Question $1$

Figure $1$ shows a reciprocating cutting tool driven by a parallelogram four$-$bar chain whose crank rotates in the CCW direction at a speed of $220$ RPM$.$ The tool is connected to the coupler via a prismatic pair and its motion is opposed by a $1000$ N constant resistance applied only in the range: $\backslash (\backslash$theta $2=\backslash$pi $\backslash )$ to $\backslash (2\backslash$pi $\backslash ).$

I. Calculate the number of degrees of freedom of the mechanism.

II$.$ Explain the degree of freedom answer above.

III. Derive expressions for the velocity and acceleration of the tool as functions of $\backslash (\backslash$theta $2\backslash )$ and the flywheel's constant angular velocity, $\backslash (\backslash$omega $\backslash ).$

IV$.$ Find the rated torque of the motor needed to drive the system.

V$.$ Find also the mass moment of inertia of the flywheel if the fluctuation of its speed is not to exceed $\backslash (\backslash$pm $3\backslash \%\backslash )$ of a given average value.