A thin wire of length L and uniform linear ma | vedclass

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

Question

(D) The linear mass density of the wire is $\rho$. The total mass of the wire of length $L$ is $M = \rho L$. Since the wire of length $L$ is bent into

Vedclass · Accepted Answer

$\frac{3\rho L^3}{8\pi^2}$

Vedclass · Answer

(D) The linear mass density of the wire is $\rho$. The total mass of the wire of length $L$ is $M = \rho L$. Since the wire of length $L$ is bent into a circular loop,its circumference is $2\pi R = L$,which gives the radius $R = \frac{L}{2\pi}$. The moment of inertia of a circular ring about its diameter is $I_{diam} = \frac{1}{2}MR^2$. Using the parallel axis theorem,the moment of inertia about the axis $XX'$ (which is tangent to the loop and parallel to the diameter) is $I = I_{cm} + MR^2$,where $I_{cm} = I_{diam} = \frac{1}{2}MR^2$. Thus,$I = \frac{1}{2}MR^2 + MR^2 = \frac{3}{2}MR^2$. Substituting $M = \rho L$ and $R = \frac{L}{2\pi}$: $I = \frac{3}{2}(\rho L)\left(\frac{L}{2\pi}\right)^2 = \frac{3}{2}\rho L \left(\frac{L^2}{4\pi^2}\right) = \frac{3\rho L^3}{8\pi^2}$.

$A$ thin wire of length $L$ and uniform linear mass density $\rho$ is bent into a circular loop (as shown in the figure) with center $O$. The moment of inertia of the loop about the axis $XX'$ is:

Similar Questions

In which of the following reactions,the concentration of product is higher than the concentration of reactant at equilibrium? $(K = \text{equilibrium constant})$

The coordinates of the point $P$ on the curve $x=a(\theta+\sin \theta), y=a(1-\cos \theta)$,where the tangent is inclined at an angle $\frac{\pi}{4}$ to the $x$-axis,are

The coefficient of $x^3 y^4 z^5$ in the expansion of $(x y+y z+x z)^6$ is

Four particles,each of mass $M$,are placed at the corners of a square of side $L$. The radius of gyration of the system about an axis perpendicular to the square and passing through its centre is:

The radius of which of the following orbits is the same as that of the $1^{st}$ orbit of $H$ atom?

Vedclass Test Series

Exam Paper Generator

Online Exam Module