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I want to calculate the equation of motion of a planar pendulum of length l with a mass m at its end and a pivot point that moves uniformly along a vertical circular path (radius a) with a constant frequency ω.
The Lagrangian and the equation of motion for a planar pendulum with a moving pivot point can be derived using the Lagrange formulation.
- Length of the pendulum: ##l##
- Mass of the pendulum bob: ##m ##
- Angle of the pendulum from the vertical: ## \theta ##
- The pivot point moves along a vertical circular path with a radius ## a ## and a constant angular frequency ## \omega ##.
- Therefore, the coordinates of the moving pivot point can be described as:
$$
(x_0, y_0) = (a \sin(\omega t), -a \cos(\omega t))
$$
The total kinetic energy ##T## of the system is the sum of the kinetic energy of the pendulum bob and the motion of the moving pivot. The position of the bob in terms of the angle ## \theta ## and the moving pivot is given by:
$$
x = x_0 + l \sin(\theta) = a \sin(\omega t) + l \sin(\theta)
$$
$$
y = y_0 - l \cos(\theta) = -a \cos(\omega t) - l \cos(\theta)
$$
The velocity of the pendulum bob is the time derivative of these coordinates:
$$
\dot{x} = \frac{d}{dt}(a \sin(\omega t) + l \sin(\theta)) = a \omega \cos(\omega t) + l \dot{\theta} \cos(\theta)
$$
$$
\dot{y} = \frac{d}{dt}(-a \cos(\omega t) - l \cos(\theta)) = a \omega \sin(\omega t) + l \dot{\theta} \sin(\theta)
$$
The kinetic energy of the pendulum bob is:
$$
T = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2)
$$
The potential energy ##U## of the system is due to gravity acting on the pendulum bob. The height ##y## of the bob (with the pivot at ## y_0 = -a \cos(\omega t) ##) is:
$$
y = -a \cos(\omega t) - l \cos(\theta)
$$
The gravitational potential energy is:
$$
U = m g y = -m g \left(a \cos(\omega t) + l \cos(\theta)\right)
$$
The Lagrangian ## L## is the difference between the kinetic and potential energies:
$$
L = T - U
$$
No problem, but to derive the equation of motion, I have to apply the Euler-Lagrange equation:
$$
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\theta}} \right) - \frac{\partial L}{\partial \theta} = ???
$$
I don't know what I should put in place of the ??? I assume it is not zero as there is a driving force. How can I calculate that driving force?
The Lagrangian and the equation of motion for a planar pendulum with a moving pivot point can be derived using the Lagrange formulation.
- Length of the pendulum: ##l##
- Mass of the pendulum bob: ##m ##
- Angle of the pendulum from the vertical: ## \theta ##
- The pivot point moves along a vertical circular path with a radius ## a ## and a constant angular frequency ## \omega ##.
- Therefore, the coordinates of the moving pivot point can be described as:
$$
(x_0, y_0) = (a \sin(\omega t), -a \cos(\omega t))
$$
The total kinetic energy ##T## of the system is the sum of the kinetic energy of the pendulum bob and the motion of the moving pivot. The position of the bob in terms of the angle ## \theta ## and the moving pivot is given by:
$$
x = x_0 + l \sin(\theta) = a \sin(\omega t) + l \sin(\theta)
$$
$$
y = y_0 - l \cos(\theta) = -a \cos(\omega t) - l \cos(\theta)
$$
The velocity of the pendulum bob is the time derivative of these coordinates:
$$
\dot{x} = \frac{d}{dt}(a \sin(\omega t) + l \sin(\theta)) = a \omega \cos(\omega t) + l \dot{\theta} \cos(\theta)
$$
$$
\dot{y} = \frac{d}{dt}(-a \cos(\omega t) - l \cos(\theta)) = a \omega \sin(\omega t) + l \dot{\theta} \sin(\theta)
$$
The kinetic energy of the pendulum bob is:
$$
T = \frac{1}{2} m (\dot{x}^2 + \dot{y}^2)
$$
The potential energy ##U## of the system is due to gravity acting on the pendulum bob. The height ##y## of the bob (with the pivot at ## y_0 = -a \cos(\omega t) ##) is:
$$
y = -a \cos(\omega t) - l \cos(\theta)
$$
The gravitational potential energy is:
$$
U = m g y = -m g \left(a \cos(\omega t) + l \cos(\theta)\right)
$$
The Lagrangian ## L## is the difference between the kinetic and potential energies:
$$
L = T - U
$$
No problem, but to derive the equation of motion, I have to apply the Euler-Lagrange equation:
$$
\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{\theta}} \right) - \frac{\partial L}{\partial \theta} = ???
$$
I don't know what I should put in place of the ??? I assume it is not zero as there is a driving force. How can I calculate that driving force?
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