Question

A rod of length 2m at 0°C and having expansion coefficient alpha= (3x+2) *10^-6 where X is the distance in cm from one end of the rod. What will the length of the rod at 20°C be?​

Answer 1

Answer:

Final length = 2.0002 m

Explanation:

Coefficient of linear expansion α is defined as the fractional change in length per unit change in temperature.

Consider an element dx of the rod of length Lat a distance x from one of its end. We are interested to find what change in length (δ) of this element will take place due to change in temperature ΔT

By definition of temperature coefficient of linear expansion, we have for this element,

α=δdxΔT

Or, δ=ΔTαdx

The total change in length (ΔL)

in the rod can then be found by integrating this with proper limits

ΔL=∫L0ΔTαdx

If α=(ax+b)

then

ΔL=∫_0^L▒〖ΔT(ax+b)〗 dx

=LΔT(aL/2+b)

Now to find the change, let us put values;

A=3, b=2, L=2, ΔT= 20

= 2(20)[(3x2)/2 + 2]*10^-6

ΔL= 0.0002

Final length = 2 + 0.0002

Final length = 2.0002 m