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
1. \(\sqrt{2} : 1\)
2. \(1 : \sqrt{2}\)
3. 4 : 1
4. 1 : 4
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Since length, load, and extension are the same: \(Y = \frac{FL}{\pi R^2 \Delta L} ⇒ R^2 \propto \frac{1}{Y} ⇒ \frac{R_B}{R_S} = \sqrt{\frac{Y_S}{Y_B}} = \sqrt{\frac{2 \times 10^{11}}{1 \times 10^{11}}} = \sqrt{2} : 1\).
Assertion (A): Bernoulli’s theorem is based on energy conservation.
Reason (R): Bernoulli’s theorem holds good for all liquids.
1. Both Assertion & Reason are true and the reason is the correct explanation of the assertion.
2. Both Assertion & Reason are true but the reason is not the correct explanation of the assertion.
3. Assertion is true statement but Reason is false.
4. Both Assertion and Reason are false statements.
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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.
1. Both (A) and (R) are true and (R) is the correct explanation of (A)
2. Both (A) and (R) are true but (R) is not the correct explanation of (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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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).
Assertion (A): Bernoulli’s theorem is based on energy conservation.
Reason (R): Bernoulli’s theorem holds good for all liquids.
1. Both Assertion & Reason are true and the reason is the correct explanation of the assertion.
2. Both Assertion & Reason are true but the reason is not the correct explanation of the assertion.
3. Assertion is true statement but Reason is false.
4. Both Assertion and Reason are false statements.
View Answer
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).
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
1. \(\sqrt{2} : 1\)
2. \(1 : \sqrt{2}\)
3. 4 : 1
4. 1 : 4
View Answer
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): 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.
1. Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (A) is true but (R) is false
4. Both (A) and (R) are false
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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.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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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.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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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.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
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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 stream of water emerging from a water tap “necks down” as it falls.
Reason (R): The volume flow rate at different levels is same.
1. (1) Both (A) & (R) are true and the (R) is the correct explanation of the (A)
2. (2) Both (A) & (R) are true but the (R) is not the correct explanation of the (A)
3. (3) (A) is true but (R) is false
4. (4) Both (A) and (R) are false
View Answer
Concept: Continuity equation for fluid flow. For an incompressible fluid in steady flow, the volume flow rate \(AV\) (Area × Velocity) must remain constant. As water falls, its velocity \(V\) increases due to gravity, therefore, its cross-sectional area \(A\) must decrease, causing it to 'neck down'. Both Assertion and Reason are true, and Reason is the correct explanation for Assertion.