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under_par_00
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Homework Statement
A point mass m slides without friction on a horizontal table at one end of a massless spring of natural length a and spring constant k. The other end of the spring is attached to the table so that it can rotate freely without friction. The spring is driven by a motor beneath the table so that the spring and mass are constrained to move around the origin with angular frequency [tex]\omega[/tex] (ignore bending of the spring and assume it always remains radially outward from the origin).
a) Using Cartesian coordinates, write down expressions for the kinetic energy of the system.
b) Change to polar coordinates. Also give the expression for potential energy and the Lagrangian.
c) How many degrees of freedom do you have for this system? Name them(it).
d) Give the equation of motion for the mass
Homework Equations
Please note this is my first time with LaTeX, so it might be a little bumpy!
L = T - U
Spring Potential: [tex]U = \frac{1}{2} k \ast x^2[/tex]
The Attempt at a Solution
The kinetic energy is going to be given by the rotating mass plus the mass extending (or compressing) with the spring. Since a is the natural length of the spring, I let b be the distance it extends (or compresses).
[tex] T = \frac{1}{2} m[\dot{x}^2 + \dot{y}^2 + (b')^2] [/tex]
[tex] U = \frac{1}{2} k(r-a)^2 [/tex] <--Note the potential energy doesn't have to be in Cartesian.
Now change T from Cartesian to polar:
[tex]\dot{x} = -r \omega sin(\omega t)[/tex]
[tex]\dot{y} = r \omega cos(\omega t)[/tex]
[tex]T = \frac{1}{2} m (r^2 \omega^2 sin^2 (\omega t) + r^2 \omega^2 cos^2 (\omega t) + (r')^2)[/tex]
[tex]T = \frac{1}{2} m (r^2 \omega^2 + (r')^2)[/tex]
[tex]L = T - U = r^2 \omega^2 + (r')^2 - \frac{1}{2} k(r^2 -2ra + a^2)[/tex]
[tex]\frac{dL}{dr} = 2 \omega^2 r - (rk - ra) = 2\omega^2 r -rk +ra[/tex]
[tex]\frac{dL}{dr'} = 2r'[/tex]
[tex]\frac{d}{dt} \frac{dL}{dr'} = 2r''[/tex]
[tex]\frac{dL}{dr} - \frac{d}{dt} \frac{dL}{dr'} = 0[/tex]
[tex]2 \omega^2 r - rk + ra - 2r'' = 0
[/tex]
So the last line gives the equation of motion for the mass. There is one degree of freedom in the radial distance r.
I am unsure if I set the problem up correctly. Can I just say let b be the distance the spring stretches or compresses? And then eventually make that r' (since it is the rate of change of r)