Question

A. A point mass m moves frictionlessly on a horizontal plane. An unusual, anharmonic spring with unstretched length r0 is attached between a pivot at the origin and the mass. Let the radial force exerted by the spring be given by Fr = -c(r - r0)^3 where c is a positive constant. Using plane polar coordinates r and ?, write down the Lagrangian L(r, ?, ?, ??) and use Lagrange’s method to find the equations of motion for the mass on the plane. B. Reanswer the question above using the x,y coordinate system. (i.e. consider L(x, y, ?, ?)) C. Now suppose the x-y plane on which the mass m moves in the question above is tilted at a fixed angle ? with respect to the horizontal, so that the height h at a position (x, y) on the plane is given by h = y sin ?. Reanswer the question (B.) above for this case.

          A. A point mass m moves frictionlessly on a horizontal plane. An unusual, anharmonic spring with unstretched length r0 is attached between a pivot at the origin and the mass. Let the radial force exerted by the spring be given by Fr = -c(r - r0)^3 where c is a positive constant. Using plane polar coordinates r and ?, write down the Lagrangian L(r, ?, ?, ??) and use Lagrange’s method to find the equations of motion for the mass on the plane.
B. Reanswer the question above using the x,y coordinate system. (i.e. consider L(x, y, ?, ?))
C. Now suppose the x-y plane on which the mass m moves in the question above is tilted at a fixed angle ? with respect to the horizontal, so that the height h at a position (x, y) on the plane is given by h = y sin ?. Reanswer the question (B.) above for this case.

A. A point mass m moves frictionlessly on a horizontal plane. An unusual, anharmonic spring with unstretched length r0 is attached between a pivot at the origin and the mass. Let the radial force exerted by the spring be given by Fr = -c(r - r0)^3 where c is a positive constant. Using plane polar coordinates r and ?, write down the Lagrangian L(r, ?, ?, ??) and use Lagrange’s method to find the equations of motion for the mass on the plane.
B. Reanswer the question above using the x,y coordinate system. (i.e. consider L(x, y, ?, ?))
C. Now suppose the x-y plane on which the mass m moves in the question above is tilted at a fixed angle ? with respect to the horizontal, so that the height h at a position (x, y) on the plane is given by h = y sin ?. Reanswer the question (B.) above for this case.

Added by Trinity B.

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