Problem $1.$ Laplace Transforms

$(20$ points$)$ The diagram shows an inverted pendulum on a cart that is driven by an external

force $F.$ Assume that the mass of the pendulum is concentrated at the ball $($the rod with

length $l$ is massless$).$ Let $m$ denote the mass of the pendulum and $M$ denote the mass of the

cart. Let $x$ denote the position of the cart in an inertial reference frame. Let $$ denote the

angle between the rod and the vertical line $($as shown in the diagram$).$ Using $F$ as the input

of the system, the linearized equations of motion are the following:

$x^{}=\frac{mg}{M}+\frac{F}{M}$

${}^{}=\frac{M+m}{Ml}g+\frac{F}{Ml}$

Derive the transfer function $H(s)=\frac{x(s)}{F(s)},$ where $x(s)$ and $F(s)$ are the Laplace domain

representations of the position of the cart $x(t)$ and the external force $F(t),$ respectively.

Note: You do not need to derive the equations of motion or the $($above$)$ linearized equations

of motion. You only need to derive the transfer function based on the $($above$)$ linearized

equations of motion.