Consider a two$-$ramp system shown in Fig. $1.$ The left and right ramps make an angle of $\backslash$theta $\_(1)$

and $\backslash$theta $\_(2)$ with the horizontal plane, respectively. At the top where the two ramps meet is a

massless pulley that is free to rotate. On the left ramp, a cart is supported by a spring and

the length of the ramp is l$.$ The cart has mass m$\_(1)$ and slides on the left ramp without friction.

In addition, the position of the cart from the pulley is x$\_(1).$ The spring is linear with stiffness

k and has a free length l$.$ Also, the cart is subjected to a horizontal force P$.$ On the right

ramp, a cylinder of radius r rolls without slipping. The cylinder is uniform and has mass m$\_(2)$;

therefore, its centroidal moment of inertia is $(1)/(2)$mr$^(2).$ The motion of the cylinder is described

via its center position x$\_(2)$ from the pulley as well as an angle of rotation $\backslash$phi in the clockwise

sense. When the spring is at its free length, the corresponding $\backslash$phi $=0.$ An inextensible string

of constant length d$,$ wrapping around the pulley, connects a cart on the left ramp and a

cylinder on the right ramp. Answer the following questions.

$($a$)$ Derive constraint equations of the two$-$ramp systems. Are the constraints rheonomic or

scleronomic? How do virtual displacements $\backslash$delta x$\_(1),\backslash$delta x$\_(2),$ and $\backslash$delta $\backslash$phi are related to one another?

$($b$)$ Apply the principle of virtual work to derive an equation that governs equilibrium of the

cart, the spring, the cylinder, and the force P$.$ Figure $1$: A two$-$ramp system