = -2 GM m - r if 8 žr -x x - 172722)"} b) u =( 6 Man ) + ) 2 = -2 G Mm √2+22 dz lets verify f= - du f - - 1 1 - 2 G Mm 3 ) = 29mm f/ 2 (z 2 + 22 %) - Just like any simple harmonie motion. force should be restoring as displaced from equilibrium position) Here when planetoid is displaced upward Gravitational force will pull it toward centre but momenturm of planetoid will take it below centre point and again force will act towards centre as it more downwards creating an hormon motion f = - ka - en as (2) where k = + 2 G Mme (72+22) 2 TZ F -2 GMm (2) Teacher's Signature ... = 2G Mm £ x 27 1 2 3 4 = 2 G Mm (2²+22) / so restoring g3 a feak z