First slide

Change in internal energy

Question

The following four wires are made of the same material. Which of these will have the largest extension when the same tension is applied?

Easy

A

length= 200 cm, diameter= 2 mm
B

length= 300 cm, diameter= 3 mm
C

length= 50 cm, diameter= 0.5 mm
D

length= 100 cm, diameter= 1 mm

Solution

Young s modulus, $Y = \frac{F L}{A Δ L} = \frac{4 F L}{π D^{2} Δ L} o r Δ L = \frac{4 F L}{π D^{2} Y}$
where F is the force applied, L is the length, D is the diameter and $Δ L$ is the extension of the wire respectively.
As each wire is made up of same material therefore their Young's modulus is same for each wire.
For all the four wires, Y, F (= tension) are the same.
$∴ Δ L \propto \frac{L}{D^{2}} I n (a) \frac{L}{D^{2}} = \frac{200 c m}{{(0.2 c m)}^{2}} = 5 \times 10^{3} c m^{- 1} I n (b) \frac{L}{D^{2}} = \frac{300 c m}{{(0.3 c m)}^{2}} = 3.3 \times 10^{3} c m^{- 1} I n (c) \frac{L}{D^{2}} = \frac{50 c m}{{(0.05 c m)}^{2}} = 20 \times 10^{3} c m^{- 1} I n (d) \frac{L}{D^{2}} = \frac{100 c m}{{(0.1 c m)}^{2}} = 10 \times 10^{3} c m^{- 1}$
Hence, $Δ L$ is maximum in (c).

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