We have to assume that the objects roll down the inclined plane without slipping.
Apply conservation of mechanical energy
m*g*h = 0.5*m*v^2 + 0.5*I*(v/R)^2
where h is the height of the inclined plane, m is the mass of the object, v is the velocity of the center of mass of the object, I – moment of inertia about the center of mass and R – radius.
The moment of inertia can be expressed as I = β*m*R^2 , where β characterizes the geometry of the object.
β(solid sphere) = 0.4 ; β(solid disc) = 0.5 ; β(ring) = 1;
Substitute I into the equation for mechanical energy conservation
m*g*h = 0.5*m*v^2 + 0.5*β*m*R^2*(v/R)^2
Solve for v and we should get
v = √(2*g*h/(1+β))
It is important to mention that the final velocity is independent of the object’s mass and size, and the winner of a rolling race can be predicted by β.
Thereby, the order of finish is the solid sphere, the solid disc and the ring.
Image Source and demonstration: You Tube