You have a pumpkin of mass m and radius r. the pumpkin has the shape of a sphere, but it is not uniform inside so you do not know its moment of inertia. in order to determine the moment of inertia, you decide to roll the pumpkin down an incline that makes an angle θ with the horizontal. the pumpkin starts from rest and rolls without slipping. when it has descended a vertical height h, it has acquired a speed of v = √(5gh/4).

As it is descended from a vertical height h, The lost Potential Energy = Mgh The gained Kenetic Energy = (1/2)Mv^2; The rotational KE = (1/2)Jw^2 The angular speed w = speed/ Radius = v/R So Rotational KE = (1/2)Jw^2 = (1/2)J(v/R)^2; J is moment of inertia Now Mgh = (1/2)Mv^2 + (1/2)J(v/R)^2 => 2gh/v^2 = 1 + (J/MR^2) As v = (5gh/4)^1/2, (J/MR^2) = 2gh/v^2 - 1 => (J/MR^2) = (8gh/5gh) - 1 so (J/MR^2) = 3/5 and therefore J = (3/5)MR^2.