Assertion (A): Bernoulli’s theorem is based on energy conservation.
Reason (R): Bernoulli’s theorem holds good for all liquids.
Bernoulli's theorem is based on energy conservation (Assertion is true). However, it only holds for ideal (non-viscous, incompressible) fluids, not all liquids (Reason is false).
Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion (A): The unit of stress is same as that of pressure.
Reason (R): Stress is a vector quantity.
In the light of above statements, select the correct option.
Assertion (A) is true as both stress and pressure are measured in \( \text{N/m}^2 \) (or Pa). Reason (R) is false because stress is a tensor quantity (neither a scalar nor a simple vector).
The Young’s modulus of brass and steel are \(1 \times 10^{11}\text{ N/m}^2\) and \(2 \times 10^{11}\text{ N/m}^2\) respectively. If wires of both materials, having same length, are loaded with same weight, then they both extend by 4 mm. Ratio of the radii of two wires \(R_B : R_S\) is
Using \(Y = \frac{FL}{\pi R^2 \Delta L}\), for constant force, length, and extension, \(R^2 \propto \frac{1}{Y}\). Thus, \(\frac{R_B}{R_S} = \sqrt{\frac{Y_S}{Y_B}} = \sqrt{\frac{2 \times 10^{11}}{1 \times 10^{11}}} = \sqrt{2} : 1\).
Assertion (A): Bernoulli’s theorem is based on energy conservation.
Reason (R): Bernoulli’s theorem holds good for all liquids.
Bernoulli's theorem is derived from the law of conservation of energy (Assertion is true). It only holds good for ideal (non-viscous, incompressible) fluids, not all liquids (Reason is false).
Assertion (A): Identical springs of steel and copper are equally stretched. More work will be done on the steel spring.
Reason (R): Steel is more elastic than copper.
Assertion (A) is true. Work done to stretch a spring is \(W = \frac{1}{2} k x^2\). Steel has a higher Young's modulus than copper, implying a higher spring constant \(k\) for identical dimensions.
Thus, more work is done on the steel spring.
Reason (R) is true. Steel is indeed more elastic than copper (possesses a higher Young's modulus).
Reason (R) correctly explains Assertion (A).
Assertion (A): Blood pressure of heart is same whether you lie down or stand up.
Reason (R): Pressure varies with height in a fluid under gravity.
Assertion (A) is true due to physiological regulation of blood pressure at the heart level. Reason (R) is true as pressure in a fluid varies with depth, \(P = \rho gh\). However, (R) describes a physical effect that (A) counteracts, so (R) is not the correct explanation for (A).
Assertion (A): A raindrop after falling through some height attains a constant velocity.
Reason (R): At constant velocity the viscous drag plus buoyant force is just equal to its weight.
Assertion (A) is true, as falling objects in a fluid reach terminal velocity when resistive forces balance gravity.
Reason (R) is true, stating the force balance condition for constant velocity: Weight = viscous drag + buoyant force. (R) correctly explains (A).
Assertion (A): In streamline flow streamlines never intersect each other.
Reason (R): If streamline intersect then their must two velocities of fluid particle at the point of intersection, which is impossible.
Assertion (A) is true by definition of streamline flow.
Reason (R) is true because intersection would imply multiple velocity vectors at a single point, which is physically impossible.
(R) correctly explains (A).
Assertion (A): The shape of a liquid drop is spherical.
Reason (R): The pressure inside the drop is greater than that of outside.
Concept: Surface tension and pressure difference. Assertion (A) is true because surface tension tends to minimize the surface area of a liquid for a given volume, and a sphere has the minimum surface area. Reason (R) is true; due to surface tension, there is an excess pressure inside a spherical liquid drop, given by \(P_{in} - P_{out} = \frac{2T}{R}\). This excess pressure balances the inward pull of surface tension, thus R correctly explains A.
Assertion (A): Surface energy of an oil drop is same whether placed on glass or water surface.
Reason (R): Surface energy is dependent only on the properties of oil.
Concept: Interfacial surface energy. Surface energy (or surface tension) is a property of the interface between two phases. It depends on the intermolecular forces between the molecules of *both* substances forming the interface. Therefore, an oil-glass interface will have different surface energy than an oil-water interface. Both Assertion and Reason are false.