# Pendulum motion lagrange's equation

• newtoquantum
In summary, the conversation was about solving a past exam problem involving a simple pendulum with a massless support that is moving horizontally with constant acceleration. The goal was to determine the Lagrange's equations of motion and the period of small oscillations. The individual provided their solution for the Lagrange's equation and asked for confirmation on its correctness. They also mentioned still trying to figure out the next part and asked for clarification on whether the pendulum is attracted or attached to the massless support. The expert suggests that the coordinates used in the solution seem strange and may not be correct, and suggests using a different approach to set up the coordinates.
newtoquantum
i have been trying to solve this past exam problem, a simple pendulum of length l and bob with mass m is attracted to a massless support moving horizontally with constant acceleration a. Determine the lagrange's equations of motion and the period of small oscillations.

here's what i solved for lagrange's equation:
Coordinates of mass
x = l cos θ
y = l sin θ + f(t)
Velocity of mass
x˙ = l(−sin θ)θ˙
y˙ = (cosθ)θ˙ + f˙
Kinetic energy
T =1/2m( ˙ x2 + ˙y2)
=1/2m[l2θ˙2 + (2lf˙ cos θ)θ˙ + f˙2]
Potential energy
U = −mgx
= −mgl cos θ
∂T/∂θ=1/2m · 2lf˙θ˙(−sin θ) = −mlf˙θ˙ sin θ
∂T/∂θ˙=1/2m[2l2θ˙ + 2lf˙ cos θ] = ml2θ˙ + mlf˙ cos θ
∂U/∂θ= −mgl(−sin θ) = mgl sin θ
Lagrangean
L = T − U =1/2m[l2 ˙ θ2 + (2l ˙ f cos θ)˙θ + f˙2] + mgl cos θ
Lagrange’s eqs.
∂L/∂θ −d/dt(∂L/∂θ)= 0
∂T/∂θ −∂U/∂θ −d/dt∂T/∂θ˙= 0
−mlf˙θ˙ sin θ − mgl sin θ −d/dt[ml2˙θ + mlf˙ cos θ] = 0
mlf˙θ˙ sin θ + mgl sin θ + ml2θ¨+ mlf¨cos θ + mlf˙(−sin θ)θ˙ = 0
Finally,
ml2 ¨θ + mgl sin θ + ml ¨ f cos θ = 0

I am not sure if i have it done correctly and aslo still trying to figure out the next part... Thanks for your concern...

Is it attracted to a massless support or attached to a massless support? It makes a difference?

You never stated how you set up your coordinates, but they look a bit strange. It is pretty common in pendulum problems to measure the angle from the downward vertical, in which case we get something like
x = L sin(theta)
y = L cos(theta)
where x is pos to the right, y is pos downward.
I'm having a lot of difficulty visualizing how your coordinates work, so perhaps you could explain them.

I rather think this is a moving support problem, in which case, you really need something like
Xm = Xs + L*sin(theta)
where
Xm is the position of the mass
Xs is the position of the support
and the last term is the motion of the pendulum bob relative to the support.

Once the kinematics is cleared up, then we can talk about the Lagrange equation. Without the correct kinematics, Lagrange is just a case of GIGO.

## What is Pendulum Motion Lagrange's Equation?

Pendulum Motion Lagrange's Equation is a mathematical equation that describes the motion of a pendulum. It is derived from the Lagrangian formalism, which is a mathematical framework for analyzing the dynamics of a system.

## Why is Pendulum Motion Lagrange's Equation important?

Pendulum Motion Lagrange's Equation is important because it allows us to accurately predict the motion of a pendulum under various conditions. It takes into account factors such as the mass, length, and initial position of the pendulum, making it a more comprehensive and accurate model than simpler equations.

## What are the variables in Pendulum Motion Lagrange's Equation?

The variables in Pendulum Motion Lagrange's Equation include the mass of the pendulum (m), the length of the pendulum (l), the angle of the pendulum (θ), and the gravitational constant (g).

## How is Pendulum Motion Lagrange's Equation derived?

Pendulum Motion Lagrange's Equation is derived using the principle of least action, which states that the motion of a system will follow the path that minimizes the action. The Lagrangian function is then used to express the kinetic and potential energy of the pendulum, and the equation is derived using the Euler-Lagrange equations.

## What are some practical applications of Pendulum Motion Lagrange's Equation?

Pendulum Motion Lagrange's Equation has various practical applications, such as predicting the motion of a swing or a clock pendulum. It is also used in engineering for designing and analyzing mechanical systems, such as pendulum clocks and suspension bridges.

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