A rod of length 2 m at 0^circ C has a linear | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(C) The length of the rod is $L = 2 \ m = 200 \ cm$. The coefficient of linear expansion is $\alpha(x) = (3x + 2) 	imes 10^{-6} \ ^\circ C^{-1}$.For

Vedclass · Accepted Answer

$2.01$

Vedclass · Answer

(C) The length of the rod is $L = 2 \ m = 200 \ cm$. The coefficient of linear expansion is $\alpha(x) = (3x + 2) 	imes 10^{-6} \ ^\circ C^{-1}$.For a small element $dx$ at distance $x$,the change in length $dL$ for a temperature change $\Delta T = 20^\circ C$ is given by $dL = \alpha(x) \cdot dx \cdot \Delta T$.Integrating this from $x = 0$ to $x = 200 \ cm$:$\Delta L = \int_{0}^{200} (3x + 2) 	imes 10^{-6} \cdot 20 \cdot dx$$\Delta L = 20 	imes 10^{-6} \int_{0}^{200} (3x + 2) \cdot dx$$\Delta L = 20 	imes 10^{-6} [\frac{3x^2}{2} + 2x]_{0}^{200}$$\Delta L = 20 	imes 10^{-6} [\frac{3(200)^2}{2} + 2(200)]$$\Delta L = 20 	imes 10^{-6} [60000 + 400] = 20 	imes 10^{-6} 	imes 60400 = 1.208 \ cm$.Since $1.208 \ cm = 0.01208 \ m$,the new length is $L' = L + \Delta L = 2 + 0.01208 = 2.01208 \ m \approx 2.01 \ m$.

$A$ rod of length $2 \ m$ at $0^\circ C$ has a linear expansion coefficient $\alpha = (3x + 2) \times 10^{-6} \ ^\circ C^{-1}$,where $x$ is the distance (in $cm$) from one end of the rod. Find the length of the rod at $20^\circ C$ in meters.

Similar Questions

$A$ hole is drilled in a copper sheet. The diameter of the hole is $4.24 \,cm$ at $27.0^{\circ} C$. What is the diameter of the hole when it is heated to $35^{\circ} C$?

$A$ solid ball of metal has a concentric spherical cavity within it. If the ball is heated,the volume of the cavity will

$A$ solid metallic cube having a total surface area of $24 \; m^{2}$ is uniformly heated. If its temperature is increased by $10 \; ^{\circ}C$,calculate the increase in volume of the cube. (Given: $\alpha = 5.0 \times 10^{-4} \; ^{\circ}C^{-1}$)

$A$ solid rectangular sheet has two different coefficients of linear expansion $\alpha_{1}$ and $\alpha_{2}$ along its length and breadth respectively. The coefficient of surface expansion is (for $\alpha_{1} \Delta t \ll 1, \alpha_{2} \Delta t \ll 1$):

$A$ solid expands upon heating because

Vedclass Test Series

Exam Paper Generator

Online Exam Module