Mechanical properties of solids practice · Mixed difficulty

Option A (4 rad/sec) is correct because the maximum angular velocity is calculated using the breaking stress, area, mass, and length.

Option C ($\mathrm{0}\mathrm{.03}$) is correct because bulk strain is the volumetric strain, which is three times the linear strain for a cube.

Option C ($2 \times {10^9}$) is correct because bulk modulus is the ratio of pressure to volumetric strain.

Option C (Stress) is correct because stress is defined as the restoring force per unit area within a deformable body.

Option C ($K = \frac{{Y\eta }}{{9\eta - 3Y}}N/{m^2}$) is the correct relation between Young’s modulus, bulk modulus and modulus of rigidity.

The correct answer is D (Tensor) because stress is a tensor quantity that describes forces acting on a cross-sectional area.

Option B (${W_2} = 2\,{W_1}$) is correct because the work done is proportional to the force constant, given the same stretch.

Option D (Decrease by 0.75 mL) is correct because the volume of water decreases under pressure, and the calculated decrease is 0.75 mL based on the given compressibility and pressure.

Option A ($12.5 \times 10^6 \mathrm{~N} / \mathrm{m}^2$) is correct because it is the calculated bulk modulus using the given Young's modulus and coefficient of rigidity.

Option B (Same) is correct because the load a wire can hold depends on the breaking stress of the material and the cross-sectional area, not the length of the wire.

Option D (80 kg-wt) is correct because the breaking force is proportional to the cross-sectional area, which is proportional to the square of the diameter.

Option C ($1 \times {10^{12}}$) is correct because it is the calculated bulk modulus of the rubber based on the given pressure and volume reduction.

Option A ($1 \times {10^{ - 4}}$) is correct because strain is defined as the change in length divided by the original length, which is 0.01% or 0.0001.

Option B (If both Assertion & Reason are True but Reason is not a correct explanation of the Assertion) is correct because elastic restoring forces can be conservative, and hysteresis explains the difference in strain for the same stress during loading and unloading, but hysteresis doesn't explain why restoring forces *may* be conservative.

Option D ($\mathrm{[}\,\mathrm{M}\,{\mathrm{L}^{ - 1}}\,{\mathrm{T}^{ - 2}}\,\mathrm{]}$) is correct because stress is force per unit area, and the dimensions of force are $[MLT^{-2}]$ and area are $[L^2]$, so stress is $[ML^{-1}T^{-2}]$.

Option D ($\frac{{Fl}}{2}$) is correct because the work done in stretching a wire is given by half the product of force and extension.

Option D (Rubber, glass, copper, steel) is correct because it lists the materials in order of increasing Young's modulus (coefficient of elasticity).

Option A (0.7) is the correct answer because Poisson's ratio is generally between 0 and 0.5 for most materials, and 0.7 exceeds this range.

Option B (${10^4}\,\mathrm{N}$) is correct because the force required is calculated using the formula $F = YA\alpha \Delta T$, where Y is Young's modulus, A is the area, $\alpha$ is the coefficient of linear expansion, and $\Delta T$ is the change in temperature.

Option C (Tensile stress and longitudinal strain) is correct because Young's modulus is defined as the ratio of tensile stress to longitudinal strain.

Option B is correct because Young's modulus is defined as the stress required to produce unit strain.

Option B ($\frac{L \rho g}{2}$) is correct because the stress at the middle point depends on the weight of the lower half of the bar.

Option A is correct because Invar has a very low coefficient of thermal expansion, making it suitable for balance wheels in watches to maintain constant time.

Option C ($1400 \mathrm{~m} / \mathrm{s}$) is correct because it is the approximate speed of sound in water, calculated using the bulk modulus and density of water.

Option A (Statement I is true and statement II is true) is correct because the wire stretches due to its own weight when held vertically.

Option C (If Assertion is true but Reason is false) is correct because while the assertion about half the work being stored is true, the reason provides an incorrect formula for potential energy.

The question is incomplete and does not provide enough information to determine the ratio of elongations. It needs additional information such as the applied load, length, and Young's modulus to calculate the elongations and their ratio.

Option C (4 cm) is correct because elongation is inversely proportional to the square of the diameter.

Option A ($Y$) is correct because Young's modulus is a material property and does not depend on the dimensions of the wire.

Option C ($4\,F$) is correct because the force needed to break a wire is proportional to the cross-sectional area, which is proportional to the square of the radius.