A uniform solid cylinder with radius R and le | vedclass

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

Question

(C) Let $\rho$ be the density of the material of the cylinder.The mass of the original cylinder is $m_1 = \rho \cdot \pi R^2 L$.The moment of inertia

Vedclass · Accepted Answer

$32$

Vedclass · Answer

(C) Let $\rho$ be the density of the material of the cylinder.The mass of the original cylinder is $m_1 = \rho \cdot \pi R^2 L$.The moment of inertia of the original cylinder about its axis is $I_1 = \frac{1}{2} m_1 R^2 = \frac{1}{2} (\rho \pi R^2 L) R^2 = \frac{1}{2} \rho \pi R^4 L$.The mass of the carved-out cylinder is $m_2 = \rho \cdot \pi (R')^2 L' = \rho \cdot \pi (\frac{R}{2})^2 (\frac{L}{2}) = \rho \cdot \pi \frac{R^2}{4} \cdot \frac{L}{2} = \frac{1}{8} \rho \pi R^2 L$.The moment of inertia of the carved-out cylinder about its axis is $I_2 = \frac{1}{2} m_2 (R')^2 = \frac{1}{2} (\frac{1}{8} \rho \pi R^2 L) (\frac{R}{2})^2 = \frac{1}{2} \cdot \frac{1}{8} \rho \pi R^2 L \cdot \frac{R^2}{4} = \frac{1}{64} \rho \pi R^4 L$.Therefore,the ratio is $\frac{I_1}{I_2} = \frac{\frac{1}{2} \rho \pi R^4 L}{\frac{1}{64} \rho \pi R^4 L} = \frac{64}{2} = 32$.

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?

$A$ uniform square plate $S$ (side $c$) and a uniform rectangular plate $R$ (sides $b$,$a$) $(a > b)$ have identical areas and masses. Based on the figure,which of the following is correct?
$(a)\ I_{xR}/I_{xS} < 1$
$(b)\ I_{yR}/I_{yS} > 1$
$(c)\ I_{zR}/I_{zS} > 1$

The moment of inertia of a semicircular plate of radius $R$ and mass $M$ about an axis $AA'$ in its plane passing through its centre is given by:

The moment of inertia of a ring of mass $M$ and radius $R$ about an axis passing through its center and perpendicular to its plane is:

Two rings of the same material have radii $R$ and $nR$ respectively. The ratio of their moments of inertia about an axis passing through their centers and perpendicular to their planes is $1 : 8$. The value of $n$ is:

Vedclass Test Series

Exam Paper Generator

Online Exam Module