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Question
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
\[T = mg - m \omega_1^2 R\]
\[ = T_0 - \frac{\left( v - \omega R \right)^2}{R^2}R\]
\[ = T_0 - \frac{\left( v^2 + \omega^2 R^2 - 2\omega Rv \right)}{R}R\]
\[ \therefore T = T_0 + 2 \ \omega { vm }\]
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