A satellite circles a spherical planet of unknown mass in a circular orbit of radius 1.9×107m . The magnitude of the gravitational force exerted on the satellite by the planet is 140N What would be the magnitude of the gravitational force exerted on the satellite by the planet if the radius of the orbit were increased to 3.2×107m ?

To find the magnitude of the gravitational force exerted on the satellite by the planet when the radius of the orbit is increased, we can use the principle of conservation of angular momentum.

The angular momentum of an object in circular motion is given by the equation:

L = mvr

Where:
L - angular momentum
m - mass of the satellite
v - velocity of the satellite
r - radius of the orbit

The gravitational force between two objects is given by Newton's law of universal gravitation:

F = (G * m * M) / r^2

Where:
F - magnitude of the gravitational force
G - gravitational constant
m - mass of the satellite
M - mass of the planet
r - distance between the satellite and the planet

Since the mass of the satellite remains constant, we can equate the angular momentum of the satellite in the initial and final orbits:

m * v1 * r1 = m * v2 * r2

Simplifying the equation, we get:

v2 = (v1 * r1) / r2

Now, to find the magnitude of the gravitational force in the second orbit, we can substitute the value of v2 in the gravitational force equation:

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

Substituting the value of v2:

F2 = (G * m * M) / ((v1 * r1 / r2)^2)

Now, we know the value of the gravitational force in the first orbit (F1 = 140N) and the radius of the first orbit (r1 = 1.9×10^7 m).

We can substitute these values into the equation to find F2:

F2 = (G * m * M) / ((v1 * 1.9×10^7 m / r2)^2)

Now, we need to find the value of v1, the velocity of the satellite in the initial orbit. To do this, we can use the centripetal force equation:

F1 = m * v1^2 / r1

Rearranging the equation to solve for v1:

v1 = sqrt(F1 * r1 / m)

Substituting the known values:

v1 = sqrt(140N * 1.9×10^7 m / m)

Now, substitute the value of v1 into the equation for F2:

F2 = (G * m * M) / ((sqrt(140N * 1.9×10^7 m / m) * 1.9×10^7 m / r2)^2)

Simplifying the equation and calculating the value will give you the magnitude of the gravitational force exerted on the satellite by the planet when the radius of the orbit is increased to 3.2×10^7 m.