Two masses, $m_1$ and $m_2,$ are connected by a massless, inextensible rope over a frictionless pulley. Mass

$m_1$ is on a frictionless horizontal surface, while mass $m_2$ hangs vertically.

Determine the acceleration of the system.

Calculate the tension in the rope.

Problem Statement $2$:

A roller coaster car with $m=500kg$ is moving along a track defined by the following equations in the plane:

$x(s)=T*cos(\frac{s}{L})$;$y(s)=R*sin(\frac{s}{L})$

where $s$ is the arc length measured from the top of the loop, with radius of $20$ m is the radius of the loop,

and $L=40m$ is the total length of the loop.

Determine the curvature kkappak of the track at any points.

Calculate: $a_n$ and $a_t$ components of acceleration when the car is moving at a speed $v=15\frac{m}{s}.$

If the car needs to maintain a minimum speed $v_{\text{m}\text{i}\text{n}}$ at the top of the loop to ensure it remains on the

track $($i$.$e$.,$ the normal force is zero$),$ determine $V_{\text{m}\text{i}\text{n}\text{.}}.$

Problem Statement $3$:

Consider a double pendulum system consisting of two masses, $m_1$ and $m_2,$ each attached to the end of

massless rigid rods of lengths $L_1$ and $L_2,$ respectively. The first rod is fixed at one end to a pivot point,

allowing it to swing freely in a vertical plane without friction. The second rod is connected to the mass $m_1$

and can also swing freely in the same vertical plane.

Determine the number of degrees of freedom for the double pendulum system.

Identify the independent coordinates required to fully describe the position of both masses.

Describe any constraints present in the system.

Explain how these constraints affect the degrees of freedom of the system.

Using Newton's Third Law, explain the internal forces acting between the two masses and how they

influence the system's overall motion.