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1. A wheel rolls down a flat inclined surface that makes an angle $\alpha$ with the horizontal. The wheel is constrained so that its plane is always perpendicular to the inclined plane, but it may rotate about the axis normal to the surface. Obtain the solution for two-dimensional motion of the wheel, using Lagrange's equations and the method of undetermined multipliers.
The kinetic energy has two components in a two dimensional motion:
\begin{equation}
T = T_{trans} + T_{rot} = \frac{m \dot x^2}{2} + \frac{m r^2 \dot \theta^2}{2} \label{eq:301}
\end{equation}
and the potential energy:
\begin{equation}
V = mg y = mg x \sin \alpha \label{eq:302}
\end{equation}
Therefore, the Lagrangian is given by:
\begin{equation}
L = \frac{m \dot x^2}{2} + \frac{m r^2 \dot \theta^2}{2} - -mg x \sin \alpha \label{eq:303}
\end{equation}
The Lagrange's equations and the method of undefined multipliers use the relation:
\begin{equation}
\frac{d}{dt}\left( \frac{\partial L}{\partial \dot q} \right) - \frac{\partial L}{\partial q} = \lambda_q \frac{\partial f}{\partial q} \label{eq:304
}
\end{equation}
Applying this in the Lagrangian :
\begin{equation} m \ddot x - mg \sin \alpha = \lambda_x \frac{\partial f}{\partial x} \end{equation}
\begin{equation} m r^2 \ddot \theta = \lambda_\theta \frac{\partial f}{\partial \theta} \end{equation}
where the constrain equation is $r \theta = x$ and therefore $f = r \theta - x = 0$. Then:
\begin{equation} m \ddot x - mg \sin \alpha = - \lambda \end{equation}
\begin{equation} m r^2 \ddot \theta = r \lambda \end{equation}
\begin{equation} m r \ddot \theta = \lambda \end{equation}
But, from the constrain we can found: $r \ddot \theta = \ddot x$. Then,
\begin{equation} \lambda = \frac{mg \sin \alpha}{2} \end{equation}
\begin{equation} \ddot x = \frac{mg \sin \alpha}{2} \end{equation}
\begin{equation} \ddot \theta = \frac{mg \sin \alpha}{2r} \end{equation}
The general solution for the equation of motion are:
\begin{equation} x(t) = \frac{1}{4} g t^2 \sin \alpha + At + B \end{equation}
\begin{equation} \theta(t) = \frac{1}{4r} g t^2 \sin \alpha + Ct + D \end{equation}
where $A,B,C,D$ are constants.
Apparently it is not the right answer. What am I doing wrong?
Homework Equations
The Attempt at a Solution
The kinetic energy has two components in a two dimensional motion:
\begin{equation}
T = T_{trans} + T_{rot} = \frac{m \dot x^2}{2} + \frac{m r^2 \dot \theta^2}{2} \label{eq:301}
\end{equation}
and the potential energy:
\begin{equation}
V = mg y = mg x \sin \alpha \label{eq:302}
\end{equation}
Therefore, the Lagrangian is given by:
\begin{equation}
L = \frac{m \dot x^2}{2} + \frac{m r^2 \dot \theta^2}{2} - -mg x \sin \alpha \label{eq:303}
\end{equation}
The Lagrange's equations and the method of undefined multipliers use the relation:
\begin{equation}
\frac{d}{dt}\left( \frac{\partial L}{\partial \dot q} \right) - \frac{\partial L}{\partial q} = \lambda_q \frac{\partial f}{\partial q} \label{eq:304
}
\end{equation}
Applying this in the Lagrangian :
\begin{equation} m \ddot x - mg \sin \alpha = \lambda_x \frac{\partial f}{\partial x} \end{equation}
\begin{equation} m r^2 \ddot \theta = \lambda_\theta \frac{\partial f}{\partial \theta} \end{equation}
where the constrain equation is $r \theta = x$ and therefore $f = r \theta - x = 0$. Then:
\begin{equation} m \ddot x - mg \sin \alpha = - \lambda \end{equation}
\begin{equation} m r^2 \ddot \theta = r \lambda \end{equation}
\begin{equation} m r \ddot \theta = \lambda \end{equation}
But, from the constrain we can found: $r \ddot \theta = \ddot x$. Then,
\begin{equation} \lambda = \frac{mg \sin \alpha}{2} \end{equation}
\begin{equation} \ddot x = \frac{mg \sin \alpha}{2} \end{equation}
\begin{equation} \ddot \theta = \frac{mg \sin \alpha}{2r} \end{equation}
The general solution for the equation of motion are:
\begin{equation} x(t) = \frac{1}{4} g t^2 \sin \alpha + At + B \end{equation}
\begin{equation} \theta(t) = \frac{1}{4r} g t^2 \sin \alpha + Ct + D \end{equation}
where $A,B,C,D$ are constants.
Apparently it is not the right answer. What am I doing wrong?
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