Question

A metal block of density $6000kg{m}^{-3}$ and mass $1.2kg$ is suspended through a spring of spring constant $200N{m}^{-1}$. The spring-block system is dipped in water kept in a vessel. The water has a mass of $260g$ and the block is at a height $40cm$ above the bottom of the vessel. If the support to the spring is broken, what will be the rise in the temperature of the water? Specific heat capacity of the block is $250Jk{g}^{-1}{K}^{-1}$ and that of water is $4200Jk{g}^{-1}{K}^{-1}$. Heat capacities of the vessel and the spring are negligible.

• A.
• B.
• C.
• D.

Answer 1

The correct option is C

Initially, when the block is suspended from the spring inside water it is in mechanical equilibrium. Also, the water and block are in thermal equilibrium. The extension of the spring be $x$, and $m$ be the mass of the block, $\rho$ density.

$k$ is the spring constant, ${\rho }_w\to$ density of water

$V\to$ volume of block

Net force on the block along the vertical direction is zero, as block is at rest.

$\therefore kx=mg-{\rho }_w{v}_g$

$\Rightarrow x=\frac{mg(1-\frac{{\rho }_w}{\rho })}{k}$

$\Rightarrow kx=12g-\frac{1}{6}\times 1.2g=g$

$\Rightarrow x=\frac{g}{k}=\frac{10}{200}=\frac{1}{20}m$

Now, after the support breaks, the block falls and hits the floor, and comes to rest, the spring comes to its natural length, gravitational potential energy is lost too.

$\therefore$ M.E. lost = $\frac{1}{2}k{x}^2+m{g}_{apparent}Y$ ............(i)

[$g_{apparent}$ = $\frac{Gravitationalforce-Buoyancyforce}{m}$

$g_{apparent}$ = $\frac{Gravitationalforce-Buoyancyforce}{m}$ = $\frac{mg-\frac{{\rho }_w}{\rho }mg}{m}$ = $\frac{5g}{6}$]

Also, now the water and block have both risen to the equal temperature, higher by $\Delta T$

$h\to$ height block falls, $M\to$ mass of water; sblock, swater the specific heats

$\therefore$ Heat gained = ($m{s}_{block}$ + $M{s}_{water}$)$\Delta T$ . . . .(ii)

(i) & (ii) are equal

$\Rightarrow$ $DeltaT$ = ${0.003}^{\circ }C$

[Note: Don't think of this as some different concept, remember work - energy theorem problems with dissipative forces (like viscosity in this question) friction etc. The work done by viscous forces here is what appears as heat, and changes the temperature]