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 ?
F₁=G•m•M/R₁²
F₂ =G•m•M/R₂²
F₁/F₂=R₂²/R₁²
F₂=F₁•R₁²/R₂²= …
To find the magnitude of the gravitational force exerted on the satellite by the planet if the radius of the orbit were increased, we can use Kepler's Third Law, which states that the square of the period of revolution of an object is directly proportional to the cube of its average distance from the center of attraction.
Kepler's Third Law equation can be expressed as:
T^2 ∝ r^3
Since we know that the satellite is in a circular orbit, the period of revolution (T) is the same for both cases.
Therefore, we can set up the following proportion:
(r1)^3 / (r2)^3 = (F1 / F2)^2
Where:
r1 = initial radius = 1.9×107m
r2 = final radius = 3.2×107m
F1 = initial gravitational force = 140N
F2 = unknown final gravitational force
Substituting the given values into the equation:
(1.9×107)^3 / (3.2×107)^3 = (140 / F2)^2
Simplifying the equation:
(1.9/3.2)^3 = 140^2 / F2^2
Solving for F2:
F2^2 = 140^2 / (1.9/3.2)^3
F2^2 = 140^2 / (1.9^3 / 3.2^3)
F2^2 = 140^2 * (3.2^3 / 1.9^3)
F2^2 = 140^2 * (32 / 17)^3
Calculating the value of F2:
F2^2 = 62100 * (32 / 17)^3
F2^2 = 62100 * 32^3 / 17^3
F2^2 = 62100 * 32 * 32 * 32 / 17 * 17 * 17
F2^2 ≈ 1.17 * 10^6
Taking the square root of both sides:
F2 ≈ √(1.17 * 10^6)
F2 ≈ 1080 N
Therefore, the magnitude of the gravitational force exerted on the satellite by the planet if the radius of the orbit were increased to 3.2x10^7m would be approximately 1080 N.
To find the magnitude of the gravitational force exerted on the satellite by the planet when the radius of the orbit is changed, we can use the formula for the gravitational force:
F = G * (m1 * m2) / r^2
Where:
F is the gravitational force
G is the gravitational constant (6.67430 × 10^-11 N m^2 / kg^2)
m1 and m2 are the masses of the two objects (in this case, the satellite and the planet)
r is the distance between the center of the two objects (in this case, the radius of the orbit)
We are given the magnitude of the gravitational force, F = 140 N, and the initial radius of the orbit, r = 1.9 × 10^7 m. We need to calculate the magnitude of the gravitational force when the radius of the orbit is changed to r = 3.2 × 10^7 m.
First, we need to find the unknown mass of the planet. We can rewrite the formula as:
m2 = (F * r^2) / (G * m1)
Using the given values, we can substitute them into the equation:
m2 = (140 N * (1.9 × 10^7 m)^2) / (6.67430 × 10^-11 N m^2 / kg^2 * m1)
Now we can solve for the mass of the planet, m2.
Next, we can use the mass of the planet, m2, and the new radius of the orbit, r = 3.2 × 10^7 m, to find the new magnitude of the gravitational force. Using the same formula:
F_new = G * (m1 * m2) / r_new^2
Substituting the known values, we can calculate the new magnitude of the gravitational force, F_new.
This will give us the answer to the question.
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 ?