A pendulum having a bob of mass m is hanging in a ship sailing along the equator from east to west. When the ship is stationary with respect to water the tension in the string is T₀. (a) Find the speed of the ship due to rotation of the earth about its axis. (b) Find the difference between T₀ and the earth's attraction on the bob. (c) If the ship sails at speed v, what is the tension in the string? Angular speed of earth's rotation is ω and radius of the earth is R.

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Solution

(a) Speed of the ship due to rotation of the Earth is v = ωR, where R is the radius of the Earth and ω is its angular speed.

(b) The tension in the string is given by
T₀ = mg − mω²R
∴ T₀ − mg = mω²R

(c) Let the ship move with a speed v.
Then the tension in the string is given by

$T = m g - m ω_{1}^{2} R = T_{0} - \frac{{(v - ω R)}^{2}}{R^{2}} R = T_{0} - \frac{(v^{2} + ω^{2} R^{2} - 2 ω R v)}{R} R ∴ T = T_{0} + 2 ω v m$

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