Linear expansion

Hello students, we know that when heat is provided to substance, most substances gets expanded on heating and contracted on cooling. Due to change in temperature strain is produced in substance which c causes change in its dimensions. The increase in the dimensions of a body due to the increase in its temperature is called thermal expansion.

When heat is given to substance three important types of expansion are produced, here we are going to discuss the expansion related to change in length of substance, i.e. linear expansion.

Let’s understand concept of linear expansion in details….!

Increase in length of substance under action of heat in known as linear expansion.

Consider a thin uniform metal rod of initial length L0 at 0o C. Let that rod is being heated to t0 C, so that its final length becomes Lt. As we know that, here the expansion is along the length of rod, then the increase in length of rod is given as,

∴ increase in length = Lt – L0 ……………………….(1)

We can easily observed that when this increase in length depends upon following factors,

  1. Increase in length is directly proportional to length c i.e. L0
  2. Increase in length is directly proportional to rise in temperature i.e. ∆t

Then mathematically we can write as,

∴ Lt – L0  α  L0

∴L t – L0  α ∆t

∴Lt – L0 = α L0.∆t…………… (2)

Where α is constant of proportionality, here known as coefficient of linear expansion.

Then from above equation we can write as,

Hence ‘the coefficient of linear expansion is defined as the increase in length of substance per unit length at 00 C per unit rise in temperature ∆t’

SI unit of coefficient of linear expansion is per 0C  or /0C

From equation (2) we can get, expression for final length of rod as,

∴ Lt – L0 = α L0.∆t

∴ Lt = L0 + α L0.∆t

∴ Lt = L0 (1 +α .∆t) ……………………….(4)

The above formula is for the final length of material of rod when its temperature is increased by t0 C.

If L1 and L2 be the initial and final lengths of rod when heated from at temperature t10 C to  t20 C, then we can write,

∴ L2 = L1 [1 + α .(t2 – t1)]

Lets solve some numericals…..!

Ex:1) Length of metal rod is increased by 5 mm when heated from 600 C to  1500 C. If the coefficient of linear expansion of material of rod is 2.5 × 10-4 /0C, find the length of rod at 600 C.

Solution:

Here, L2 – L1 =5 mm=5 × 10-3 m, t1 = 600 C , t2 = 1200 C,

∴∆t = t2 – t1 = 600 C

α= 2.5 × 10-4  /0C

We have,

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