JEE Physics Mechanical Properties Of Solids MCQs Set 11

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Chapter-wise Objective Questions: Mechanical Properties Of Solids

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Multiple Choice Questions

Question 1: A wire of length 1 m and cross-sectional area 10⁻⁶ m² is suspended from a roof with a load W at the other end. The extension is 1 × 10⁻⁴ m for W = 20 N and 4 × 10⁻⁴ m for W = 80 N, in the linear region. The Young's modulus of the material of the wire is
(a) \( 2\times10^{11}\ \text{N/m}^2 \)
(b) \( 2\times10^{-11}\ \text{N/m}^2 \)
(c) \( 2\times10^{-12}\ \text{N/m}^2 \)
(d) \( 2\times10^{-13}\ \text{N/m}^2 \)
Show Answer & Explanation

Answer: (a) \( 2\times10^{11}\ \text{N/m}^2 \)

Explanation:
1. Y = FL/(AΔl) = (20 × 1)/(10⁻⁶ × 1 × 10⁻⁴).
2. = 2 × 10¹¹ N/m².

Teacher's Note:
1. Use a point on the linear part.

Question 2: An iron bar of length l cm and cross-section A cm² is pulled by a force of F dynes from its ends so as to produce an elongation Δl cm. Which of the following statements is correct?
(a) Elongation is inversely proportional to length
(b) Elongation is directly proportional to cross-section A
(c) Elongation is inversely proportional to cross-section
(d) Elongation is directly proportional to Young's modulus
Show Answer & Explanation

Answer: (c) Elongation is inversely proportional to cross-section

Explanation:
1. Δl = Fl/(AY).
2. So elongation is directly proportional to the length and inversely proportional to A and to Y.

Teacher's Note:
1. Read each dependence from Δl = Fl/AY.

Question 3: Hooke's law states that
(a) stress is directly proportional to strain
(b) stress is inversely proportional to strain
(c) stress is equal to strain
(d) stress and strain are independent of each other
Show Answer & Explanation

Answer: (a) stress is directly proportional to strain

Explanation:
1. Within the elastic limit, stress ∝ strain.
2. The constant of proportionality is the modulus of elasticity.

Teacher's Note:
1. Hooke's law.

Question 4: If a mass M produces an elongation Δl in a wire of radius r and length L, then the Young's modulus of the material of the wire is given by
(a) \( Y=\frac{Mg}{\pi r^2\Delta l} \)
(b) \( Y=\frac{Mg\,\Delta l}{\pi r^2 L} \)
(c) \( Y=\frac{Mg\,L}{\pi r^2\Delta l} \)
(d) \( Y=\frac{M\Delta l}{\pi r^2 L} \)
Show Answer & Explanation

Answer: (c) \( Y=\frac{Mg\,L}{\pi r^2\Delta l} \)

Explanation:
1. Y = (F/A)/(Δl/L) = (Mg/πr²)(L/Δl).
2. So Y = MgL/(πr²Δl).

Teacher's Note:
1. Stress divided by strain.

Question 5:
Assertion (A): A hollow shaft is found to be stronger than a solid shaft made of the same material and the same mass.
Reason (R): The torque required to produce a given twist in a hollow cylinder is greater than that required to twist a solid cylinder of the same length and material.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (a) Both A and R are true, and R is the correct explanation of A.

Explanation:
1. For equal mass and length, the hollow shaft has a larger radius, so the torsional rigidity (∝ r⁴) is greater.
2. The reason is exactly that property.

Teacher's Note:
1. Mass farther from the axis resists twisting better.

Question 6: Which of the following statements is/are true? I. Water is more elastic than air. II. The modulus of elasticity is more for steel than for copper. III. Young's modulus of elasticity for a perfectly rigid body is infinite.
(a) I only
(b) II only
(c) I and II only
(d) I, II and III
Show Answer & Explanation

Answer: (d) I, II and III

Explanation:
1. Water is much less compressible than air (larger bulk modulus).
2. Y(steel) > Y(copper).
3. A perfectly rigid body has infinite Y.

Teacher's Note:
1. All three statements are true.

Question 7: If Y is Young's modulus and G is the modulus of rigidity, then
(a) \( Y \lt G \)
(b) \( Y \gt G \)
(c) \( Y=G \)
(d) none of these
Show Answer & Explanation

Answer: (b) \( Y \gt G \)

Explanation:
1. Y = 2G(1 + σ) where Poisson's ratio σ is positive for ordinary materials.
2. So Y is greater than G (between 2G and 3G).

Teacher's Note:
1. Y = 2G(1 + σ).

Question 8:
Assertion (A): Stress is the internal force per unit area of a body.
Reason (R): Rubber is less elastic than steel.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (b) Both A and R are true, but R is not the correct explanation of A.

Explanation:
1. Stress = restoring force per unit area, so the assertion is true.
2. The reason, in the sense of Young's modulus, is also true but it does not explain the definition of stress.

Teacher's Note:
1. Both true, unrelated.

Question 9: When a pressure of 100 atmosphere is applied on a spherical ball, its volume reduces by 0.01%. The bulk modulus of the material of the rubber in dyne/cm² is
(a) \( 10\times10^{12} \)
(b) \( 100\times10^{12} \)
(c) \( 1\times10^{12} \)
(d) \( 0.1\times10^{12} \)
Show Answer & Explanation

Answer: (c) \( 1\times10^{12} \)

Explanation:
1. P = 100 × 1.013 × 10⁷ Pa ≈ 10⁹ ×1.013 N/m² and ΔV/V = 10⁻⁴.
2. B = P/(ΔV/V) ≈ 1.013 × 10¹¹ N/m².
3. In CGS, 1 N/m² = 10 dyne/cm², so B ≈ 1 × 10¹² dyne/cm².

Teacher's Note:
1. Convert units carefully.

Question 10: The property of a body by virtue of which it tends to regain its original size and shape when the applied force is removed is called
(a) elasticity
(b) plasticity
(c) rigidity
(d) compressibility
Show Answer & Explanation

Answer: (a) elasticity

Explanation:
1. This is the definition of elasticity.
2. Plasticity is the opposite: the body keeps its deformed shape.

Teacher's Note:
1. Elastic bodies recover.

Question 11: A wire elongates by l mm when a load W is hung from it. If the wire goes over a pulley and two weights W each are hung at the two ends, the elongation of the wire will be (in mm)
(a) l
(b) 2l
(c) zero
(d) l/2
Show Answer & Explanation

Answer: (a) l

Explanation:
1. The tension in the wire is W in both cases (the pulley merely redirects it).
2. So the elongation is the same, l.

Teacher's Note:
1. Tension equals W.

Question 12: If a spring extends by x on loading, then the energy stored by the spring is (T is the tension in the spring and k the spring constant)
(a) \( \frac{T^2}{2x} \)
(b) \( \frac{T^2}{2k} \)
(c) \( \frac{2x}{T^2} \)
(d) \( \frac{2T^2}{k} \)
Show Answer & Explanation

Answer: (b) \( \frac{T^2}{2k} \)

Explanation:
1. U = ½kx² with x = T/k.
2. U = T²/2k.

Teacher's Note:
1. Express the energy through the tension.

Question 13: Which of the following statements is/are true? I. Stress is not a vector quantity. II. A deforming force in one direction can produce strains in other directions also.
(a) I only
(b) II only
(c) Both I and II
(d) none of these
Show Answer & Explanation

Answer: (c) Both I and II

Explanation:
1. Stress is a tensor-like quantity defined as force per area and is not a vector.
2. A force in one direction produces a lateral strain too (Poisson's effect).

Teacher's Note:
1. Longitudinal stress causes lateral strain.

Question 14: A steel wire is suspended vertically from a rigid support. When loaded with a weight in air, it extends by ℓₐ and when the weight is immersed completely in water, the extension is reduced to ℓ_w. Then the relative density of the material of the weight is
(a) \( \ell_a/\ell_w \)
(b) \( \frac{\ell_a}{\ell_a-\ell_w} \)
(c) \( \ell_w/(\ell_a-\ell_w) \)
(d) \( \ell_w/\ell_a \)
Show Answer & Explanation

Answer: (b) \( \frac{\ell_a}{\ell_a-\ell_w} \)

Explanation:
1. Extension ∝ apparent weight: ℓₐ ∝ W and ℓ_w ∝ W(1 − 1/d).
2. ℓₐ − ℓ_w ∝ W/d, so d = ℓₐ/(ℓₐ − ℓ_w).

Teacher's Note:
1. Buoyancy reduces the apparent weight by W/d.

Question 15: If the ratio of the radii of two wires of the same material is 2 : 1 and the ratio of their lengths is 4 : 1, then the ratio of the normal forces that will produce the same extension in the length of the two wires is
(a) 2 : 1
(b) 4 : 1
(c) 1 : 4
(d) 1 : 1
Show Answer & Explanation

Answer: (d) 1 : 1

Explanation:
1. F = YAΔl/l ∝ r²/l for the same extension.
2. Ratio = (4/4) = 1, so 1 : 1.

Teacher's Note:
1. F ∝ r²/l.

Question 16: From a steel wire of density ρ a brass block of density ρ_b is suspended. The extension of the steel wire comes to e. If the brass block is now fully immersed in a liquid of density ρ_l, the extension becomes e′. The ratio e/e′ will be
(a) \( \frac{\rho_b}{\rho_b-\rho_l} \)
(b) \( \frac{\rho_b-\rho_l}{\rho_b} \)
(c) \( \frac{\rho_b-\rho}{\rho_l-\rho} \)
(d) \( \frac{\rho_l}{\rho_b-\rho_l} \)
Show Answer & Explanation

Answer: (a) \( \frac{\rho_b}{\rho_b-\rho_l} \)

Explanation:
1. The extension is proportional to the tension.
2. In air T = ρ_b V g; in liquid T′ = (ρ_b − ρ_l)Vg.
3. So e/e′ = ρ_b/(ρ_b − ρ_l).

Teacher's Note:
1. Buoyancy reduces the tension.

Question 17: The extension in a string obeying Hooke's law is x. The speed of sound in the stretched string is v. If the extension in the string is increased to 1.5x, the speed of sound will be
(a) \( 1.22v \)
(b) \( 0.61v \)
(c) \( 1.50v \)
(d) \( 0.75v \)
Show Answer & Explanation

Answer: (a) \( 1.22v \)

Explanation:
1. Tension T ∝ x and v = √(T/μ).
2. v ∝ √x, so v′ = v√1.5 = 1.22v.

Teacher's Note:
1. Wave speed depends on the square root of tension.

Question 18: The correct increasing order of the coefficient of elasticity (Young's modulus) of copper, steel, glass and rubber is
(a) Steel, Rubber, Copper, Glass
(b) Rubber, Copper, Glass, Steel
(c) Rubber, Glass, Steel, Copper
(d) Rubber, Glass, Copper, Steel
Show Answer & Explanation

Answer: (d) Rubber, Glass, Copper, Steel

Explanation:
1. Approximate Young's moduli: rubber 10⁷–10⁸, glass about 7 × 10¹⁰, copper 1.1 × 10¹¹, steel 2 × 10¹¹ N/m².
2. The increasing order is rubber, glass, copper, steel.

Teacher's Note:
1. Remember typical Y values.

Question 19: When a fluid compresses an object, then Hooke's law is expressed as (magnitudes)
(a) \( F=kV \)
(b) \( P=\frac{\Delta V}{V} \)
(c) \( P=B\left(\frac{\Delta V}{V}\right) \)
(d) \( F=B\left(\frac{\Delta V}{V}\right) \)
Show Answer & Explanation

Answer: (c) \( P=B\left(\frac{\Delta V}{V}\right) \)

Explanation:
1. Bulk modulus B = P/(ΔV/V).
2. So P = B(ΔV/V) in magnitude.

Teacher's Note:
1. Pressure is B times the volume strain.

Question 20: Which of the following is/are correct statement(s) about shearing strain? I. Shearing strain is produced by applying normal force. II. Shearing strain = tan θ.
(a) I only
(b) II only
(c) Both I and II
(d) none of these
Show Answer & Explanation

Answer: (b) II only

Explanation:
1. Shearing strain is produced by a tangential force, so I is false.
2. Shearing strain = Δx/L = tan θ, so II is true.

Teacher's Note:
1. Shear needs tangential force.

Question 21: The isothermal bulk modulus of a gas at atmospheric pressure is
(a) 1 mm of Hg
(b) 13.6 mm of Hg
(c) \( 1.013\times10^5\ \text{N/m}^2 \)
(d) \( 2.026\times10^5\ \text{N/m}^2 \)
Show Answer & Explanation

Answer: (c) \( 1.013\times10^5\ \text{N/m}^2 \)

Explanation:
1. For an isothermal change PV = constant, so B = P.
2. At atmospheric pressure B = 1.013 × 10⁵ N/m².

Teacher's Note:
1. Isothermal B equals P.

Question 22:
Assertion (A): Girders are given an I shape.
Reason (R): To bear more pressure, the depth is increased as per P = hρg.
(a) Both A and R are true, and R is the correct explanation of A.
(b) Both A and R are true, but R is not the correct explanation of A.
(c) A is true, but R is false.
(d) A is false, but R is true.
Show Answer & Explanation

Answer: (c) A is true, but R is false.

Explanation:
1. The depth is increased to reduce the bending (depression ∝ 1/d³) while saving material, so the assertion is true.
2. The pressure formula has nothing to do with it, so the reason is false.

Teacher's Note:
1. I-section resists bending.

Question 23: The ratio of stress to strain is called
(a) elastic limit
(b) plastic deformation
(c) modulus of elasticity
(d) tensile strength
Show Answer & Explanation

Answer: (c) modulus of elasticity

Explanation:
1. Modulus of elasticity = stress/strain.
2. It is a constant of the material within the elastic limit.

Teacher's Note:
1. Modulus = stress/strain.

Question 24: An iron rod of length 2 m and cross-sectional area 50 mm² is stretched by 0.5 mm when a mass of 250 kg is hung from its lower end. The Young's modulus of the iron rod is (g = 9.8 m/s²)
(a) \( 19.6\times10^{20}\ \text{N/m}^2 \)
(b) \( 19.6\times10^{18}\ \text{N/m}^2 \)
(c) \( 19.6\times10^{10}\ \text{N/m}^2 \)
(d) \( 19.6\times10^{15}\ \text{N/m}^2 \)
Show Answer & Explanation

Answer: (c) \( 19.6\times10^{10}\ \text{N/m}^2 \)

Explanation:
1. Y = FL/(AΔl) = (250 × 9.8 × 2)/(50 × 10⁻⁶ × 0.5 × 10⁻³).
2. = 4900/(2.5 × 10⁻⁸) = 1.96 × 10¹¹ = 19.6 × 10¹⁰ N/m².

Teacher's Note:
1. Convert mm² to m².

Question 25: The reciprocal of the bulk modulus is called
(a) modulus of rigidity
(b) volume stress
(c) volume strain
(d) compressibility
Show Answer & Explanation

Answer: (d) compressibility

Explanation:
1. Compressibility k = 1/B.
2. It measures the fractional change in volume per unit pressure.

Teacher's Note:
1. k = 1/B.

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