There are a couple of challenges in connecting the engine to the wheels.

1. Gearing is required.  This is similar to a bicycle's gearing.  The wheels must spin much slower than the engine.  Interesting math can be used to calculate how fast your kart will go based on a the wheel diameter, sproket size ratios, and engine speed.
2. You need a clutch which permits the engine to run without spinning the wheels.  This permits starting the engine.  Without a clutch you couldn't start the engine with the go-kart standing still and the wheels on the ground.

 The cylinder on the axle of this engine is a centrifugal clutch.  The axle can spin freely up to about 600 RPM.  Above that, weights push out against the cylinder and the chain sprocket begins to spin. These are the simplest and least expensive form of clutch.  If you're going to purchase one part, this is probably the one to purchase.  Make sure that it fits the axle of your engine and the sprocket's teeth spacing matches your chain. They're about \$30. This clutch is from a lawn mower.  A belt drives a shaft which turns the chain sprocket.  In this case the shaft moves forward and backward to loosen and tighten the belt.  The white pipe going diagonally in front of the engine is dropped by the driver to engage the engine.This particular clutch tended to engage suddenly, with a low gear ratio, which resulted in much smoke and tire wear. No Photo You can make a clutch similar to the above with an axle that doesn't move.  Instead an idler pulley tightens the belt when the operator lifts on a lever.  A big advantage of this is that if one of the pulleys is a cone pulley then it is feasible to have gears.  A cone pulley is a pulley with three to five grooves, each with a different circumference.   The driver wiggles the idler until the belt is in the smallest groove.  This is first gear.  When the go-kart is up to about 10 mph the driver eases the idler and wiggles it until the belt slips onto the second groove.  The idler pulley can be simply a belt pulley spinning on a bolt, though bearings are preferable. No Photo The world's simplest clutch is to not have one.  The wheel is directly tied to the engine with a chain.  The rear axle of the go-kart is put up on blocks while it is started.  Once it's running you shove it off the blocks.  While not ideal this can work, particularly if headed down hill. It's often hard to keep the chain tight.  A relatively easy solution to this is to mount the engine on a platform which is bolted on only one side of the go-kart so that it can move backward and forward a few inches on the chain side.  Then a piece of wire such as coat hanger from the front of the go-kart to the engine mount and back.  A stick between the two pieces of wire is twisted around to tighten the wire and pull the engine mount forward.  This is a technique cowboys use to tighten fences. On this go-kart we had continual problems with the chain falling off.  When it accelerated the front portion of the chain vibrated and fell off.  We tried using various idler pulleys including the derailleur from a bicycle.  The boys came up with this final solution, a  simple piece of plastic tubing that reduces chain vibration.

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