A speedboat of mass 531 kg (including the driver) is tethered to a fixed buoy by

Question

A speedboat of mass 531 kg (including the driver) is tethered to a fixed buoy by a strong 28.1-m cable. The boat's owner loves high speed, but doesn't really want to go anywhere. So he revs up the boat's engine, makes a lot of noise, and runs the boat in circles around the buoy with the cable supplying all the necessary centripetal force. When the tension of the cable is steady at 1.31 × 104 N, with what force is the boat's engine pushing the boat? Different physics textbooks treat drag force somewhat differently and use different formulas. For the present purpose, take the water's drag force on the boat to be (450 kg/m)× v2, where v denotes the boat's speed. Ignore any drag force on the cable.

Explanation / Answer

let tangential speed of the boat be v m/s

then centripetal force=mass*speed^2/radius=531*v^2/28.1=18.8968*v^2

as given centripetal force=tension in the cable

==>18.8968*v^2=1.31*10^4

==>v=26.3294 m/s

then drag force=450*v^2=3.1195*10^5 N

hence total force by which the engine is pushing the boat=1.31*10^4+3.1195*10^5=3.25*10^5 N

hence the boat's engine is pushing with 3.25*10^5 N force.

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