JEE Physics Mechanical Properties Of Fluids MCQs Set 37

Answer: (a) 300 m

Explanation:
1. \( \Delta p=\rho g(d_2-d_1) \).
2. \( d_2-d_1=\dfrac{8.08\times10^6-5.05\times10^6}{10^3\times10}=303\approx300 \) m.

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (b) 0.6

Explanation:
1. Density of the block (relative to water) \( =\dfrac45=0.8 \).
2. With half in water and half in oil: \( 0.8=0.5(1)+0.5\rho_{oil}\Rightarrow\rho_{oil}=0.6 \).

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (c) \( \dfrac{5}{4} \) d

Explanation:
1. Equilibrium of weight and buoyancy: \( D\,L=d\cdot\dfrac{3L}4+2d\cdot\dfrac L4 \).
2. \( D=\dfrac{3d}4+\dfrac{d}2=\dfrac{5d}4 \).

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (d) 7.2

Explanation:
1. Weight = buoyancy: \( \rho V=\tfrac V2(13.6)+\tfrac V2(0.8) \).
2. \( \rho=\dfrac{13.6+0.8}2=7.2 \) g/cm³.

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (a) 1.1

Explanation:
1. The liquid I surface on the side without II has risen 2 cm, so on the other side it has dropped 2 cm; the column of II is 4 cm high above the original level, and the free surfaces are level.
2. At the lower interface: pressure of II column \( =\rho_{II}g(4) \) equals the extra I column \( \rho_Ig(4) \) on the other side, so \( \rho_{II}=\rho_I=1.1 \).

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (b) zero

Explanation:
1. In free fall the effective gravity is zero, so the pressure in the liquid has no depth variation.
2. Upthrust arises from the pressure difference over depth, hence it is zero.

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (c) (a), (b) and (d) only

Explanation:
1. For a taut string, A tends to rise (lighter than the fluid) and B tends to sink (heavier): (a) and (b) hold.
2. The pair as a whole is in equilibrium: \( (d_A+d_B)Vg=2d_FVg\Rightarrow d_A+d_B=2d_F \): (d) holds; (c) is false.

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (d) the balance A will read less than 2 kg and B will read more than 5 kg

Explanation:
1. The liquid exerts an upthrust on the block, so the spring balance A reads less than 2 kg.
2. By Newton's third law the block pushes down on the liquid with the same force, so the balance B reads more than 5 kg.

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (a) falls

Explanation:
1. While floating, the ice with lead displaces water equal to its total weight (ice + lead).
2. After melting, the water from ice occupies the same volume as its weight in water, but the lead sinks and displaces only its own volume, which is less. So the level falls.

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (b) 0.25 cm

Explanation:
1. Let the cylinder length be \( L \) with area S. Equilibrium: \( 0.8L=0.7(1.2)+1.2(0.8)=1.8 \).
2. \( L=2.25 \) cm, so \( h=2.25-1.2-0.8=0.25 \) cm.

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (c) g/6

Explanation:
1. Now 1.2 cm lies in A and 1.05 cm in B. Buoyant force per unit area \( =g(0.7\times1.2+1.2\times1.05)=2.1g \) (g cm/cm³ units).
2. Weight \( =0.8\times2.25g=1.8g \). Net upward \( =0.3g \), so \( a=\dfrac{0.3g}{1.8}=\dfrac g6 \).

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (d) \( \pi R^2L\left(\sqrt{\rho\sigma}-\rho\right) \)

Explanation:
1. Let x be the submerged length: \( \pi R^2L\rho+m=\pi R^2x\sigma \). Centre of gravity (from the bottom) \( =\dfrac{\rho L^2}{2\sigma x} \); centre of buoyancy at \( \dfrac x2 \).
2. Vertical equilibrium is stable (for a thin stick) when the centre of buoyancy is above the centre of gravity: \( \dfrac x2\ge\dfrac{\rho L^2}{2\sigma x}\Rightarrow x\ge L\sqrt{\dfrac\rho\sigma} \).
3. Minimum m: \( \pi R^2L\sqrt{\rho/\sigma}\cdot\sigma=\pi R^2L\rho+m\Rightarrow m=\pi R^2L\left(\sqrt{\rho\sigma}-\rho\right) \).

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (a) \( 45^\circ \)

Explanation:
1. Submerged length along the plank \( \ell=\dfrac{0.5}{\cos\theta} \). Torques about the hinge: buoyancy \( \rho_wA\ell g\cdot\dfrac\ell2\sin\theta \) = weight \( 0.5\rho_wA(1)g\cdot\dfrac12\sin\theta \).
2. \( \ell^2=0.5\Rightarrow\ell=0.707 \) m, \( \cos\theta=\dfrac{0.5}{0.707}\Rightarrow\theta=45^\circ \).

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (b) The level falls

Explanation:
1. When the stones are in the boat, they displace water equal to their weight (stone density is greater than water).
2. After unloading, they sink and displace only their own volume, which is smaller, so the level falls.

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (c) 2.95 cm

Explanation:
1. Each air column initially \( 45 \) cm at 76 cm Hg. After turning, the upper column is \( 45+x \) (pressure \( p_1 \)) and the lower \( 45-x \) (pressure \( p_1+10 \)).
2. \( p_1(45+x)=76(45) \) and \( (p_1+10)(45-x)=76(45) \) give \( x^2+684x-2025=0 \).
3. \( x=2.95 \) cm.

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (d) 9

Explanation:
1. \( v_t\propto r^2 \). Volume conservation: \( 27r^3=R^3\Rightarrow r=\dfrac R3 \).
2. \( \dfrac{v_1}{v_2}=\dfrac{R^2}{r^2}=9 \).

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (a) \( 5\times10^{-5} \) m²

Explanation:
1. \( v^2=1^2+2(10)(0.15)=4\Rightarrow v=2 \) m/s.
2. Continuity: \( A\cdot2=10^{-4}\cdot1\Rightarrow A=5\times10^{-5} \) m².

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (b) 2.0

Explanation:
1. \( \omega=2\pi(2)=4\pi \) rad/s. \( \Delta h=\dfrac{\omega^2r^2}{2g} \).
2. \( \Delta h=\dfrac{(157.9)(0.05)^2}{2(9.8)}=0.0201 \) m \( =2.0 \) cm.

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (c) \( \dfrac{3}{4} \) \( \rho v^2 \)

Explanation:
1. Mass hitting per unit time \( =\rho Av \). Momentum change rate: \( 0.25\rho Av\cdot v+0.25\rho Av\cdot2v=0.75\rho Av^2 \).
2. Pressure \( =\dfrac{\text{force}}A=\tfrac34\rho v^2 \).

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (d) 5.1 cm

Explanation:
1. Outflow speed \( v=\dfrac{10^{-4}}{10^{-4}}=1 \) m/s.
2. \( h=\dfrac{v^2}{2g}=\dfrac1{19.6}=0.051 \) m \( =5.1 \) cm.

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (a) 4.8 m

Explanation:
1. Flow rate \( =\dfrac{0.74}{60}=0.01233 \) m³/s, area \( =\pi(0.02)^2=1.257\times10^{-3} \) m². \( v=9.8 \) m/s.
2. \( h=\dfrac{v^2}{2g}=\dfrac{96.3}{20}=4.8 \) m.

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (b) \( 50\ \text{m}^2/\text{s}^2 \)

Explanation:
1. Depth of the orifice below the surface \( =3-0.525=2.475 \) m. With the free surface also moving: \( v^2\left[1-(0.1)^2\right]=2gh \).
2. \( v^2=\dfrac{2(10)(2.475)}{0.99}=50\ \text{m}^2/\text{s}^2 \).

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (c) \( \dfrac{L}{\sqrt{2\pi}} \)

Explanation:
1. Equal discharge: \( L^2\sqrt{2gy}=\pi R^2\sqrt{2g(4y)}=2\pi R^2\sqrt{2gy} \).
2. \( R^2=\dfrac{L^2}{2\pi}\Rightarrow R=\dfrac L{\sqrt{2\pi}} \).

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (d) 500 Pa

Explanation:
1. Continuity: \( v_2=\dfrac{10}5(1)=2 \) m/s.
2. Bernoulli: \( p_2=p_1+\tfrac12\rho(v_1^2-v_2^2)=2000+500(1-4)=500 \) Pa.

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (a) 2 m

Explanation:
1. Continuity: \( v_{nozzle}=0.25\left(\dfrac84\right)^2\cdot\)... \( 0.25\times\dfrac{8^2}{2^2}=4 \) m/s.
2. Fall time \( t=\sqrt{\dfrac{2(1.25)}{10}}=0.5 \) s, so range \( =4\times0.5=2 \) m.

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (b) 29025 J/m³

Explanation:
1. \( v_Q=\dfrac{4\times10^{-3}}{8\times10^{-3}}(1)=0.5 \) m/s.
2. Bernoulli: \( p_P-p_Q=\rho g(h_Q-h_P)+\tfrac12\rho(v_Q^2-v_P^2)=29400+500(0.25-1)=29025 \) Pa.
3. Work per unit volume by pressure forces \( =p_P-p_Q=29025 \) J/m³ (gravity does \( -29400 \) J/m³).

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.

Answer: (c) \( \dfrac g{50} \)

Explanation:
1. Efflux speed \( v=\sqrt{2gh} \); thrust \( F=\rho\dfrac A{100}v^2=\rho\dfrac A{100}\cdot2gh=\dfrac{\rho Agh}{50} \).
2. The weight of the liquid is \( m_0g=\rho Ag h \) and the container is massless, so \( a=\dfrac{F}{m_0}=\dfrac g{50} \).

Teacher's Note:
1. Use the pressure formula P = P0 + rho g h for liquids at rest and the equation of continuity A1 v1 = A2 v2 with Bernoulli's equation for flow.
2. Check that the forces or pressures balance at a chosen reference level, or that the energy per unit volume is conserved.