Universal gravitation 4-determine the weight of astronaut on planet Z

[dani123]

Homework Statement

An astronaut weighs 833N on the surface of the Earth. Determine the weight of the astronaut on Planet Z if the planet's mass is 50.0 times the mass of the Earth and has a radius of 10.0 times the radius of the Earth.

Homework Equations

Kepler's 3rd law: (T_a/T_b)²=(R_a/R_b)³

motion of planets must conform to circular motion equation: F_c=4∏²mR/T²

From Kepler's 3rd law: R³/T²=K or T²=R³/K

Gravitational force of attraction between the sun and its orbiting planets: F=(4∏²K_s)*m/R²=Gm_sm/R²

Gravitational force of attraction between the Earth and its orbiting satelittes: F=(4∏²K_e)m/R²=Gm_em/R²

Newton's Universal Law of Gravitation: F=Gm₁m₂/d²

value of universal gravitation constant is: G=6.67x10^-11N*m²/kg²

weight of object on or near Earth: weight=F_g=m_og, where g=9.8 N/kg
F_g=Gm_om_e/R_e²

g=Gm_e/(R_e)²

determine the mass of the Earth: m_e=g(R_e)²/G

speed of satellite as it orbits the Earth: v=√GM_e/R, where R=R_e+h

period of the Earth-orbiting satellite: T=2∏√R³/GM_e

Field strength in units N/kg: g=F/m

Determine mass of planet when given orbital period and mean orbital radius: M_p=4∏²R_p³/GT_p[Kepler's 3rd law: (T_a/T_b)²=(R_a/R_b)³

motion of planets must conform to circular motion equation: F_c=4∏²mR/T²

From Kepler's 3rd law: R³/T²=K or T²=R³/K

Gravitational force of attraction between the sun and its orbiting planets: F=(4∏²K_s)*m/R²=Gm_sm/R²

Gravitational force of attraction between the Earth and its orbiting satelittes: F=(4∏²K_e)m/R²=Gm_em/R²

Newton's Universal Law of Gravitation: F=Gm₁m₂/d²

value of universal gravitation constant is: G=6.67x10^-11N*m²/kg²

weight of object on or near Earth: weight=F_g=m_og, where g=9.8 N/kg
F_g=Gm_om_e/R_e²

g=Gm_e/(R_e)²

determine the mass of the Earth: m_e=g(R_e)²/G

speed of satellite as it orbits the Earth: v=√GM_e/R, where R=R_e+h

period of the Earth-orbiting satellite: T=2∏√R³/GM_e

Field strength in units N/kg: g=F/m

Determine mass of planet SUP]2[/SUP]

The Attempt at a Solution

F_g=weight=833N on surface of the Earth

m_p=50 * (5.98*10²⁶ kg)
R_p=10*(6.38*10⁶m)= 6380000 m

I used g_p=GM_p/(R_p)²=489.95 N/kg

I also used F_g=gxm_o and manipulated the equation to solve for m_o=1.7 kg

Just wondering if someone would be able to have a look at my attempt and let me know if its wrong and if it is maybe point out where it is that I made my mistake. It would be greatly appreciated! Thanks again so much in advance!

 

[Doc Al]

Wow, that is one complicated pile of equations! (Most of which are irrelevant.)

All you need is this:

F_g=Gm_om_e/R_e²

To find Fg on the planet, just replace the mass and radius of the Earth with the mass and radius of the planet and compare. (Hint: Almost no calculation is required.)

 

[cepheid]

A lot of the relevant equations that you posted are actually totally irrelevant. In fact, Newton's universal law of gravitation is all you need, and a couple of the other equations you have there are just Newton's universal law of gravitation applied to specific situations.

For me, step 1 would be to find the mass of the astronaut. It is certainly not 1.7 kg! Think about that for a second. How can a human being have a mass of only 1.7 kg?

Getting the mass shouldn't be too hard, since you know the weight on Earth, and you know g on Earth.

Then the second step, once you have the mass, would be to multiply it by the g_p value that you computed to get the weight on the planet.

EDIT: Er, yeah, or you could do what Doc Al said, which is even smarter.

 

[dani123]

do I multiply the g_p with the mass of the astronaut on Earth in order to find out the value of his mass on the planet Z? Just want to make sure I understood that correctly...

 

[Doc Al]

do I multiply the g_p with the mass of the astronaut on Earth in order to find out the value of his mass on the planet Z?

You can certainly do it that way. The mass is the same everywhere, of course. But you'll have to redo your calculation of g_p. You can do that easily by just comparing the calculation to that of g_e. (Again, hardly any calculation needed. But more than if you followed my original method.)

 

[dani123]

ok so i tried to do what you said but i just ended up getting mass=84.99kg... which is only 0.1kg off from what the mass is on earth...

 

[Doc Al]

ok so i tried to do what you said but i just ended up getting mass=84.99kg... which is only 0.1kg off from what the mass is on earth...

I don't quite understand what you mean. There's only one mass. If you calculate the astronaut's mass on Earth then that is also his mass on the planet. (But that value for mass looks OK.)

 

[dani123]

so why do they even ask what the mass on planet Z is, if they are both the same?

 

[BruceW]

The question asked for weight, not mass. Weight and mass are two different things in physics, even though in everyday language they are often used interchangeably. You just need to use the physics definition of mass and weight.

 

[Doc Al]

so why do they even ask what the mass on planet Z is, if they are both the same?

As BruceW already pointed out, they don't ask for the mass on planet Z. They ask for the weight on planet Z.

The only purpose in finding the mass would be to use it to find the weight via W_p = mg_p. Which is OK, but, as I have tried to point out, that's the hard way of doing the problem.

 

[dani123]

so is it just g of the planet which I found to be 489.95N/kg, and times that by 85kg (which is the mass of the astronaut)? If I do that I get weight=force due to gravity=41646.12N

 

[Doc Al]

so is it just g of the planet which I found to be 489.95N/kg, and times that by 85kg (which is the mass of the astronaut)? If I do that I get weight=force due to gravity=41646.12N

As I said earlier, you'll need to redo your calculation of g on the planet as it is incorrect. But once you have the correct value of g, then you can multiply it by the mass to find the weight on the planet. (That's the hard way, but perfectly OK.)

 

[dani123]

I don't see how my value for g of the planet is wrong though...

 

[Doc Al]

Just for fun, try the easy way. Answer this: If the mass of the planet is increased by a factor of 50, what would happen to the weight of the astronaut? Would it go up or down? By what factor? (Imagine everything else is held fixed.)

Note that this is equivalent to asking: If the mass of the planet is increased by a factor of 50, what would happen to the value of g at the surface?

 

[Doc Al]

I don't see how my value for g of the planet is wrong though...

m_p=50 * (5.98*10²⁶ kg)
R_p=10*(6.38*10⁶m)= 6380000 m

Check the number of zeroes in that last calculation.

 

[dani123]

OPS! thank you for noticing that!

 

[dani123]

so the answer should be weight=416.46N?

 

[Doc Al]

so the answer should be weight=416.46N?

Sounds about right.

But I encourage you to follow the reasoning and answer the question I posed in my last post (#14).