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Pi-Bond
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Homework Statement
A train consists of a locomotive and five identical carriages, connected via massless ropes. Initially the train is moving at a speed V. At this speed, the tension in the rope, F between the locomotive and the first carriage exactly balances the resistive drag, so the train does not accelerate.
1. Find the tension in all other ropes in terms of F.
2. A single carriage has mass M and length L. The resistive force is given by kV, where k is a constant. If the train now moves along a bend of radius R, (R>>L) show that the velocity for which minimum force is exerted by the tracks on the central carriage is given by
[itex]V_m = \frac{5 k L}{2M}[/itex]
Homework Equations
Centripetal acceleration
[itex]a_c=\frac{v^2}{c}[/itex]
Newton's Second Law
The Attempt at a Solution
1. Label the carriages 1 to 5. (1:closest to locomotive) Let Fi be the coupling between the ith an (i-1)th carriage. If I start from the last carriage, then Newton's second law will give: F5= MkV.
Similarly for the second carriage, F4=F5+MkV=2MkV.
Continuing all the way to the first carriage, I find F=5MkV
So
[itex]F_5=\frac{F}{5}[/itex]
[itex]F_4=\frac{F}{4}[/itex]
etc.
2. The central carriage is the third one. At a velocity V, the drag on this carriage will be MkV. The other forces will be the tensions of the two connecting ropes. I'm confused whether to include a force due to the track? The resultant of the forces must be the centripetal force. Also I am unsure of how to include the length L in the equations.
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