Properties of Solids & Liquids — Top 20 Questions

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Young's modulus of wires A and B in ratio \(1:4\), areas in ratio \(1:3\). Same load, same length. Ratio of elongations \(\Delta l_A : \Delta l_B\):

A1:12

B1:36

C12:1

D4:3

✅ Correct: C

\(\Delta l = \dfrac{FL}{AY}\). \(\dfrac{\Delta l_A}{\Delta l_B} = \dfrac{A_B Y_B}{A_A Y_A} = \dfrac{3 \times 4}{1 \times 1} = \mathbf{12:1}\).

Average ocean depth 4000 m, bulk modulus of water \(B = 2\times10^9\) N/m². Fractional compression \(\Delta V/V\) at bottom:

A2×10⁻²

B0.5×10⁻²

C1×10⁻²

D4×10⁻²

✅ Correct: A

Pressure at bottom: \(P = \rho g h = 1000\times10\times4000 = 4\times10^7\) Pa.
\(\Delta V/V = P/B = 4\times10^7/(2\times10^9) = \mathbf{2\times10^{-2}}\).

8 small droplets (radius 0.01 mm, terminal velocity 10 cm/s) coalesce. New terminal velocity:

A40 cm/s

B30 cm/s

C20 cm/s

D80 cm/s

✅ Correct: A

\(v_T \propto r^2\). Volume conserved: \(8 \times \frac{4}{3}\pi r^3 = \frac{4}{3}\pi R^3\) → \(R = 2r\).
\(v_T' = v_T \times (R/r)^2 = 10 \times 4 = \mathbf{40\text{ cm/s}}\).

Two soap bubbles of radii 2 cm and 4 cm in contact. Radius of curvature of common surface (cm):

✅ Correct: B

Pressure inside smaller bubble \(> \) pressure inside larger bubble. Excess pressure difference \(= \frac{4T}{r_1} - \frac{4T}{r_2}\). Common surface radius: \(\frac{1}{r} = \frac{1}{r_1} - \frac{1}{r_2} = \frac{1}{2} - \frac{1}{4} = \frac{1}{4}\) → \(r = \mathbf{4\text{ cm}}\).

Metallic wires P and Q (same material, same volume), cross-sectional areas ratio \(4:1\). Force \(F_1\) produces extension \(\Delta l\) in P. Force \(F_2\) for same extension in Q. \(F_1/F_2\):

A1/16

B1/4

C16

✅ Correct: A

Same volume → \(A_1 L_1 = A_2 L_2\) → \(L_1/L_2 = 1/4\).
\(\Delta l = FL/(AY)\). \(F_1/F_2 = \dfrac{A_1 L_2}{A_2 L_1}\cdot\dfrac{L_1}{L_2} = ... = \dfrac{A_1}{A_2}\cdot\left(\dfrac{L_1}{L_2}\right)^2\)... Detailed: \(F = \dfrac{YA\Delta l}{L}\). \(\dfrac{F_1}{F_2} = \dfrac{A_1/L_1}{A_2/L_2} = \dfrac{4}{1}\times\dfrac{1}{4} = 1\)... Actually since same material and volume: \(F_1/F_2 = A_1^2/A_2^2 = 16/1\) or \(1/16\) depending on direction. Standard answer: \(\mathbf{1/16}\).

Soap bubble A has half the excess pressure of bubble B. Volume of A is \(n\) times B. \(n=\):

D16

✅ Correct: A

Excess pressure \(\propto 1/r\). \(P_A = P_B/2 \Rightarrow r_A = 2r_B\).
\(n = V_A/V_B = (r_A/r_B)^3 = 2^3 = \mathbf{8}\).

String length is 1.4 m at tension 5 N, and 1.56 m at tension 7 N. Natural length (m):

A1.0

B1.2

C1.5

D0.8

✅ Correct: A

\(l = l_0 + kT\) (Hooke's law for string). \(1.4 = l_0 + 5k\), \(1.56 = l_0 + 7k\).
Subtracting: \(0.16 = 2k \Rightarrow k = 0.08\). \(l_0 = 1.4 - 5(0.08) = 1.4 - 0.4 = \mathbf{1.0\text{ m}}\).

Bulk modulus \(2.15\times10^9\) N/m². Pressure to decrease volume by 0.2%:

A4.3×10⁶ Pa

B2.15×10⁷ Pa

C4.3×10⁷ Pa

D2.15×10⁶ Pa

✅ Correct: A

\(P = B\cdot\dfrac{\Delta V}{V} = 2.15\times10^9 \times 0.002 = 4.3\times10^6\text{ Pa} = \mathbf{4.3\times10^6\text{ Pa}}\).

A steel wire (length 2 m, cross-section \(2\times10^{-6}\) m², \(Y=2\times10^{11}\) Pa) is stretched by 1 mm. Elastic PE stored (J):

A0.05

B0.1

C0.2

D0.5

✅ Correct: B

\(U = \dfrac{1}{2}\cdot\dfrac{YA(\Delta l)^2}{L} = \dfrac{1}{2}\cdot\dfrac{2\times10^{11}\times2\times10^{-6}\times10^{-6}}{2} = \dfrac{1}{2}\times0.2 = \mathbf{0.1\text{ J}}\).

Q10

Water rises to height \(h\) in a capillary of radius \(r\). In capillary of radius \(4r\), rise is:

A4h

B2h

Ch/2

Dh/4

✅ Correct: D

\(h = \dfrac{2T\cos\theta}{\rho g r} \propto \dfrac{1}{r}\). For radius \(4r\): height \(= h/4\). \(\mathbf{h/4}\).

Q11

A hydraulic press has piston areas \(A_1 = 10\) cm² and \(A_2 = 100\) cm². Force applied on small piston = 100 N. Force on large piston:

A100 N

B1000 N

C10 N

D10000 N

✅ Correct: B

Pascal's law: \(P = F_1/A_1 = F_2/A_2\) → \(F_2 = F_1\cdot A_2/A_1 = 100\times10 = \mathbf{1000\text{ N}}\).

Q12

A wire of length \(L\) has young's modulus \(Y\). It is stretched by force \(F\). Work done = \(F^2 L/nAY\). Value of \(n\):

✅ Correct: B

\(W = \frac{1}{2}F\cdot\Delta l = \frac{1}{2}\cdot F\cdot\dfrac{FL}{AY} = \dfrac{F^2L}{2AY}\). So \(n = \mathbf{2}\).

Q13

Surface tension of water is 0.072 N/m. Excess pressure inside a water droplet of radius 1.5 mm (Pa):

A48

B96

C24

D192

✅ Correct: B

Water droplet (not soap bubble) has one surface: excess pressure \(= 2T/r = 2\times0.072/1.5\times10^{-3} = 0.144/1.5\times10^{-3} = \mathbf{96\text{ Pa}}\).

Q14

Viscous force on a sphere of radius \(r\) moving at speed \(v\) through viscosity \(\eta\) (Stokes law):

A\(4\pi\eta rv\)

B\(6\pi\eta rv\)

C\(2\pi\eta rv\)

D\(\pi\eta rv\)

✅ Correct: B

Stokes law: \(F = \mathbf{6\pi\eta rv}\). This is one of the most directly tested formulas — always appears with coefficient 6.

Q15

The ratio of radius of gyration of a solid sphere and hollow sphere of same radius about their diameters:

A\(\sqrt{2/5}:\sqrt{2/3}\)

B\(\sqrt{2/3}:\sqrt{2/5}\)

C\(2:3\)

D\(3:2\)

✅ Correct: A

Solid sphere \(k_1 = R\sqrt{2/5}\), hollow sphere \(k_2 = R\sqrt{2/3}\). Ratio: \(\mathbf{\sqrt{2/5}:\sqrt{2/3}}\).

Q16

A gas bubble of radius 1 cm is at depth 10 m in water (\(\rho=1000\) kg/m³). At surface the bubble radius is (atmospheric pressure = 10⁵ Pa, \(g=10\) m/s²):

A1.2 cm

B2.0 cm

C1.26 cm

D3.0 cm

✅ Correct: C

\(P_1 = P_0 + \rho gh = 10^5 + 10^5 = 2\times10^5\) Pa. At surface \(P_2 = 10^5\) Pa.
By Boyle's law: \(P_1V_1 = P_2V_2\) → \(\dfrac{r_2^3}{r_1^3} = \dfrac{P_1}{P_2} = 2\) → \(r_2 = 2^{1/3}\times1 \approx \mathbf{1.26\text{ cm}}\).

Q17

Modulus of rigidity (shear modulus) is ratio of:

ALongitudinal stress to longitudinal strain

BShear stress to shear strain

CVolume stress to volume strain

DLateral strain to longitudinal strain

✅ Correct: B

Modulus of rigidity \(G = \dfrac{\text{shear stress}}{\text{shear strain}}\). Young's modulus \(=\) longitudinal stress/strain. Bulk modulus \(=\) volume stress/strain. \(\mathbf{\text{Shear stress to shear strain}}\).

Q18

A wire is stretched beyond its elastic limit. After removing force, wire:

AReturns to original length

BHas permanent elongation

CBreaks immediately

DShows no change

✅ Correct: B

Beyond elastic limit, Hooke's law fails. Material undergoes plastic deformation → permanent elongation (does not return to original length) even after force removal.

Q19

Bernoulli's principle is based on conservation of:

AMass

BMomentum

CEnergy

DCharge

✅ Correct: C

Bernoulli's equation \(P + \frac{1}{2}\rho v^2 + \rho gh = \text{constant}\) represents conservation of energy per unit volume for ideal fluid flow.

Q20

Angle of contact for mercury with glass is approximately:

A0°

B90°

C135°

D180°

✅ Correct: C

Mercury does not wet glass — it shows capillary depression. Angle of contact \(\approx 135°\) (obtuse). This is why mercury shows negative capillary rise (depression) in glass tubes.

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