The dimensions of four wires if the same material are given below. The increase in length is maximum in the wire of :
The dimensions of four wires if the same material are given below. The increase in length is maximum in the wire of :
Length 100 cm, Diameter 1 mm
Length 200 cm, Diameter 2 mm
Length 300 cm, Diameter 3mm
Length 50 cm, Diameter 0.5 mm
The Correct Option is D
Approach Solution - 1
To solve the problem, we need to determine which wire among the four shows the maximum increase in length when subjected to the same force (load), and all are made of the same material.
1. Formula for Elongation in a Wire:
The extension (increase in length) $\Delta L$ in a wire under tension is given by:
$ \Delta L = \frac{F L}{A Y} $
Where:
- $F$ = force applied (same for all),
- $L$ = original length of the wire,
- $A$ = cross-sectional area of the wire,
- $Y$ = Young’s modulus (same material ⇒ constant).
So, $ \Delta L \propto \frac{L}{A} $
2. Cross-Sectional Area:
Since the wire is cylindrical, $ A = \frac{\pi d^2}{4} $
So, $ \Delta L \propto \frac{L}{d^2} $
3. Calculating $\frac{L}{d^2}$ for Each Option:
- Option 1: $ \frac{100}{1^2} = 100 $
- Option 2: $ \frac{200}{2^2} = \frac{200}{4} = 50 $
- Option 3: $ \frac{300}{3^2} = \frac{300}{9} \approx 33.33 $
- Option 4: $ \frac{50}{(0.5)^2} = \frac{50}{0.25} = 200 $
4. Conclusion:
Since elongation is directly proportional to $ \frac{L}{d^2} $, the maximum value is 200 for Option 4.
Final Answer:
The wire with Length 50 cm, Diameter 0.5 mm has the maximum increase in length.
Approach Solution -2
The correct option is: (D) Length 50 cm, Diameter 0.5 mm.
Given: Y = Δl / Fl aΔl / Fl = Δl / Fl a = Δl / Δl a = 1
a. Δl / l₂ = 1 / 100 Δl = l₂ / 100 Δl = 100 / 100 Δl = 1
b. Δl / 4 = 1 / 200 Δl = 4 / 200 Δl = 1 / 50 Δl = 0.02
c. Δl / 9 = 1 / 300 Δl = 9 / 300 Δl = 3 / 100 Δl = 0.03
d. Δl / (1/2)² = 1 / 50 Δl / (1/4) = 1 / 50 Δl = 1 / 12.5 Δl = 0.08
The options can be summarized as: a. Δl = 1 b. Δl = 0.02 c. Δl = 0.03 d. Δl = 0.08
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Concepts Used:
Thermodynamics
Thermodynamics in physics is a branch that deals with heat, work and temperature, and their relation to energy, radiation and physical properties of matter.
Important Terms
System
A thermodynamic system is a specific portion of matter with a definite boundary on which our attention is focused. The system boundary may be real or imaginary, fixed or deformable.
There are three types of systems:
- Isolated System – An isolated system cannot exchange both energy and mass with its surroundings. The universe is considered an isolated system.
- Closed System – Across the boundary of the closed system, the transfer of energy takes place but the transfer of mass doesn’t take place. Refrigerators and compression of gas in the piston-cylinder assembly are examples of closed systems.
- Open System – In an open system, the mass and energy both may be transferred between the system and surroundings. A steam turbine is an example of an open system.
Thermodynamic Process
A system undergoes a thermodynamic process when there is some energetic change within the system that is associated with changes in pressure, volume and internal energy.
There are four types of thermodynamic process that have their unique properties, and they are:
- Adiabatic Process – A process in which no heat transfer takes place.
- Isochoric Process – A thermodynamic process taking place at constant volume is known as the isochoric process.
- Isobaric Process – A process in which no change in pressure occurs.
- Isothermal Process – A process in which no change in temperature occurs.
Laws of Thermodynamics
Zeroth Law of Thermodynamics
The Zeroth law of thermodynamics states that if two bodies are individually in equilibrium with a separate third body, then the first two bodies are also in thermal equilibrium with each other.
First Law of Thermodynamics
The First law of thermodynamics is a version of the law of conservation of energy, adapted for thermodynamic processes, distinguishing three kinds of transfer of energy, as heat, as thermodynamic work, and as energy associated with matter transfer, and relating them to a function of a body's state, called internal energy.
Second Law of Thermodynamics
The Second law of thermodynamics is a physical law of thermodynamics about heat and loss in its conversion.
Third Law of Thermodynamics
Third law of thermodynamics states, regarding the properties of closed systems in thermodynamic equilibrium: The entropy of a system approaches a constant value when its temperature approaches absolute zero.