Where is the Point of Suspension to Keep a Suspended Rod Fixed?

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The discussion focuses on determining the distance 'a' from the point of suspension where the impulse J' at the top of the stick becomes zero. The problem involves applying Lagrange's equations to analyze the dynamics of a vertically suspended stick when a horizontal impulse J is applied. Participants discuss the relationship between torque and impulse, suggesting that if the torque is zero, no impulse is needed, leading to the conclusion that a=0 might be the solution. There is also mention of including a constraint term in the Lagrangian to fully solve for 'a'. Overall, the conversation emphasizes the need for a correct application of torque and impulse principles to find the solution.
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For prelim preparation. I've also included a figure of the problem.

Homework Statement


A stick of length l is suspended by one end at point P so that it hangs vertically and so that the top end of the stick does not move. A horizontal impulse J is applied perpendicular to the stick a distance a below the point of suspension. In general there will be an opposite impulse J' that must be given at the top of the stick (point P) to keep its point of suspension fixed. Find the distance a such that J'=0.

Homework Equations


\tau=r\hspace{1 mm} x\hspace{1 mm} F

\tau=I\alpha

L=T-U

The Attempt at a Solution


L=\frac{1}{2}I\dot{\theta}^{2} +mg\frac{l}{2}cos\theta

Using Lagrange's Equations
I\alpha=-mg\frac{l}{2}sin\theta

Equating the two equations for torque and setting r=a
I\alpha = aFsin\theta

Equate the last to equations
aFsin\theta=-mg\frac{l}{2}sin\theta

Solving for F
F=-\frac{mgl}{2a}

Therefore J is
J=-\frac{mglt}{2a}

So J' is
J'=\frac{mglt}{2a}

Is this looking correct? Gracias!
 

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I don't think that's right.

you have torque T=Fa since it applied perpendicular.
since it gives an impulse to balance the torque. if torque is zero no impulse. then a=0 is the point you are looking for i assume.

Your langrangian should have the constraint term too coming from the applied force.
 
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You seemingly haven't solved for a, have you? a(J' = 0)?

I'm not up on Lagrangians any more but might look at the classical solution later.

OK, looked at it, seems simple if I did it right.

If you get an answer I'll tell you if it's what I got.

Hint: torques about the c.g. ?
 
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