JEE MainPhysicsProperties of MatterMCQ+4 / −1
Two steel wires having same length are suspended from a ceiling under the same load. If the ratio of their energy stored per unit volume is 1 : 4, the ratio of their diameters is:
- A1 : 2
- B2 : 1
- C
- D
View written solutionFree
Correct answer: D
- Strain energy per unit volume for a stretched wire
For a wire under तनाव (stress) within elastic limit,
Using Hooke’s law,
So,
= \frac{\text{stress}^2}{2Y}$$ Since both are **steel** wires, their Young’s modulus $Y$ is same. Hence, $$u \propto (\text{stress})^2$$ 2. **Stress in each wire** Stress is $$\text{stress} = \frac{F}{A}$$ Both wires carry the **same load**, so $F$ is same. Therefore, $$\text{stress} \propto \frac{1}{A}$$ Thus, $$u \propto \frac{1}{A^2}$$ 3. **Relating area to diameter** For a circular wire, $$A = \frac{\pi d^2}{4} \Rightarrow A \propto d^2$$ So, $$u \propto \frac{1}{d^4}$$ 4. **Using the given ratio** Given energy stored per unit volume ratio: $$u_1 : u_2 = 1:4$$ Using $u \propto \frac{1}{d^4}$, $$\frac{u_1}{u_2} = \frac{d_2^4}{d_1^4} = \frac{1}{4}$$ So, $$\left(\frac{d_2}{d_1}\right)^4 = \frac{1}{4}$$ Taking fourth root, $$\frac{d_2}{d_1} = \frac{1}{\sqrt{2}}$$ Hence, $$\frac{d_1}{d_2} = \sqrt{2}$$ Therefore, the ratio of diameters is $$d_1:d_2 = \sqrt{2}:1$$ 5. **Option check** - A: $1:2$ ❌ - B: $2:1$ ❌ - C: $1:\sqrt2$ ❌ - D: $\sqrt2:1$ ✅ So the correct option is **D**.More from Properties of Matter
- A small spherical droplet of density d is floating exactly half immersed in a liquid of density and surface tension T. The radius of the droplet is (take note that the surface tension applies an upward force on the droplet) :2020 · MCQ
- A steel wire having a radius of 2.0 mm, carrying a load of 4 kg, is hanging from a ceiling. Given that g = 3.1 p ms–2, what will be the tensile stress that would be developed in the wire ?2019 · MCQ
- Water from a pipe is coming at a rate of 100 litres per minute. If the radius of the pipe is 5 cm, the Reynolds number for the flow is of the order of : (density of water = 1000 kg/m3, coefficient of viscosity of water = 1mPas)2019 · MCQ
- A boy's catapult is made of rubber cord which is 42 cm long, with 6 mm diameter of cross-section and of negligible mass. The boy keeps a stone weighing 0.02kg on it and stretches the cord by 20 cm by applying a constant force. When…2019 · MCQ
- Young's moduli of two wires A and B are in the ratio 7 : 4. Wire A is 2 m long and has radius R. Wire B is 1.5 m long and has radius 2 mm. If the two wires stretch by the same length for a given load, then the value of R is close to :-2019 · MCQ
- If 'M' is the mass of water that rises in a capillary tube of radius 'r', then mass of water which will rise in a capillary tube of radius '2r' is :2019 · MCQ
- A simple pendulum oscillating in air has period T. The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is 1/16 th of the material of the bob. If the bob is inside liquid all the time, its…2019 · MCQ
- A wooden block floating in a bucket of water has 4/5 of its volume submerged. When certain amount of an oil is poured into the bucket, it is found that the block is just under the oil surface with half of its volume under water and half in…2019 · MCQ