Question

$\huge\sf{Question:-}$ A rod of metal X of length 50.0cm elongates by 0.10 cm when It is heated from 0°C to 100°C Another rod of metal Y of length 80.0cm enolagates by 0.08cm for the same rise in temperature.A third rod length 50cm made by Weldjng process of rods X and Y placed end to end enolagates by 0.03 cm when Its temperature is rasied from 0° to 50° NOW SOLVE all three questions. 1) The coefficient of linear expansion of metal X and of metal Y are in ratio. ? 2) The length of rod of metal X in composite piece ? 3) The length of metal rod of metal Y im composite piece​

Answer 1

Answer:

1) 2 : 1

2) 10 cm

3) 40 cm

Explanation:

Let α₁ be the coefficient of linear expansion of metal X and α₂ be the coefficient of linear expansion of metal Y

A rod of metal X of length 50.0 cm elongates by 0.10 cm when It is heated from 0°C to 100°C

Initial length, l= 50 cm

Change in length, ∆l = 0.10 cm

Change in temperature, ∆T = 100°C – 0°C = 100°C

We know,

∆l = lα∆T

where α is the coefficient of linear expansion

0.1 = 50 × α₁ × 100

α₁ = 0.1/5000

α₁ = 2 × 10⁻⁵ /°C

Another rod of metal Y of length 80.0cm elongates by 0.08cm for the same rise in temperature

Initial length, l= 80 cm

Change in length, ∆l = 0.08 cm

Change jn temperature, ∆T = 100°C

∆l = lα∆T

0.08 = 80 × α₂ × 100

α₂ = 0.08/8000

α₂ = 1 × 10⁻⁵ /°C

The ratio of coefficient of linear expansion of metal X and of metal Y

= α₁ : α₂

= 2 × 10⁻⁵ /°C : 1 × 10⁻⁵ /°C

= 2 : 1

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A third rod length 50 cm made by Welding process of rods X and Y placed end to end elongates by 0.03 cm when its temperature is raised from 0° to 50°

Let x cm and y cm be the respective lengths of rod X and rod Y in the composite rod.

Total length of rod, x + y = 50 cm

Change in temperature, ∆T= 50° – 0° = 50°

The change in length of the composite rod is equal to the sum of the changes in rods X and Y

Let ∆l be the change in length in the composite rod and ∆l₁ and ∆l₂ are the changes in the rods X and Y

∆l = ∆l₁ + ∆l₂

0.03 = xα₁∆T + yα₂∆T

0.03 = (x × 2 × 10⁻⁵ × 50) + (y × 1 × 10⁻⁵ /°C × 50)

0.03 = 10⁻⁵ × 50 (2x + y)

2x + y = 0.03/10⁻⁵ × 50

2x + y = 300/5

2x + y = 60 cm –(1)

x + y = 50 cm –(2)

Subtract equation (2) from equation (1),

2x + y – (x + y) = 60 – 50

2x + y – x – y = 10

x = 10 cm

=> y = 50 – 10 = 40 cm

Therefore,

The length of rod of metal X in composite piece is 10 cm

The length of metal rod of metal Y in composite piece is 40 cm

Answer 2

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