Planetary motion the mass of the sun

[prospec]

Homework Statement

Derive the equation using Newton's Law of gravitation and the equation for circular motion.

Homework Equations

V = $$\sqrt{\stackrel{GM}{r}}$$

Where G is the universal gravitational constant, M is the mass of the central body and r is the radius of the orbit

The Attempt at a Solution

F= GM1*m2/r2


F= GMp*Ms/r2

Fc= MpV2/r

F=FC

GMpMS/r2 = mpV2/r

I'm not sure how to derive this equation ?

 

[Janus]

Your heading in the right direction, now all you do is re-arrange the last equation until you've solved for V. (hint: some of the variables at least partially cancel out.)

 

[ogb p]

I would suggest breaking down the problem into smaller, more manageable steps. First, we can start with Newton's Law of Gravitation, which states that the force of gravity between two objects is directly proportional to the product of their masses and inversely proportional to the square of the distance between them. Mathematically, this can be represented as:

F = G * (m1 * m2)/r^2

Where F is the force of gravity, G is the universal gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.

Next, we can look at the equation for circular motion, which states that the centripetal force (Fc) required to keep an object moving in a circular path is equal to the mass of the object (m) multiplied by its velocity (V) squared, divided by the radius of the orbit (r). Mathematically, this can be represented as:

Fc = m * V^2 / r

Since we know that the force of gravity (F) is equal to the centripetal force (Fc), we can set these two equations equal to each other and solve for V:

F = Fc

G * (m1 * m2)/r^2 = m * V^2 / r

Solving for V, we get:

V = √(G * (m1 * m2)/r)

This is the equation for the velocity of an object in circular motion around a central body, in this case, the sun (represented by M). So, the mass of the sun (Ms) would be included in the equation as the central body's mass, and the mass of the planet (Mp) would be included as the orbiting object's mass. Plugging these values into the equation, we get:

V = √(G * (Ms * Mp)/r)

Which is the same equation as the one provided in the homework statement. Therefore, using Newton's Law of Gravitation and the equation for circular motion, we can derive the equation for planetary motion and its dependence on the mass of the sun.