Question

Two equal masses, m, are joined by a massless string of length L that passes through a hole in a frictionless horizontal table. The first mass slides on the table while the second hangs below the table and moves up and down in a vertical line. (string length constraint) a. Assuming the string remains taut, write the Lagrangian for the system in terms of the polar coordinates $(r, \phi)$ of the mass on the table. (6 points) b. Find the two Lagrange equations and interpret the $\phi$ equation in terms of the angular momentum $l$ of the first mass. (6 points) c. Find the value $r_0$ at which the first mass is moving in a circular path (Hint: express $\dot{\phi}$ in terms of $l$ and eliminate it from the $r$ equation.) (6 points) d. Suppose the first mass is moving in this circular path and is given a small radial nudge. Use $r(t) = r_0 + \epsilon(t)$ to rewrite the $r$ equation in terms of $\epsilon(t)$ dropping all powers of $\epsilon(t)$ higher than linear. Show that the circular path is stable and that $r(t)$ oscillates sinusoidally about $r_0$. What is the frequency of its oscillations? (7 points)

          Two equal masses, m, are joined by a massless string of length L that passes through a hole in
a frictionless horizontal table. The first mass slides on the table while the second hangs below
the table and moves up and down in a vertical line. (string length constraint)
a. Assuming the string remains taut, write the Lagrangian for the system in terms of the
polar coordinates \((r, \phi)\) of the mass on the table. (6 points)
b. Find the two Lagrange equations and interpret the $\phi$ equation in terms of the angular
momentum $l$ of the first mass. (6 points)
c. Find the value $r_0$ at which the first mass is moving in a circular path (Hint: express $\dot{\phi}$ in
terms of $l$ and eliminate it from the $r$ equation.) (6 points)
d. Suppose the first mass is moving in this circular path and is given a small radial nudge.
Use $r(t) = r_0 + \epsilon(t)$ to rewrite the $r$ equation in terms of $\epsilon(t)$ dropping all powers of
$\epsilon(t)$ higher than linear. Show that the circular path is stable and that $r(t)$ oscillates
sinusoidally about $r_0$. What is the frequency of its oscillations? (7 points)

Two equal masses, m, are joined by a massless string of length L that passes through a hole in
a frictionless horizontal table. The first mass slides on the table while the second hangs below
the table and moves up and down in a vertical line. (string length constraint)
a. Assuming the string remains taut, write the Lagrangian for the system in terms of the
polar coordinates (r, ϕ) of the mass on the table. (6 points)
b. Find the two Lagrange equations and interpret the ϕ equation in terms of the angular
momentum l of the first mass. (6 points)
c. Find the value r0 at which the first mass is moving in a circular path (Hint: express ϕ̇ in
terms of l and eliminate it from the r equation.) (6 points)
d. Suppose the first mass is moving in this circular path and is given a small radial nudge.
Use r(t) = r0 + ϵ(t) to rewrite the r equation in terms of ϵ(t) dropping all powers of
ϵ(t) higher than linear. Show that the circular path is stable and that r(t) oscillates
sinusoidally about r0. What is the frequency of its oscillations? (7 points)

Added by Tony S.

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