Question A portion of a quick$-$return mechanism for a slotting machine is shown in the

figure below. The crank $($link $2)$ rotates CCW about the fixed point O$1$ at a constant

angular velocity of $300$ rpm$.$ Through the slider P $($link $3),$ it drives the tooth sector gear

$($link $4),$ which rotates about the fixed point O$2.$ Link $4$ in turn drives the rack $($link $5).$

The gear sector $($link $4)$ has a mass of $3$ kg$,$ with its center of mass being located at O$2,$

and its radius of gyration about O$2$ is $100$ mm$.$ The mass of the crank and that of the

slider are negligible. Ignore the friction force on the slider.

$(1)$ Determine the velocity of point A on the gear sector $($link $4)$ at the instant shown.

$(2)$ Determine the tangential acceleration of point A on the gear sector $($link $4)$ at the

instant shown.

$(3)$ If the force at A from the rack on the gear sector is $1500$ N opposite to the velocity of

A$,$ determine the torque that must be applied to the crank at the instant to maintain

this motion.