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 .

Part A
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 ?

Well, well, well, looks like our satellite is going on a cosmic joyride with different orbit radii! Let's calculate the magnitude of the gravitational force for the new radius, shall we?

First things first, we need to know that the gravitational force is inversely proportional to the square of the distance between two objects. So, if we increase the radius of the orbit from 1.9×10^7m to 3.2×10^7m, we can calculate the change in the magnitude of the gravitational force.

Now, since you said the original force was 140N, let's call it F1, and the force with the new radius F2.

According to the law of gravity, we have the equation:

F1/F2 = (r2/r1)^2

Substituting the values we know:

140N/F2 = (3.2×10^7m/1.9×10^7m)^2

Time for some math magic!

140N/F2 = (3.2/1.9)^2

Now let's do some calculating:

(3.2/1.9)^2 = 2.778

So, 140N/F2 = 2.778

To find F2, we can rearrange the equation:

F2 = 140N/2.778

And the moment of truth:

F2 ≈ 50.4N

Therefore, if the radius of the orbit were increased to 3.2×10^7m, the magnitude of the gravitational force exerted on the satellite by the planet would be approximately 50.4N. Happy orbiting!

To calculate the magnitude of the gravitational force exerted on the satellite by the planet when the radius of the orbit is increased to 3.2×10^7m, we can use the formula for gravitational force:

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

Where:
F is the magnitude of the gravitational force
G is the gravitational constant (approximately 6.67 × 10^-11 N m^2/kg^2)
M is the mass of the planet
m is the mass of the satellite
r is the radius of the orbit

Given that the radius of the initial orbit is 1.9×10^7m and the magnitude of the gravitational force is 140N, we can rearrange the formula to solve for M:

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

Let's plug in the values:

M = (140N * (1.9×10^7m)^2) / (6.67 × 10^-11 N m^2/kg^2)

Simplifying:

M = (140N * 3.61×10^14m^2) / (6.67 × 10^-11 N m^2/kg^2)

M = (5.1764×10^16 N m^2) / (6.67 × 10^-11 N m^2/kg^2)

M ≈ 7.76×10^26 kg

Now we can calculate the magnitude of the gravitational force when the radius of the orbit is increased to 3.2×10^7m using the same formula:

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

Where r' is the new radius of the orbit (3.2×10^7m).

Plugging in the values:

F' = (6.67 × 10^-11 N m^2/kg^2 * 7.76×10^26 kg * m) / (3.2×10^7m)^2

Simplifying:

F' ≈ (5.15792×10^23 N) / (1.024×10^15 m^2)

F' ≈ 5.04 N

Therefore, the magnitude of the gravitational force exerted on the satellite by the planet when the radius of the orbit is increased to 3.2×10^7m would be approximately 5.04N.

To find the magnitude of the gravitational force exerted on the satellite when the radius of the orbit is increased, we can use the formula for gravitational force:

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

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

Given that the magnitude of the gravitational force when the radius of the orbit is 1.9×10^7 m is 140 N, and we want to find the magnitude of the gravitational force when the radius of the orbit is 3.2×10^7 m.

Let's assume the mass of the satellite m2 remains the same (since it's not explicitly mentioned), and we need to determine the mass of the planet m1.

We can rearrange the formula to solve for the mass of the planet:

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

Substituting the values into the formula, we have:

m1 = (140 N * (3.2×10^7 m)^2) / (6.67×10^-11 N*m^2/kg^2 * m2)

Here, we don't know the mass of the satellite (m2). So we cannot calculate the exact mass of the planet or the exact magnitude of the gravitational force. However, we can determine the ratio between the two magnitudes of gravitational force by dividing the second formula by the first one.

F2 = (F1 * r2^2) / (r1^2)

Where F2 is the gravitational force when the radius is 3.2×10^7 m, F1 is the gravitational force when the radius is 1.9×10^7 m, r2 is the new radius, and r1 is the original radius.

Dividing the second formula by the first one, we obtain:

F2 / F1 = (r2^2) / (r1^2)

Substituting the values into the equation:

F2 / 140 N = (3.2×10^7 m)^2 / (1.9×10^7 m)^2

Calculating the values, we find:

F2 / 140 N ≈ 3.21

Therefore, the magnitude of the gravitational force exerted on the satellite by the planet would be approximately 3.21 times the initial magnitude of 140 N when the radius of the orbit is increased to 3.2×10^7 m.

F₁=G•m•M/R₁²

F₂ =G•m•M/R₂²

F₁/F₂=R₂²/R₁²
F₂=F₁•R₁²/R₂²= …