A cylindrical object of outer diameter 20 cm height 20 cm and density 8000 `kgm^-3` is supported by a vertical spring and is half dipped in water as shown in figue. a. Figure the elongation of the spring in equilibrium condition. b. If the object is sllightlly depressed and relation, find the time period of resulting oscillations of the objcet. The spring constant `=500 Nm^-1`.

Answer

A, B, C

Explanation

`d=10cm`
`r=5cm`
`rarr h=20cm`
`rho_b=8000kg/m^3=8gm/cc`
`k=500N/m=500xxx10^3dyne/cm`
a. Here `F+U=mg [whee Fkx]`
`rarr kx+Vrho_ug=mg`
`rarr 500xx10^3xx(x)+(pir^2)xx(h/2)xx1xx1000`
`=pir^2xhxxrho_bxx1000
`rarr 500xx10^3xx(x)` brgt`=pir^2xxhxx1000(rho-1/2)`
`=pixx(5)^2xx20xx1000(rho_b-1/2)`
`rarr 50x=pixx25xx2xx(rho_b-1/2)`
`=x=pi(8-0.5)`
or `x=pixx7.5=23.5cm`
b. If Xrarr displacement of the block form the equilibrium position,
Driving force
`F=kX+Vrho_wxxg`
`rarr ma=kX+pir^2xx(X)xxrho_wxxg`
`=(k+pir^2xxrho_wxxg)X`
`rarr omega^2xx(X)=((k+pir^2+rho_wxxg))/m xx(X)`
`[because a=w^2XinSHM]`
`rarr T=2i m/(k+pir^2+rho_wxxg) `,
`=2pi(pixx25xx20xx8)/(500+10^3+pix25xx1xx1000)`
`=0.935s.