Solving Universal Gravity: Height of Chicago's Sears Tower

[tnutty]

Homework Statement

A sensitive gravimeter is carried to the top of Chicago's Sears Tower, where its reading for the acceleration of gravity is 0.00136 m/s^2 lower than at street level.

Find the height of the building.

Homework Equations

F = GMm/(r^2)

and maybe

K + U = K_o + U_o

The Attempt at a Solution

don't know where to start!

 

[RoryP]

consider other units for m/s^2

 

[tnutty]

Well I think I could use (g - 0.00136) = g_t.

I don't know how I could use this though.

 

[tnutty]

wait could I do this :

(9.8 - 0.00136) = GMm/r^2 ?

 

[LowlyPion]

Yes, so you can then write ...

g*r₁² = G*M= (g - .00136)*r₂²

 

[tnutty]

Yes, so you can then write ...

g*r₁² = G*M= (g - .00136)*r₂²

what happened to small m

(9.8 - 0.00136)(r^2) = GM (m?)

 

[LowlyPion]

m*g*r₁² = G*M*m= m*(g - .00136)*r₂²

If that makes you happier. It isn't a factor.

F = m*g = GM*m/r²

g = GM/r²

 

[tnutty]

so,

let G_t = (9.8-0.00136);

then,

G_t*r^2 = GM

M = G_t*r^2 / G

and r is the radius of the Earth and G is the universal gravity constant.

m ~ 9.3579 * 10 ^ 17

But what does this mean? This is the mass of th building and...

 

[LowlyPion]

In case you didn't notice you don't need G and you don't need M and you don't need m.

You just need r_e and g.

Building height will be Δr = r₂ - r₁

where r₁ = r_e

 

[tnutty]

why is that?

 

[tnutty]

so

(9.8-0.00136) r^2 = GM

and GM is just dropped? why?

 

[LowlyPion]

so

(9.8-0.00136) r^2 = GM

and GM is just dropped? why?

The answer is because you can.

g*r₁² = (g - .00136)*r₂²

You just need r_e and g.

Building height will be Δr = r₂ - r₁

where r₁ = r_e

 

[tnutty]

OK, i guess.

so

(9.8-0.00136) ( 6.37 * 10^6)^2 = R1;

R1 - R2 = height of tower

R1 ~ 3.976 * 10^14
R2 = (6.37 * 10^6)

right I got 6.37 * 10^6 as the radius of the Earth from my book.

 

[LowlyPion]

No.

Try being more careful.

(6378)² km = 40678884

now multiply by 9.81/(9.81 - .00136) = ...

then take the square root. Then subtract one from the other. Keep as much precision as you can.

Your answer will be less than a km. Multiply by 1000 for meters.

 

[tnutty]

(6378)^2 = 40678884

ans * 9.81 / (9.81-0.00136) =
40684524.26

sqrt(ans) = 6378.44
??

 

[tnutty]

confused even more. Dont know where 6378 came from and why
sqrt ( ans * 9.81/(9.81-0.00136) )
...
...

 

[LowlyPion]

confused even more. Dont know where 6378 came from and why
sqrt ( ans * 9.81/(9.81-0.00136) )
...
...

I used radius of Earth as 6378 km. And g you recognize as 9.81

sqrt(ans) = 6378.44

Right.

Now 6378.44 - 6378 = .44 km = 440 m = height of building.

 

[tnutty]

could you do it out in variables first so I can see what you have done.
Sorry.

 

[LowlyPion]

https://www.physicsforums.com/showpost.php?p=2111606&postcount=12

It's all there.

 

[tnutty]

ok thanks