If mass distorts the curvature of space time which produces gravity and since it is a attractive force,it must attract smaller masses to the larger ones but instead the smaller mass assumes a stable round the larger mass.
What force is responsible for the orbit, Explain in detail along with mathematical expressions.
The force responsible for the orbit of a smaller mass around a larger one is the gravitational force. According to Newton's law of universal gravitation, the gravitational force between two masses is given by:
F = G * (m1 * m2) / r^2
where F is the gravitational force between the two masses, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers of mass.
In the case of a smaller mass orbiting a larger one, such as a planet orbiting a star, the gravitational force provides the centripetal force necessary for the circular motion.
The centripetal force required for circular motion can be calculated using the formula:
F = (m * v^2) / r
where F is the centripetal force, m is the mass of the smaller object, v is its orbital velocity, and r is the radius of its orbit.
For a circular orbit, the gravitational force acts as the centripetal force, so we can equate the two expressions:
F = G * (m1 * m2) / r^2 = (m * v^2) / r
Rearranging the equation, we can solve for the orbital velocity v:
v^2 = (G * (m1 * m2)) / r
Taking the square root, we find the expression for the orbital velocity:
v = sqrt((G * (m1 * m2)) / r)
This equation explains the relationship between the masses of the two objects, the radius of the orbit, and the orbital velocity required for a stable circular orbit. It demonstrates that the gravitational force acting between the two masses provides the necessary centripetal force to keep the smaller object in a stable orbit around the larger object.
For elliptical or non-circular orbits, the equation would be slightly different. Instead of using the radius of the orbit, the distance between the centers of the two objects at a given point in the orbit should be used. The concept remains the same though, with the gravitational force providing the necessary centripetal force for the orbital motion.
The force responsible for the orbit of smaller masses around larger ones is known as the gravitational force. This force is based on Isaac Newton's law of universal gravitation, which states that every particle with mass attracts every other particle with mass by a force proportional to the product of their masses and inversely proportional to the square of the distance between their centers.
Mathematically, the force of gravity between two objects can be expressed as:
F = (G * m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant (approximately 6.674 * 10^-11 N*m^2/kg^2), m1 and m2 are the masses of the two objects, and r is the distance between their centers.
In the context of orbital motion, let's consider a smaller mass, such as a satellite, orbiting around a much larger mass, such as a planet. The gravitational force between these two objects provides the centripetal force necessary to keep the smaller mass in its orbit.
To understand this, we can equate the gravitational force to the centripetal force:
F = m * (v^2 / r)
Here, m represents the mass of the smaller object, v is its velocity, and r is the radius of its orbit. By substituting the gravitational force expression from Newton's law of universal gravitation, we get:
(G * m1 * m2) / r^2 = m * (v^2 / r)
Simplifying this equation, we find:
v^2 = (G * m2) / r
This equation shows that the square of the satellite's velocity is directly proportional to the mass of the larger object and inversely proportional to the radius of its orbit.
From this equation, we can deduce that an increase in the mass of the larger object would result in an increase in the required velocity for the smaller mass to maintain its orbit. Similarly, decreasing the radius of the orbit would require the smaller mass to have a higher velocity to compensate.
Overall, it is the balance between the gravitational force pulling the smaller mass towards the larger mass and the centrifugal force due to the object's velocity that allows it to remain in a stable orbit around the larger mass.