Please help me with this problem, I have no background in this subject so I$\text{'}$d appreciate it if you can explain each step in detail. Thank you!

A mass M is connected to another mass m$.$ Mass M moves without

friction along a circle of radius r on the horizontal surface of a table

$($Figure $5.$ The two masses are connected by a massless string of length

l that passes through a hole in the table. At given time the mass M is

located by r and theta. Derive the equations of motion by Lagrange.

Consider a pendulum made of a spring with mass m on the end $($Figure $6).$

The spring is arranged to lie in a straight line $($which we can arrange by$,$

Figure $5$: A mass $\backslash ($ m $\backslash )$ attached to a string which is threaded through a table with another mass $\backslash ($ M $\backslash )$ attached at the other end

say, wrapping the spring around a rigid massless rod$).$ The equilibrium length of the spring is $\backslash ($ l $\backslash ).$ Let the spring has length $\backslash ($ l$+$x$($t$)\backslash ),$ and let its angle with the vertical be $\backslash (\backslash$theta$($t$)\backslash ).$ Assuming that the motion takes place in a vertical plane, find the equations of motion for $\backslash ($ x $\backslash )$ and $\backslash (\backslash$theta $\backslash )$ using the Lagrangian.

Figure $6$: A mass $\backslash ($ m $\backslash )$ attached to a pendulum made of a spring

$4.$ A cart and pendulum, shown in Figure $7,$ consists of a cart of mass, $\backslash ($ m$\_\{1\}\backslash ),$ moving on a horizontal surface, acted upon by a spring with spring constant $\backslash ($ k $\backslash ).$ From the cart is suspended a pendulum consisting of a uniform rod of length, $\backslash ($ l $\backslash ),$ and mass, $\backslash ($ m$\_\{2\}\backslash ),$ pivoting about point $\backslash ($ A $\backslash ).$ Derive the equations of motion by Lagrange.