A plumb bob (a mass m hanging on a string) is deflected from the vertical by an

Question

A plumb bob (a mass m hanging on a string) is deflected from the vertical by an angle ? due to a massive mountain nearby

Part A)

Find an approximate formula for ? in terms of the mass of the mountain, mM, the distance to its center, DM, and the radius and mass of the Earth.

Part B)

Make a rough estimate of the mass of mountain, assuming it has the shape of a cone 3500 m high and base of diameter 3000 m . Assume its mass per unit volume is 3000 kg per m3.

Part C)

Estimate the angle ? of the plumb bob if it is 4.5 km from the center of mountain.

Explanation / Answer

a) in horizontal,

Tsin@ = Fm

Fm = GMm * m / Dm^2

Tsin@ = GMm * m / Dm^2

in vertical,

Tcos@ = mg

mg = GMe*m / Re^2

Tsin@ / Tcos@ = Mm* Re^2 / Me*Dm^2

@ = tan^-1 [ Mm * Re^2 / Me * Dm^2 ]

b)

volume = pi r^2 h /3 = pi x 1500^2 x 3500 = 8.25 x 10^9 m^3

mass = volume x density

= 3000 x 8.25 x 10^9
= 2.47 x 10^13 kg

c) Mm * Re^2 / Me * Dm^2

= (2.47 x 10^13 x (6371 x 10^3)^2) / (5.972 x 10^24 x (4.5 x 10^3)^2)

= 8.29 x 10^-6

angle = tan-1(8.29x 10
^-6) = 4.75 x 10^-4 deg

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