Heating A Metallic Scale
A metallic scale (linear) expands in length when heated. As a result all the paging is displaced from their usual (correct) positions.
A reading of l point on a heated scale is equivalent to an actual length of l (1+aΔt), where is coefficient of linear expansion of material of scale, and Δt is the temperature of the heated scale.
If the reading is x, actual length
actual length = reading (1+aΔt)
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![375_Error in mesurement due to thermal expansion.png](https://www.expertsmind.com/CMSImages/375_Error%20in%20mesurement%20due%20to%20thermal%20expansion.png)
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Diffrence of length of two rods:
Consider two rods 1 and 2 of length l1 and l2 . Lest they be heated through a temperature . If l′1 and l′2 are their larger lengths, then:
l′2 = l2 (1+a2Δt)
l′1=l1 (1+a1Δt)
Since the difference of size of two rods is constant
l′2 -l′1 = l2 -l1 and l1a1=l2a2
Time peried of Pendulum:
Time period (T) of a simple pendulum of length l is given by T= 2π√(l/g)
If there is a rise in temperature by Δt, length of the pendulum increases and hence the time period increase. As a given result the clock slows down.
If lo be the length of the pendulum and corresponding time period be T0 then
To = 2π√(lo/g)
If the pendulum be heated by Δt (rise in temperature), the new time section T1 is:
![651_Error in mesurement due to thermal expansion1.png](https://www.expertsmind.com/CMSImages/651_Error%20in%20mesurement%20due%20to%20thermal%20expansion1.png)
as α is very small ⇒ ΔT/To = 1/2αΔt
The above relation gives the time lost per second by the pendulum clock. If Δt is the fall in temperature, same equation will provide the time gained by the clock per second as its oscillation will become faster due to reduction in its length.
Stress in Objects:
If cross-sectional area of wire is A and F be the tension developed in wire due to stretching. Thus the stress created in the wire due to this tension F is given as
Stress = Force/Area= F / A ... (i)
Strain given in wire due to its elastic properties is Strain = ΔL/L = αΔT ... (ii)
If young's modulus of the material of wire is Y, we require
Υ = Stress/strain or Υ = (F/A)/αΔT or F = ΥAαΔT ... (iii)
Equation - (iii) gives the expression for tension in the wire due to decrease in its temperature by ΔT. This result gives the tension in wire if initially wire is just taught between rods. If it already has some force in it then this expression will give the increment in tension in the wire.
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