The original definition of the metre was such that the distance along the Earth\

Question

The original definition of the metre was such that the distance along the Earth's surface between the North Pole and the equator was 10 000 km. (a) Determine the percentage error between this original definition of the metre and the modern definition, which is used when quoting the radius of the Earth. You can assume the Earth is a sphere. The Moon has a mass 81 times smaller than that of the Earth, and is 3.84 times 10^5 km away. (b) Determine, using Newton's law of gravitation, the gravitational force exerted by the Earth on the Moon. (c) Determine the acceleration of the Moon due to the force in (b), and write down the direction of this acceleration. (d) Deduce, using your answer to (c), the angular velocity of the Moon about the Earth (e) Calculate the period of the Moon's orbit around the Earth in Earth days.

Explanation / Answer

(A) R' = 5000 km

1m original definition = d / (5000 x 10^3)

R = 6371 km

modern definition = d / (6371 x 10^3)

%error = (original - modern) / original x 100

= (1/5000 - 1/6371) / (1/5000) x 100

= 21.5%

(b) F = G m1 m2 / d^2

= (6.67 x 10^-11) (5.972 x 10^24) (5.972 x 10^24/ 81) / (3.84 x 10^8)^2

= 2 x 10^20 N

(c) a = F/m = (2 x 10^20) / (5.972 x 10^24/81)

= 2.7 x 10^-3 m/s^2

towards the earth

(d) a = w^2 r

2.7 x 10^-3 = w^2 (3.84 x 10^8)

w = 2.65 x 10^-6 rad/s

(e) T = 2pi / w = 2.37x 10^7 sec

T = 27.4 days

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