Question

A child swings a yo-yo of weight mg in a horizontal circle so that the cord makes an angle of 30 degrees with the vertical. Find the centripetal acceleration.

Answer 1

You did not say what the mass M is. As turns out, you don't need to know M to get the answer. First convert the mass to weight, W.
W = Mg
The vertical component of string tension is T cos30 = Mg
The horizontal component of string tension is the centripetal force
T sin30 = Ma,
where a is the centripetal acceleration

Divide one by the other and you get
sin30/cos30 = tan30 = sqrt3 = a/g

Solve for a.

Answer 2

Ok, the radius of rotation is length*sin30 where length is the length of string.

looking at vectors, mg is down, and mv^2/r is outward, so tan30=mv^2/(r*mg)

tan 30= v^2/(length*sin30*g)

but v= 2PI(length*sin30) square that, put it in the equation

tan30=4PI^2*length*sin30/g check that. solve for length of string.

finally, centacc= v^2/r and you can do that algebra.