Quick Lagrangian of a pendulum question

In summary, the conversation discusses the use of the E-L equation to calculate the period of oscillation of a simple pendulum, and the effect of vertical acceleration of the pendulum support on its period of oscillation. The generalized coordinate θ is used to represent the angle of the pendulum, and the period of oscillation is found to be T=2pi√(l/g). The equations for the pendulum's motion with the support's acceleration are set up, with the y coordinate being -lcosθ+0.5at^2. The only freedom of the system is represented by the angle θ.
  • #1
Lucy Yeats
117
0

Homework Statement



Use the E-L equation to calculate the period of oscillation of a simple pendulum
of length l and bob mass m in the small angle approximation.

Assume now that the pendulum support is accelerated in the vertical direction at a rate
a, find the period of oscillation. For what value of a the pendulum does not
oscillate? Comment on this result.

Homework Equations


The Attempt at a Solution



I've got the first bit:
L=(m/2)(l^2)(dθ/dt)^2-mgl(1-cosθ)
E.O.M.: d2θ/dt2+(g/l)sinθ=0
d2θ/dt2+(g/l)θ=0 in the small angle approximation,
which is S.H.M. with ω^2=√(g/l) (though I'm not sure about this as there's no minus sign in the E.O.M.) so T=2pi√(l/g)

For the next bit, I just need help setting up the equations:
So the generalized coordinates are θ and a.
Are the following correct?:
x=lsinθ
y=-lcosθ+at
(taking the origin as the point from which the pendulum is swinging)
 
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  • #2
Lucy Yeats said:
I've got the first bit:
L=(m/2)(l^2)(dθ/dt)^2-mgl(1-cosθ)
E.O.M.: d2θ/dt2+(g/l)sinθ=0
d2θ/dt2+(g/l)θ=0 in the small angle approximation,
which is S.H.M. with ω^2=√(g/l) (though I'm not sure about this as there's no minus sign in the E.O.M.) so T=√(l/g)
Careful here, [itex]\omega[/itex] is that angular velocity in radians per time unit. Your T is the time to travel one radian (of the oscillatory cycle) not time per cycle. You need to multiply by [itex]2 \pi[/itex].
 
  • #3
Thanks for pointing that out, I'll correct that in the first post. :-)
 
  • #4
Any help would be great. :D
 
  • #5
Should y be -lcosθ+0.5at^2 instead?
 
  • #6
Lucy Yeats said:
Should y be -lcosθ+0.5at^2 instead?

Yes.

So the generalized coordinates are θ and a.

No.
"a" is not a coordinate.
Actually only θ is a generalized coordinate since the y coordinate of the support is constrained to be y=0.5at^2.
The angle θ is the only freedom that the system has.
 
  • #7
Great, I've got it now.

Thanks! :-)
 

Related to Quick Lagrangian of a pendulum question

1. What is the Quick Lagrangian of a pendulum?

The Quick Lagrangian of a pendulum is a mathematical formula that describes the dynamics of a pendulum system. It takes into account the mass of the pendulum, the length of the string, and the angle at which the pendulum is displaced from its equilibrium position.

2. How is the Quick Lagrangian of a pendulum different from other Lagrangian formulations?

The Quick Lagrangian takes into account the instantaneous acceleration of the pendulum, making it more accurate for describing the motion of a pendulum that is rapidly changing. Other Lagrangian formulations may only consider the pendulum's average acceleration.

3. Why is the Quick Lagrangian of a pendulum important?

The Quick Lagrangian is important because it allows scientists and engineers to accurately model and predict the behavior of pendulum systems. This is useful in a variety of fields, including physics, engineering, and robotics.

4. How is the Quick Lagrangian of a pendulum used in real-world applications?

The Quick Lagrangian is used in a variety of real-world applications, such as designing pendulum-based clocks, analyzing the stability of bridges and building structures, and creating control algorithms for robotic arms.

5. Can the Quick Lagrangian of a pendulum be applied to other systems?

Yes, the Quick Lagrangian can be applied to other systems beyond just pendulums, such as double pendulums, swinging doors, and spring-mass systems. It can also be extended to more complex systems by combining it with other mathematical concepts, such as Hamilton's equations.

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