Section $4\mathrm{.}1$

$4\mathrm{.}1.($a$)$ A pendulum with a "gravity spring" is shown in Figure P$4\mathrm{.}1.$ Derive the relation between the restoring torque $\backslash ($ T $\backslash )$ due to gravity and the angle $\backslash (\backslash$theta $\backslash ).$

$($a$)$

Figure P$4\mathrm{.}1$

$($b$)$ A cylinder with a "buoyancy spring is shown in Figure P$4\mathrm{.}1$b$."$ Derive the relation between the restoring force $\backslash ($ f $\backslash )$ due to buoyancy and the displacement $\backslash ($ x $\backslash )$ from the rest position. The liquid's mass density is $\backslash (\backslash$rho $\backslash ).($Hint: Use Archimedes' principle: the buoyancy force equals the weight of displaced liquid.$)$

$($c$)$ A wire of length $\backslash (2$ L $\backslash )$ has an an initial tension $\backslash ($ T $\backslash ),$ as shown in Figure P$4\mathrm{.}1$c$.$ The mass $\backslash ($ m $\backslash )$ in the middle moves a distance $\backslash ($ x $\backslash )$ horizontally. Derive the relation between the horizontal restoring force $\backslash ($ f $\backslash )$ and the displacement $\backslash ($ x $\backslash ).$

$4\mathrm{.}2.$ For each case, plot the force exerted on the mass by the springs as a function of the displacement $\backslash ($ x $\backslash )($Figure P$4\mathrm{.}2).$ Neglect friction.