Q$2.($a$)(10$ points$)$ An object of mass $0\mathrm{.}5$ kg is suspended from a vertical massless spring whose natural length is $8\mathrm{.}0$ cm as shown in the diagram. When the object reaches equilibrium, the length of the deformed spring is $20\mathrm{.}0$ cm $.$

$($a$).$ Draw a free$-$body diagram and find the spring constant

$($b$)(10$ points$)$ Now spring$-$mass system is placed in the beaker full of water as shown in the diagram. When the spring$-$mass system reaches equilibrium, the length of the deformed spring is $12\mathrm{.}8$ cm $.$ The volume of an object is $3\mathrm{.}0{10}^{-4}{m}^3$ and the density of the water is $1000$ kg$/$m ${?}^3.$

$(1)$ Draw a free$-$body diagram and find the mass of an object in water.

$(2)$ Find the spring constant and compare it with the one calculated in part $($a$).$