Problem involving Newton's Law of Gravitation

[collide]

1. A planet has a mass 1/2 that of Earth and a radius 2 times that of Earth. What is the acceleration due to gravity on the surface of the planet in terms of g?



2.
a)F=G*m1*m2/r$$^{2}$$
b)F=mg



3. I figured that I would let the two equations equal to each other so...

G*m1*m2/r$$^{2}$$=mg
It simplifies to be g=G*m/r$$^{2}$$
However, I don't know where to go from there to get the acceleration in terms of g.

Any suggestions would be greatly appreciated. Thanks

 

[Doc Al]

You're on the right track. Maybe rewriting it this way will give you a hint:
$$g_{earth} = G M_{earth}/R_{earth}^2$$

(Express M and R for the planet in terms of M and R for the earth.)

 

[collide]

You're on the right track. Maybe rewriting it this way will give you a hint:
$$g_{earth} = G M_{earth}/R_{earth}^2$$

(Express M and R for the planet in terms of M and R for the earth.)

So then you would get this correct:

$$g_{earth} = G (1/2)M_{earth}/(2*R_{earth})^2$$

Which you can then simplify to be:

$$g_{earth} = G M_{earth}/8R_{earth}$$

However, I don't know where to go from there to find the acceleration

 

[Doc Al]

So then you would get this correct:

$$g_{earth} = G (1/2)M_{earth}/(2*R_{earth})^2$$

Almost. You'd get:

$$g_{planet} = G M_{planet}/R_{planet}^2$$

$$g_{planet} = G (1/2)M_{earth}/(2*R_{earth})^2$$

Which simplifies to:

$$g_{planet} = (1/8) G M_{earth}/R_{earth}^2$$

I'll leave it to you to interpret the right hand side in terms of g for earth.

 

[collide]

Should I plug in the values for M_earth, R_earth, and G to find out what g_planet equals?

And then I do g_planet=(g/x) to solve for x to find out in terms of what value for g is right? Doing this method... I get accelerated is g/8. However, is the proper way to solve this problem or am I doing more work than required?

 

[Doc Al]

Should I plug in the values for M_earth, R_earth, and G to find out what g_planet equals?

And then I do g_planet=(g/x) to solve for x to find out in terms of what value for g is right? Doing this method... I get accelerated is g/8. However, is the proper way to solve this problem or am I doing more work than required?

That's definitely the hard way. We already determined that the acceleration due to gravity on the planet equals:

$$g_{planet} = (1/8) G M_{earth}/R_{earth}^2$$

But you should recognize the equation we stated with:

$$g_{earth} = G M_{earth}/R_{earth}^2$$

Substituting this last equation into the first gives:

$$g_{planet} = (1/8) g_{earth} = g/8$$

That's all you need to do--no calculations needed.

 

[collide]

Thanks for the help, Doc Al!