Newton law of universal gravitation

[Tiven white]

Homework Statement

Having problems on the method of calculating the acceleration due to gravity of the planet.

A certain planet has a diameter of 1715 km and a density of 5254 kg/m³. The planet has a moon that orbits every 4.46 earth-days.
What is the acceleration due to gravity associated with this planet? Answer in units of miles/hour

Homework Equations

V = (4/3)*∏*(r)^3
V-volume r- radius

F = (G *M1*m2)/(r^2)
G - gravitational constant
r - radius

F = (m)*(v^2)/(r)
r-radius
V--velocity

The Attempt at a Solution

so after utilizing half the diameter given as the radius the formula for volume was used to calculate the volume of this planet. since density = mass/volume the equation was rearranged to find the mass of the planet. i am a bit stuck here since i find it difficult using the information for the orbit of the planet into this problem. any insights towards the solution for the acceleration would be appreciated.

 

[haruspex]

F = (G *M1*m2)/(r^2)
G - gravitational constant
r - radius
F = (m)*(v^2)/(r)
r-radius
V--velocity

OK, but not, in general, the same radius as in the previous formula, right?

It isn't entirely clear, but I assume you're being asked for the acceleration due to the planet's gravity at the moon's distance.

so after utilizing half the diameter given as the radius the formula for volume was used to calculate the volume of this planet. since density = mass/volume the equation was rearranged to find the mass of the planet. i am a bit stuck here since i find it difficult using the information for the orbit of the planet into this problem.

Let the distance from the moon to the centre of the planet be d.
In terms of that, what would the gravity at that distance be? What period of orbit would it give you?

 

[tiny-tim]

Hi Tiven white!

A certain planet has a diameter of 1715 km and a density of 5254 kg/m³. The planet has a moon that orbits every 4.46 earth-days.
…
F = (G *M1*m2)/(r^2)
G - gravitational constant
r - radius

F = (m)*(v^2)/(r)
r-radius
V--velocity

… i find it difficult using the information for the orbit of the planet into this problem. any insights towards the solution for the acceleration would be appreciated.

you don't know v but you do know ω (the angular velocity), so use the alternative formula for centripetal force:

F = mω²r ​

(and that gives you two formulas for F, and there's only one value of r for which they're equal)

 

[SteamKing]

Acceleration due to gravity will not have units of miles per hour. Those are the units of velocity.

 

[HalfLostAndSearching]

The acceleration due to gravity of a planet can be calculated using Newton's law of universal 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. In this case, we can use the mass and radius of the planet to calculate the gravitational force acting on an object on the surface of the planet.

First, we need to calculate the mass of the planet using the given density and diameter. Using the formula for volume, we can find the volume of the planet and then use the density to find the mass.

Next, we can use the given information about the orbit of the moon to find the distance between the planet and the moon. This distance can then be used in the formula for gravitational force, along with the mass of the planet and the mass of the moon.

Finally, we can use the formula for acceleration, a = F/m, to find the acceleration due to gravity on the surface of the planet. To convert the units from m/s² to miles/hour², we can use unit conversion factors.

I hope this helps guide you towards finding the solution to your problem. Remember to always carefully consider the given information and use the appropriate equations to solve for the unknown quantity.