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Forces on the tire if it's just sitting there:

1)weight of the tire and the kid pulling down

2)support of the tree branch where it's tied on pulling the rope up

If the rope is in equilibrium, these forces will cancel.

Now if the swing is actually swinging, it's a bit more complicated. Suppose the tire is displaced by an angle theta and swinging with an angular speed omega.

Now the component of gravity along the line of the rope is:

Fg = m g cos(theta)

And you add to that the centripetal force required to keep the tire moving in a circle:

Fc = m omega^2 r

Add those up to get the tension in the rope, which will be the support required by the tree branch (neglecting drag and the weight of the rope itself):

Ftotal = m (omega^2 r + g cos(theta) ),

where m is the total weight of the kid and tire.

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Q: Describe the forces on a rope that support a tire swing when a child sits on the tire?
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