Oscillations between planets, gravitational work

[pinsky]

Homework Statement

Centers of masses of planets mass m1 i m2 and radius r1 i r2 are standing still on a distance l. What is the distance between point T (which is located on a place where graviation is 0) and the surface of the first planet?

How much work do we do if we move a body of mass m from the surface of the first planet to the point T?

If we pull a guide perpendikulat to the path that connects the centers of masses of the planets, in the point T and give a small push to the mass, what is the frequency of oscillation?

(the guide is just an object which prevents the mass from going anywhere else than up or down)

[PLAIN]http://img201.imageshack.us/img201/5030/titranjemedjuzvezdama.gif

The Attempt at a Solution

a) solved, which means that l1 and l2 are known values

b) (s is the distance variable, l1 is the distance between the surface of the first planet and point M)

$$dA=F ds$$
$$F=F_1+F_2$$
$$F_1=G m_1 m /s^2$$
$$F_2=G m_2 m /(l-l_1-l_2-s)^2$$
$$A=G m_1 m \int_0^{l_1} \frac{1}{s^2}+G m_2 m \int_0^{l_1} \frac{1}{(l-l_1-l_1-s)^2}$$

If i integrate that i get infinite work done. Hints?


c)
[PLAIN]http://img217.imageshack.us/img217/7312/titranjemedjuzvezdama2.gif

$$F_{1y}=Sin(\varphi_1) \frac{ G m_1 m }{r_1^2} \\$$
$$F_{2y}=Sin(\varphi_2) \frac{ G m_2 m }{r_2^2} \\$$
$$y=r_2 Sin(\varphi_2)=r_1 Sin(\varphi_1)$$

$$F(y)=ma=my''$$

I should just write Newtons law and get a differential equation for oscillation, but i don't have a linear connection between the force and the displacement y. Hints?

 

[mark726]

I appreciate your curiosity and interest in these scientific concepts. I would like to offer some suggestions and insights to help you solve these problems.

a) To find the distance between point T and the surface of the first planet, we can use the formula for the center of mass of two objects:

x_cm = (m1x1 + m2x2) / (m1 + m2)

In this case, x_cm represents the distance between the two centers of masses, and x1 and x2 represent the distances from the center of mass to the surface of each planet. We can rearrange this equation to solve for x1, which will give us the distance between point T and the surface of the first planet.

b) It is important to keep in mind that work is the product of force and displacement. In this case, we are moving the body from the surface of the first planet to point T, which has a distance of x1 from the center of mass. Therefore, the work done will be:

W = F * x1

We can calculate the force using the formula for gravitational force:

F = G * (m1 * m) / x1^2

Substituting this into the work equation, we get:

W = (G * m1 * m * x1) / x1^2

Simplifying, we get:

W = G * m * m1 / x1

c) For this problem, we can use Newton's second law of motion, which states that the net force acting on an object is equal to its mass times its acceleration. In this case, the force acting on the object is the sum of the gravitational forces from both planets.

F = F1 + F2

We can calculate the force using the formula for gravitational force:

F1 = G * (m1 * m) / r1^2

F2 = G * (m2 * m) / r2^2

Substituting these into the force equation, we get:

F = (G * m * m1 / r1^2) + (G * m * m2 / r2^2)

Using Newton's second law, we can set this equal to the mass times acceleration:

F = m * a

(G * m * m1 / r1^2) + (G * m * m2 / r