Twin Cavities | PhysElo

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57.Twin Cavities

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A uniform sphere of mass

M

and radius

R

has two spherical cavities, each of radius

R/2

, scooped out so that the cavities are tangent to each other at the sphere's center and tangent to the sphere's outer surface at diametrically opposite points (so the two cavities together fill an entire diameter). Point

P

lies on the sphere's outer surface along the axis perpendicular to the line joining the two cavity centers. What is the magnitude of the gravitational field at

P

, expressed in terms of the original pre-cavity mass

M

A 2D cross-section in the $xy$-plane. Draw a large solid circle of radius $R$ centered at the origin (the planet). Inside it, draw two dashed circles of radius $R/2$ centered at $(R/2, 0)$ and $(-R/2, 0)$, representing the two cavities; they should be tangent to each other at the origin and tangent to the outer circle at $(R,0)$ and $(-R,0)$ respectively. Mark the origin with a small dot labeled "$O$". Mark the point $(0,R)$ at the top of the outer circle with a dot labeled "$P$". Label the planet's radius $R$ with a small radial line segment, and label one cavity's radius $R/2$ similarly. Shade the two cavity regions lightly to distinguish them from the remaining (solid) planet material.

(0.642)\,GM/R^2

(0.750)\,GM/R^2

(0.800)\,GM/R^2

(0.821)\,GM/R^2

(0.911)\,GM/R^2

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