A uniform rod of length L has a mass per unit | vedclass

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

Question

(B) Consider a small element of length $dx$ at a distance $x$ from the lower end of the rod.The weight of the portion of the rod below this element is

Vedclass · Accepted Answer

$\frac{\lambda g L^2}{2 A Y}$

Vedclass · Answer

(B) Consider a small element of length $dx$ at a distance $x$ from the lower end of the rod.The weight of the portion of the rod below this element is $W(x) = (\lambda x)g$.This weight acts as a tensile force $F = \lambda x g$ on the element $dx$.The elongation $d(\Delta L)$ in the element $dx$ is given by $d(\Delta L) = \frac{F dx}{A Y} = \frac{(\lambda x g) dx}{A Y}$.To find the total elongation $\Delta L$,we integrate from $x = 0$ to $x = L$:$\Delta L = \int_{0}^{L} \frac{\lambda g x}{A Y} dx = \frac{\lambda g}{A Y} \int_{0}^{L} x dx = \frac{\lambda g}{A Y} \left[ \frac{x^2}{2} \right]_{0}^{L} = \frac{\lambda g L^2}{2 A Y}$.

$A$ uniform rod of length $L$ has a mass per unit length $\lambda$ and area of cross-section $A$. If the Young's modulus of the rod is $Y$,then the elongation in the rod due to its own weight is:

Similar Questions

Two wires of the same material (Young's modulus $Y$) and same length $L$ but radii $R$ and $2R$ respectively are joined end to end and a weight $W$ is suspended from the combination as shown in the figure. The elastic potential energy in the system is

$A$ metal rod has length,cross-sectional area,and Young's modulus as $L$,$A$,and $Y$ respectively. If the elongation produced in the rod is $\ell$,then the work done is proportional to:

If the work done in stretching a wire by $1 \ mm$ is $2 \ J$,the work necessary for stretching another wire of same material but with double radius of cross-section and half the length by $1 \ mm$ is $.... \ J$.

Two wires of the same material and length but diameters in the ratio $1:2$ are stretched by the same force. The elastic potential energy per unit volume for the wires,when stretched by the same force,will be in the ratio: (in $:1$)

An elastic material of Young's modulus $Y$ is subjected to a stress $S$. The elastic energy stored per unit volume of the material is

Vedclass Test Series

Exam Paper Generator

Online Exam Module