I'll make the assumption that:
- The mass of each object is #"M"#
- The radius of each is #"R"#
This means that #"I"_S=2/5"MR"^2# is the moment of inertia of the sphere about its center, and that
#"I"_D = 1/2"MR"^2# is the moment of inertia of the disc about its center.
That said,
We want to conserve the mechanical energies of each object, by saying:
The Gain in rotational #"KE = "# the Loss in #"PE"#
For the Sphere,
#=> 1/2"I"_Somega^2= "Mgh"#
#omega= "v"/"R"#
and #"I"_S= 2/5"MR"^2#
#=>1/2xx2/5"MR"^2xx("v"_1)^2/"R"^2= "Mgh"#
#=> "v"_1= sqrt(5"gh")#
For the Disc,
#=> 1/2"I"_Domega^2= "Mgh"#
#omega= "v"/"R"#
and #"I"_D= 1/2"MR"^2#
#=>1/2xx1/2"MR"^2xx("v"_2)^2/"R"^2= "Mgh"#
#=> "v"_2= sqrt(4"gh")#
Now what we wanted,
#v_1/v_2= sqrt(5"gh")/sqrt(4"gh")= sqrt(5)/2#
