An aluminium sphere is dipped into water.
Question: An aluminium sphere is dipped into water. Which of the following is true? (a) buoyancy will be less in water at 0oC than that in water at 4oC (b) buoyancy will be more in the water at 0oC than that in water at 4oC (c) buoyancy in water at 0oC will be same as that in the water at 4oC (d) buoyancy may be more or less in water at 4oC depending on the radius of the sphere Solution: The correct answer is (a) buoyancy will be less in water at 0oC than that in water at 4oC...
Read More →A uniform metallic rod rotates about
Question: A uniform metallic rod rotates about its perpendicular bisector with constant angular speed. If it is heated uniformly to raise its temperature slightly (a) its speed of rotation increases (b) its speed of rotation decreases (c) its speed of rotation remains the same (d) its speed increases because its moment of inertia increases Solution: The correct answer is (b) its speed of rotation decreases...
Read More →Find the points on the curve
Question: Find the points on the curve $x^{2}+y^{2}=13$, the tangent at each one of which is parallel to the line $2 x+3 y=7$. Solution: Let $\left(x_{1}, y_{1}\right)$ represent the required point. The slope of line $2 x+3 y=7$ is $\frac{-2}{3}$. Since, the point lies on the curve. Hence, $x_{1}^{2}+y_{1}^{2}=13 \quad \cdots$ (1) Now, $x^{2}+y^{2}=13$ On differentiating both sides w.r.t. $x$, we get $2 x+2 y \frac{d y}{d x}=0$ $\Rightarrow \frac{d y}{d x}=\frac{-x}{y}$ Slope of the tangent at $...
Read More →Find the point on the curve
Question: Find the point on the curve $y=3 x^{2}+4$ at which the tangent is perpendicular to the line whose slop is $-\frac{1}{6}$. Solution: Let $\left(x_{1}, y_{1}\right)$ be the required point. Slope of the given line $=\frac{-1}{6}$ $\therefore$ Slope of the line perpendicular to it $=6$ Since, the point lies on the curve. Hence, $y_{1}=3 x_{1}^{2}+4$ Now, $y=3 x^{2}+4$ $\therefore \frac{d y}{d x}=6 x$ Now, Slope of the tangent at $\left(x_{1}, y_{1}\right)=\left(\frac{d y}{d x}\right)_{\lef...
Read More →Surface tension is exhibited by liquids
Question: Surface tension is exhibited by liquids due to the force of attraction between molecules of the liquid. The surface tension decreases with increase in temperature and vanishes at boiling point. Give that the latent heat of vaporisation for water Lv = 540 k cal/kg, the mechanical equivalent of heat J = 4.2 J/cal, density of water w= 103 kg/l, Avogadros number NA = 6.0 1026 k/mole, and the molecular weight of water MA = 18 kg for 1 k mole. (a) Estimate the energy required for one molecul...
Read More →(a) Pressure decreases as one ascends the atmosphere.
Question: (a) Pressure decreases as one ascends the atmosphere. If the density of air is , what is the change in pressure dp over a differential height dh? (b) Considering the pressure p to be proportional to the density, find the pressure p at a height h if the pressure on the surface of the earth is o. (c) If po= 1.03 105N/m2, o= 1.29 kg/m3and g is 9.8 m/s2at what height will the pressure drop to (1/10) the value at the surface of the earth? (d) This model of the atmosphere works for relativel...
Read More →At what points on the curve
Question: At what points on the curve $y=2 x^{2}-x+1$ is the tangent parallel to the line $y=3 x+4 ?$ Solution: Let $\left(x_{1}, y_{1}\right)$ be the required point. The slope of line $y=3 x+4$ is 3 . Since, the point lies on the curve. Hence, $y_{1}=2 x_{1}^{2}-x_{1}+1$ Now, $y=2 x^{2}-x+1$ $\frac{d y}{d x}=4 x-1$ Now, Slope of the tangent at $\left(x_{1}, y_{1}\right)=\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}=4 x_{1}-1$ Slope of the tangent at $\left(x_{1}, y_{1}\right)=$ Slope...
Read More →The surface tension and vapour pressure
Question: The surface tension and vapour pressure of water at 20oC is 7.28 10-2N/m and 2.33 103Pa, respectively. What is the radius of the smallest spherical water droplet which can form without evaporating at 20oC? Solution: Surface tension of water, T = 7.28 10-2N/m Vapour pressure, P = 2.33 103Pa Radius of drop = r 2T/r is the excess pressure which is greater than the vapour pressure. Vapour pressure = excess pressure in drop Using the above relation, we can calculate r = 6.25 10-5m...
Read More →If a drop of liquid breaks into smaller droplets,
Question: If a drop of liquid breaks into smaller droplets, it results in lowering of the temperature of the droplets. Let a drop of radius R, break into N small droplets each of radius r. Estimate the drop in temperature. Solution: Volume of the liquid of drop of radius R = (N)(volume of liquid droplet of radius r) $\frac{4}{3} \pi R^{3}=N \times \frac{4}{3} \pi r^{3}$ $N=\frac{R^{3}}{r^{3}}$ Energy released is: $\Delta U=T \times \Delta A=T\left[4 \pi R^{2}-N\left(4 \pi r^{2}\right)\right]=4 \...
Read More →The sap in trees,
Question: The sap in trees, which consists mainly of water in summer, rises in a system of capillaries of radius r = 2.5 10-5m. The surface tension of sap is T = 7.28 10-2N/m and the angle of contact is 0o. Does surface tension alone account for the supply of water to the top of all tress? Solution: Radius, r = 2.5 10-5m Surface tension, T = 7.28 10-2N/m Angle of contact, = 0o Density, = 103 kg/m3 The maximum height h is given as: $h=\frac{2 S \cos \theta}{r \rho g}$ Substituting the values we g...
Read More →Find the points on the curve
Question: Find the points on the curve $y=3 x^{2}-9 x+8$ at which the tangents are equally inclined with the axes. Solution: Let $\left(x_{1}, y_{1}\right)$ be the required point. It is given that the tangent at this point is equally inclined to the axes. It means that the angle made by the tangent with the $x$-axis is $\pm 45^{\circ}$. $\therefore$ Slope of the tangent $=\tan (\pm 45)=\pm 1$...(1) Since, the point lies on the curve. Hence, $y_{1}=3 x_{1}^{2}-9 x_{1}+8$ Now, $y=3 x^{2}-9 x+8$ $\...
Read More →An iceberg floats in water with part
Question: An iceberg floats in water with part of it submerged. What is the fraction of the volume of an iceberg submerged if the density of ice is i= 0.917 g/cm3? Solution: Density of ice, ice= 0.917 g/cm3 Density of water, water= 1 g/cm3 Viis the volume of the iceberg Vwis the volume of the water displaced by the iceberg Weight of iceberg, W = iVig Upthrust, FB= wVwg At equilibrium, weight of iceberg = Vw/Vi= 0.917...
Read More →Is surface tension a vector?
Question: Is surface tension a vector? Solution: Surface tension is a scalar quantity as it has a specific direction....
Read More →Is viscosity a vector?
Question: Is viscosity a vector? Solution: Viscosity is not a vector quantity. It is a scalar quantity and it is a property of liquid with no direction....
Read More →Streamline flow is more likely for liquids with
Question: Streamline flow is more likely for liquids with (a) high density (b) high viscosity (c) low density (d) low viscosity Solution: The correct answer is (b) high viscosity (c) low density...
Read More →With increase in temperature, the viscosity of
Question: With increase in temperature, the viscosity of (a) gases decreases (b) liquids increases (c) gases increases (d) liquids decreases Solution: The correct answer is (c) gases increases (d) liquids decreases...
Read More →Pressure is a scalar quantity because
Question: Pressure is a scalar quantity because (a) it is the ratio of force to the area and both force and area are vectors (b) it is the ratio of the magnitude of the force to area (c) it is the ratio of a component of the force normal to the area (d) it does not depend on the size of the area chosen Solution: The correct answer is (b) it is the ratio of the magnitude of the force to area (c) it is the ratio of a component of the force normal to the area...
Read More →At what point of the curve
Question: At what point of the curve $y=x^{2}$ does the tangent make an angle of $45^{\circ}$ with the $x$-axis? Solution: Let the required point be (x1, y1).The tangent makes an angle of 45owith thex-axis. Slope of the tangent = tan 45o= 1 Since, the point lies on the curve. Hence, $y_{1}^{2}=x_{1}$ Now, $y^{2}=x$ $\Rightarrow 2 y \frac{d y}{d x}=1$ $\Rightarrow \frac{d y}{d x}=\frac{1}{2 y}$ Slope of the tangent $=\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}=\frac{1}{2 y_{1}}$ Give...
Read More →Solve this
Question: If $\tan \theta=\frac{a}{b}$, prove that $a \sin 2 \theta+b \cos 2 \theta=b$ Solution: Given: $\theta=\frac{a}{b}$ To Prove: a sin 2 + b cos 2 = b Given: $\theta=\frac{a}{b}$ We know that $\tan \theta=\frac{\text { Perpendicular }}{\text { Base }}=\frac{\mathrm{a}}{\mathrm{b}}$ By Pythagoras Theorem, (Perpendicular) $^{2}+(\text { Base })^{2}=(\text { Hypotenuse })^{2}$ $\Rightarrow(a)^{2}+(b)^{2}=(H)^{2}$ $\Rightarrow a^{2}+b^{2}=(H)^{2}$ $\Rightarrow H=\sqrt{a^{2}+b^{2}}$ So, $\sin \...
Read More →At what point of the curve
Question: At what point of the curve $y=x^{2}$ does the tangent make an angle of $45^{\circ}$ with the $x$-axis? Solution: Let the required point be (x1, y1).The tangent makes an angle of 45owith thex-axis. Slope of the tangent = tan 45o= 1 Since, the point lies on the curve. Hence, $y_{1}^{2}=x_{1}$ Now, $y^{2}=x$ $\Rightarrow 2 y \frac{d y}{d x}=1$ $\Rightarrow \frac{d y}{d x}=\frac{1}{2 y}$ Slope of the tangent $=\left(\frac{d y}{d x}\right)_{\left(x_{1}, y_{1}\right)}=\frac{1}{2 y_{1}}$ Give...
Read More →For a surface molecule
Question: For a surface molecule (a) the net force on it is zero (b) there is a net downward force (c) the potential energy is less than that of a molecule inside (d) the potential energy is more than that of a molecule inside Solution: The correct answer is (b) there is a net downward force (d) the potential energy is more than that of a molecule inside...
Read More →The angle of contact at the interface
Question: The angle of contact at the interface of water-glass is 0o, ethyl alcohol-glass is 0o, mercury-glass is 140o, and methyl iodide-glass is 30o. A glass capillary is put in a trough containing one of these four liquids. It is observed that the meniscus is convex. The liquid in the trough is (a) water (b) ethyl alcohol (c) mercury (d) methyl iodide Solution: The correct answer (c) mercury...
Read More →An ideal fluid flows through a pipe of circular
Question: An ideal fluid flows through a pipe of circular cross-section made of two sections with diameters 2.5 cm and 3.75 cm. The ratio of the velocities in the two pipes is (a) 9:4 (b) 3:2 (c) 3: 2 (d) 2: 3 Solution: The correct answer (a) 9:4...
Read More →Along a streamline
Question: Along a streamline (a) the velocity of a fluid particle remains constant (b) the velocity of all fluid particles crossing a given position is constant (c) the velocity of all fluid particles at a given instant is constant (d) the speed of a fluid particle remains constant Solution: The correct answer is (b) the velocity of all particles crossing a given position is constant...
Read More →At what points on the circle
Question: At what points on the circle $x^{2}+y^{2}-2 x-4 y+1=0$, the tangent is parallel to $x$-axis? Solution: Let the required point be (x1,y1).We know that the slope of thex-axis is 0.Given: $x^{2}+y^{2}-2 x-4 y+1=0$ $\left(x_{1}, y_{1}\right)$ lies on a curve. $\therefore x_{1}^{2}+y_{1}^{2}-2 x_{1}-4 y_{1}+1=0$ ....(1) Now, $x^{2}+y^{2}-2 \mathrm{x}-4 y+1=0$ $\Rightarrow 2 x+2 y \frac{d y}{d x}-2-4 \frac{d y}{d x}=0$ $\Rightarrow \frac{d y}{d x}(2 y-4)=2-2 x$ $\Rightarrow \frac{d y}{d x}=\...
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