No but it should torsion/ swivel
>Either a front or rear sway bar shouldn't have "play" in it. Not any noticeable play anyway. If you have it up on jack stands it may have additional play but nothing extreme. Check that the rubber endlinks or bushings are present and in working order.
there should be a rear sway bar
No, it should be exactly the same size.
The links go on the end on the sway bar and hook to the suspension. Bushings are used to connect the sway bar to cars frame.
sway bar
A sway bar is a torsion bar mounted laterally on a suspension and is designed to reduce side sway. The sway bar attempts to accomplish this by transferring movement from one side of the suspension to the other. This is done through the connection of a torsion bar to each side of an axle by flexible links. (Sway bar links)
A little more information. Year/model. Some sway bar links become loose because of wear and the links continue to work but make a noise when the front end hits a bump. Other sway bar links will break altogether and the vehicle will lean excessivly when going around a corner.
The sway bar is not adjustable, but you can replace the bushings.
A bad sway bar will cause vehicles to shift from side to side more easily when turning. The sway bar provides stability and limits motion.
If by "disconnected" you mean unbolted, and if by "fix" you mean re-bolting, then it's not hard. It should consist of an endlink, which is typically just a bolt that connects the end of the sway bar to the lower control arm. Endlink bushings should sit on either side of the sway bar and on either side of the lower control arm. If you look at the way the sway bar is connected on the driver's side, it should be easy to figure out.
sway bar end linksway bar end link
Broken sway bar or sway bar links.
sway bar must be disconnected so you can move it out of the way