Question

A bead is threaded onto a wire bent into a sinusoidal shape lying in the plane of this page/screen. The vertical coordinate along the wire is y and the shape of the wire is given by $x = A \sin ky$ where $A$ and $k$ are constants. The bead is subject to gravity (pointing down). Find the following using the method of Lagrange multipliers to enforce the constraints (we will find the constraint forces in the final step). Your answers may contain $m$, $g$, $k$, $A$ along with $x$, $y$ and their time derivatives. a) What is the Kinetic energy of the bead as a function of $x$, $y$ and their time derivatives? b) What is the potential energy of the bead as a function of $x$, $y$ and their time derivatives? c) What is the Lagrangian as a function of $x$, $y$ and their time derivatives? d) What are the equation(s) of constraint? e) Solve Lagrange's equations to find the equation of motion of the bead in the y coordinate and the force(s) of constraint in terms of $y$ and its time derivatives? f) What is the physical source of the force(s) of constraint?

          A bead is threaded onto a wire bent into a sinusoidal shape lying in the plane of this page/screen.
The vertical coordinate along the wire is y and the shape of the wire is given by $x = A \sin ky$
where $A$ and $k$ are constants. The bead is subject to gravity (pointing down).

Find the following using the method of Lagrange multipliers to enforce the constraints (we will
find the constraint forces in the final step). Your answers may contain $m$, $g$, $k$, $A$ along with $x$, $y$
and their time derivatives.
a) What is the Kinetic energy of the bead as a function of $x$, $y$ and their time derivatives?
b) What is the potential energy of the bead as a function of $x$, $y$ and their time derivatives?
c) What is the Lagrangian as a function of $x$, $y$ and their time derivatives?
d) What are the equation(s) of constraint?
e) Solve Lagrange's equations to find the equation of motion of the bead in the y coordinate
and the force(s) of constraint in terms of $y$ and its time derivatives?
f) What is the physical source of the force(s) of constraint?

A bead is threaded onto a wire bent into a sinusoidal shape lying in the plane of this page/screen.
The vertical coordinate along the wire is y and the shape of the wire is given by x = A sin ky
where A and k are constants. The bead is subject to gravity (pointing down).

Find the following using the method of Lagrange multipliers to enforce the constraints (we will
find the constraint forces in the final step). Your answers may contain m, g, k, A along with x, y
and their time derivatives.
a) What is the Kinetic energy of the bead as a function of x, y and their time derivatives?
b) What is the potential energy of the bead as a function of x, y and their time derivatives?
c) What is the Lagrangian as a function of x, y and their time derivatives?
d) What are the equation(s) of constraint?
e) Solve Lagrange's equations to find the equation of motion of the bead in the y coordinate
and the force(s) of constraint in terms of y and its time derivatives?
f) What is the physical source of the force(s) of constraint?

Added by Cody T.

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