I\'m completely lost can you please show me how you solve this? Thank you so muc

Question

I'm completely lost can you please show me how you solve this? Thank you so much!

3. The gravitational attraction due to a nearby mountain range might be expect a plumb bob to hang at an angle slightly different from the vertical. If a mountain range could be represented by an infinite half-cylinder of radius a and density p lying on a flat plane (see picture below), show that a plumb bob at a distance ro from the cylinder would be deflected by an angle ~2Gp/(rg). You ca Hint: Gauss' law is useful. In actual measurements of this effect, the observed deflection is much smaller. Next assume that the mountain range can be represented by a cylinder of radius a and density which is floating in a fluid of density 2p as illustrated. Show that th plumb-bob deflection due to the mountain range is zero in this model. Since the latter result is in much better agreement with observations, it is postulated that mountains, and also continents, are in isostatic equilibrium with the underlying mantle rock. n assume that > a. 4 PM PM

Explanation / Answer

3. cosndeir case 1

consider a cylinderical surface centerd on the axis of the mountain cylinder

let the radius of this cylinder be equal to the distance between the bob and the center of the moiuntain and let length of this cylinder be L

then

from gauss law

acceleration felt by the ball due to mountain is a

a*2*pi*ro*l = 4*piG*rhom*pi*a^2*l/2

hence

a = 4*piG*rhom*pi*a^2*l/2*2*pi*ro*l

a = 4*piG*rhom*a^2/4*ro

hence angle, theta = a/g

hence

theta = pi*G*rhom*a^2/ro*g

case II

in this case the mountain range will have a net upward force which will be cancelled by the force of the 2*rho material in the downward direction

and hence the deflection in this model is 0 deg

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