Gravitational Acceleration given mass and radius

[EliteCodexer]

Homework Statement

If you lived on a planet with five times the mass of Earth and twice the radius, what would be the gravitational acceleration at the surface of your planet?

Homework Equations

GM/r^2
Mass of Earth = 6.00*10^24 kg
Radius of Earth = 6.38*10^3 km

The Attempt at a Solution

So, when I started out doing this, I quickly plugged in Earth's radius (in m), mass (in kg) and the gravitational constant just to make sure it came out at 9.8 because I know that's what Earth's gravitational acceleration from the surface is, but none of the answer I get are even close. I can't even get this right and I still have to do the actual problem (using 5x the mass and 2x the radius).

 

[SteamKing]

Homework Statement

If you lived on a planet with five times the mass of Earth and twice the radius, what would be the gravitational acceleration at the surface of your planet?

Homework Equations

GM/r^2
Mass of Earth = 6.00*10^24 kg
Radius of Earth = 6.38*10^3 km

The Attempt at a Solution

So, when I started out doing this, I quickly plugged in Earth's radius (in m), mass (in kg) and the gravitational constant just to make sure it came out at 9.8 because I know that's what Earth's gravitational acceleration from the surface is, but none of the answer I get are even close. I can't even get this right and I still have to do the actual problem (using 5x the mass and 2x the radius).

Instead of making us guess where you went wrong, how about posting your work?

 

[BiGyElLoWhAt]

it does work. we did it the other day sitting around.
try
GM/r^2
Mass of Earth ≈ 5.97*10^24 kg
Radius of Earth ≈ 6.38*10^3 KM = 6.38*10^6 m
G≈ 6.67 × 10^-11

$\frac{M_{earth}}{r_{earth}^2}G ≈ \frac{5.97*10^{24}}{6.38*10^{12}}*6.67*10^{-11}$
$=\frac{5.97*6.67*10}{6.38} ≈9.805$

 

[haruspex]

I can't even get this right and I still have to do the actual problem (using 5x the mass and 2x the radius).

It's worth persevering to get that right, but you don't need it for this question.
You can solve it just by taking g = 9.8m/s² for Earth and adjusting it. What effect will 5 times the mass have on that? What effect will twice the radius have?

 

[HalfLostAndSearching]

First of all, it is important to note that gravitational acceleration is not solely determined by the mass and radius of a planet. Other factors such as the distribution of mass and the rotation of the planet also play a role. However, using the given information, we can make an estimate.

The equation GM/r^2 gives the gravitational acceleration at the surface of a planet. The gravitational constant (G) is a universal constant, so we can keep it constant for both Earth and the new planet. However, we need to convert the given values for Earth's mass and radius into standard SI units (meters and kilograms) to get an accurate result.

Using the given values for Earth's mass and radius, we get a gravitational acceleration of 9.81 m/s^2, which is close to the expected value of 9.8 m/s^2.

Now, for the new planet with 5 times the mass and twice the radius, the equation becomes:

GM/r^2 = (5*6.67*10^-11 * 6.00*10^24) / (2*6.38*10^3)^2 = 4.31 m/s^2

This means that the gravitational acceleration at the surface of the new planet would be approximately 4.31 m/s^2. This is significantly lower than Earth's gravitational acceleration, which is to be expected given the increase in mass and radius.

It is also important to remember that this is just an estimate and the actual gravitational acceleration on the new planet may vary due to other factors. As a scientist, it is important to always consider and account for all relevant variables in any calculation.