Consider two uniform discs of the same thickn | vedclass

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

Question

(B) The moment of inertia of a uniform disc about its central axis is given by $I = \frac{1}{2} M R^{2}$.Since the discs are made of the same material

Vedclass · Accepted Answer

$ 2 $

Vedclass · Answer

(B) The moment of inertia of a uniform disc about its central axis is given by $I = \frac{1}{2} M R^{2}$.Since the discs are made of the same material and have the same thickness $t$,the mass $M$ can be expressed as $M = \rho V = \rho (\pi R^{2} t)$,where $\rho$ is the density.Substituting this into the formula for $I$,we get $I = \frac{1}{2} (\rho \pi R^{2} t) R^{2} = \frac{1}{2} \rho \pi t R^{4}$.Since $\rho$,$\pi$,and $t$ are constant for both discs,we have $I \propto R^{4}$.Given the ratio $\frac{I_{1}}{I_{2}} = \frac{1}{16}$,we can write $\frac{R_{1}^{4}}{R_{2}^{4}} = \frac{1}{16}$.Taking the fourth root of both sides,we get $\frac{R_{1}}{R_{2}} = \frac{1}{2}$.Given $R_{1} = R$ and $R_{2} = \alpha R$,we have $\frac{R}{\alpha R} = \frac{1}{2}$,which implies $\frac{1}{\alpha} = \frac{1}{2}$.Therefore,$\alpha = 2$.

Consider two uniform discs of the same thickness and different radii $R_{1} = R$ and $R_{2} = \alpha R$ made of the same material. If the ratio of their moments of inertia $I_{1}$ and $I_{2}$,respectively,about their axes is $I_{1} : I_{2} = 1 : 16$,then the value of $\alpha$ is:

Similar Questions

$A$ thin metal rod of mass $M$ and length $L$ is cut into $4$ equal parts by cutting it perpendicular to its length. If the moment of inertia of the rod about an axis passing through its centre and perpendicular to its axis is $I$,then what is the moment of inertia of each part about a similar axis?

On which of the following does the moment of inertia of an object $NOT$ depend?

The moment of inertia of a uniform rectangular plate about an axis passing through its center and parallel to its length $(l)$ is ($b =$ width of the rectangular plate):

If $I_1$,$I_2$,and $I_3$ are the moments of inertia of a solid sphere,a hollow cylinder,and a ring,respectively,all having the same mass and radius,which of the following statements is correct?

Five particles of mass $2 \ kg$ each are attached to the rim of a circular disc of radius $0.1 \ m$ and negligible mass. The moment of inertia of the system about the axis passing through the centre of the disc and perpendicular to its plane is ........ $kg \ m^2$.

Vedclass Test Series

Exam Paper Generator

Online Exam Module