A rod has a variable coefficient of linear expansion α =(x/5000), where x is the distance from one end of the rod. If the length of the rod is 1m then find the increase in the length of the rod (in cm ) on increasing the temperature of the rod by 100° C.

Question

A rod has a variable coefficient of linear expansion α =(x/5000), where x is the distance from one end of the rod. If the length of the rod is 1m then find the increase in the length of the rod (in cm ) on increasing the temperature of the rod by 100° C. (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

Increase in length (d x)=α (d x)Δ T (d l)=(x/5000)× (d x)× 100=(x/50)dx ∴ total expansion, Δ l= displaystyle ∫ 01(x/50)dx=(1/100)m=1cm

Q. A rod has a variable coefficient of linear expansion $α = \frac{x}{5000},$ where $x$ is the distance from one end of the rod. If the length of the rod is $1 m$ then find the increase in the length of the rod (in $c m$ ) on increasing the temperature of the rod by $10 0^{\circ} C .$

Increase in length $(d x) = α (d x) Δ T$
$(d l) = \frac{x}{5000} \times (d x) \times 100 = \frac{x}{50} d x$
$∴$ total expansion, $Δ l = \int_{0}^{1} \frac{x}{50} d x = \frac{1}{100} m = 1 c m$

Q. A rod has a variable coefficient of linear expansion α=5000x​, where x is the distance from one end of the rod. If the length of the rod is 1m then find the increase in the length of the rod (in cm ) on increasing the temperature of the rod by 100∘C.

Increase in length (dx)=α(dx)ΔT (dl)=5000x​×(dx)×100=50x​dx ∴ total expansion, Δl=∫01​50x​dx=1001​m=1cm

Q. A rod has a variable coefficient of linear expansion $α = \frac{x}{5000},$ where $x$ is the distance from one end of the rod. If the length of the rod is $1 m$ then find the increase in the length of the rod (in $c m$ ) on increasing the temperature of the rod by $10 0^{\circ} C .$

Increase in length $(d x) = α (d x) Δ T$
$(d l) = \frac{x}{5000} \times (d x) \times 100 = \frac{x}{50} d x$
$∴$ total expansion, $Δ l = \int_{0}^{1} \frac{x}{50} d x = \frac{1}{100} m = 1 c m$