A small sphere of radius $r$ falls from rest in a viscous liquid. As a result, heat is produced due to viscous force. The rate of production of heat when the sphere attains its terminal velocity, is proportional to
NEET 2018, Diffcult
Download our app for free and get started
The viscous drag force, $F = 6\pi \eta rv;$
$where\,v = terminal\,velocity$
$\therefore \,The\,rate\,of\,production\,of\,heat = power$
$ = force \times terminal\,velocity$
$ \Rightarrow power = 6\pi \eta rv \cdot v = 6\pi \eta r{v^2}\,\,\,\,\,\,\,\,...\left( i \right)$
$Terminal\,velocity\,v = \frac{{2{r^2}\left( {\rho - \sigma } \right)}}{{9\eta }};\,\,\,\therefore v \propto {r^2}$
$Now,\,power = 6\pi \eta r\left[ {\frac{{4{r^2}{{\left( {\rho - \sigma } \right)}^2}}}{{81{\eta ^2}}}{g^2}} \right]or\,power \propto {r^5}.$
Download our appand get started for free
Experience the future of education. Simply download our apps or reach out to us for more information. Let's shape the future of learning together!No signup needed.*
Similar Questions
- 1Water falls down a $500.0 \,m$ shaft to reach a turbine which generates electricity. ................ $m^3$ water must fall per second in order to generate $1.00 \times 10^9 \,W$ of power ? (Assume $50 \%$ efficiency of conversion and $\left.g=10 \,ms ^{-2}\right)$View Solution
- 2View SolutionAir is blown through a hole on a closed pipe containing liquid. Then the pressure will
- 3A $U-$ tube in which the cross-sectional area of the limb on the left is one quarter, the limb on the right contains mercury (density $13.6 g/cm^3$). The level of mercury in the narrow limb is at a distance of $36 cm$ from the upper end of the tube. What will be the rise in the level of mercury in the right limb if the left limb is filled to the top with water ........ $cm$View Solution
- 4A ball of density $'\rho '$ is released from deep inside of a viscous liquid of density $'2\rho '$ . It will move upView Solution
- 5If water is flowing in a pipe with speed $2 \,m / s$ then its kinetic energy per unit volume is ........... $J ^2 m ^3$View Solution
- 6A tank $5\, m$ high is half-filled with water and then is filled to the top with oil of density $0.85\, g/cm^3$. The pressure at the bottom of the tank, due to these liquids is ....... $g\, dyne/cm^2$View Solution
- 7A tank with a small hole at the bottom has been filled with water and kerosene (specific gravity $0.8$). The height of water is $3\,m$ and that of kerosene $2\,m$. When the hole is opened the velocity of fluid coming out from it is nearly ........ $ms^{-1}$ .(take $g\, = 10\, m s^{-2}$ and density of water $= 10^3\, kg\, m^{-3}$)View Solution
- 8The reading of spring balance when a block is suspended from it in air, is $60\,N$. This reading is changed to $40\, N$ when the block is immersed in water. The specific density of the block isView Solution
- 9A uniform solid cylinder of density $0.8$ $g/cm^3$ floats in equilibrium in a combination of two non-mixing liquid $A$ and $B$ with its axis vertical. The densities of liquid $A$ and $B$ are $0.7$ $g/cm^3$ and $1.2$ $gm/cm^3$. The height of liquid $A$ is $h_A = 1.2$ $cm$ and the length of the part of cylinder immersed in liquid $B$ is $h_B = 0.8$ $cm$. Then the length part of the cylinder in air is ....... $cm$View Solution
- 10View SolutionIn a laminar flow the velocity of the liquid in contact with the walls of the tube is