A cylindrical object of outer diameter 10 cm, height 20 cm and density 8000 `kgm^-3` is supported by a vertical spring and is half dipped in water as shown in figure. a. Find the elongation of the spring in equilibrium condition. b. If the object is slightly depressed and released, find the time period of resulting oscillations of the object. 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.