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Assume a planet is a uniform sphere of radius R that (somehow) has a narrow radial tunnel through its center. Also assume we can position an apple anywhere along the tunnel or outside the sphere. Let FR be the magnitude of the gravitational force on the apple when it is located at the planet's surface. How far from the surface (what multiple of R) is there a point where the magnitude of the gravitational force on the apple is 0.3 FR if we move the apple (a) away from the planet and (b) into the tunnel?

Respuesta :

Answer:

Explanation:

Let mass of the apple be m . Force on it at surface of the planet can be expressed as follows

F_R  = G Mm / R²

Where M is mass of the planet.

Let at at height h the gravitational force on apple be 0.3 F_R

0.3 F_R  = G Mm / ( R +h)²

Dividing the two equation

1 / 0.3  =  ( R +h)²/ R²

( R +h)/ R = 1.825

h = .825 R

b )

If we go deep inside the planet to get force on apple equal to .3 F_R

.3 F_R = G  4/3 π ( R - h )³ d m / (R-h)²         [ d is density of planet ]

.3 F_R = G  4/3 π ( R - h ) d m

F_R  = G Mm / R²

F_R  = G 4/3πR³d m / R²

F_R  = G 4/3πR d m

dividing two equations

0.3 =  ( R - h ) / R

h = .7 R

batolisis

The answers to your question are as listed below

A) If we move the apple 0.3 FR away from the planet the value of h is ;

0.825 R

B) If we move the apple 0.3 FR into the tunnel the distance is ; 0.7 R

Assumptions :

mass of apple = m

Force on apple at planet surface = [tex]F_{r}[/tex] = GMm / R² --- ( 1 )

Determine th value of h

a) Applying equation ( 1 )

gravitational force of apple at h = 0.3 FR

therefore equation ( 1 ) becomes

0.3 FR =  GMm / (R + h)² ----- ( 2 )

divide both sides of equation by

1 / 0.3  =  ( R + h )²/ R²

therefore :

h = 0.825 R

b) When we go into the tunnel

Applying the relation below

0.3 FR = G *4/3 π *( R - h )³* dm / (R-h)²   -- ( 3 )

where : d = density of planet

resolving equation ( 3 )

FR  = G *4/3πR * dm

dividing both sides of the equation

0.3 =  ( R - h ) / R

therefore :

h = 0.7 R

Hence we can conclude that A) If we move the apple 0.3 FR away from the planet the value of h is 0.825 R  and  If we move the apple 0.3 FR into the tunnel the distance is ; 0.7 R

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