- #1
Fascheue
- 17
- 3
- Homework Statement
- A 12 kg stone is tied to a 2.7 m “massless” rope, which can deal with tensions of up to 210 N. Forcing the stone on a circular trajectory in a horizontal plane using the rope, how fast can it be rotated uniformly before the rope likely gets torn apart? What happens for a rope that is twice as long? Neglect air resistance effects.
Take into account that the rotating rope holding the stone will make an angle α < π/2 with the vertical direction (i.e. the axis of rotation). The faster the stone goes around, the closer the angle α gets to this limit.
If you solved the problem using Newton’s formulation of classical mechanics, try using the Lagrangian approach next!
- Relevant Equations
- L = T - U
pd(L)/pd(x) = d/dt(pd(L)/pd(x’))
Where pd represents a partial derivative.
In the situation described in the problem, the mass is moving on a horizontal circular path with constant velocity. Wouldn’t this make L and U both constant? Then the Lagrange equation would give 0 = 0, which isn’t what I’m looking for. Any help would be appreciated.