Let I1 be the moment of inertia about the centre of mass of a thick asymmetrical body. Let I2 be the moment of inertia about an axis parallel to I1. The distance between the two axes is ‘a’ & the mass of the body is ‘m’. Find the relation between I1 & I2.

According to parallel axis theorem, the moment of inertia of a section about an axis parallel to the axis through center of gravity (i.e. $$I$$P) is given by (where, A = Area of the section, $$I$$G = Moment of inertia of the section about an axis passing through its C.G. and h = Distance between C.G. and the parallel axis.)

A

$$I{\text{P}} = I{\text{G}} + {\text{A}}{{\text{h}}^2}$$

B

$$I{\text{P}} = I{\text{G}} - {\text{A}}{{\text{h}}^2}$$

C

$$I{\text{P}} = \frac{{I{\text{G}}}}{{{\text{A}}{{\text{h}}^2}}}$$

D

$$I{\text{P}} = \frac{{{\text{A}}{{\text{h}}^2}}}{{I{\text{G}}}}$$

According to parallel axis theorem, the moment of inertia of a section about an axis parallel to the axis through center of gravity (i.e. $${I_{\text{P}}}$$) is given by (where, A = Area of the section, $${I_{\text{G}}}$$ = Moment of inertia of the section about an axis passing through its C.G. and h = Distance between C.G. and the parallel axis.)

A

$${I_{\text{P}}} = {I_{\text{G}}} + {\text{A}}{{\text{h}}^2}$$

B

$${I_{\text{P}}} = {I_{\text{G}}} - {\text{A}}{{\text{h}}^2}$$

C

$${I_{\text{P}}} = \frac{{{I_{\text{G}}}}}{{{\text{A}}{{\text{h}}^2}}}$$

D

$${I_{\text{P}}} = \frac{{{\text{A}}{{\text{h}}^2}}}{{{I_{\text{G}}}}}$$

The moment of inertia of a body about an axis passing through the center of mass is Icm. If the mass of the body is m, find the moment of inertia about an axis parallel to the center of mass and at a distance of R from it.

A

I = Icm = MR2

B

I = I_cm + MR²

C

I = Icm × MR2

D

I = Icm - MR2

The moment of inertia of a uniform sphere of radius, r about an axis passing through its centre is given by $$\frac{2}{5}\left( {\frac{{4\pi }}{3}{r^5}\rho } \right).$$ A rigid sphere of uniform mass density $$\rho $$ and radius R has two smaller spheres of radii $$\frac{R}{2}$$ hollowed out of it as shown in the figure given below.
Classical Mechanics mcq question image

The moment of inertia of the resulting body about Y-axis is

A

$$\frac{{\pi \rho {R^5}}}{4}$$

B

$$\frac{{5\pi \rho {R^5}}}{{12}}$$

C

$$\frac{{7\pi \rho {R^5}}}{{12}}$$

D

$$\frac{{3\pi \rho {R^5}}}{4}$$

A spherical body is symmetrical about its perpendicular axis. According to Routh's rule, the moment of inertia of a body about an axis passing through its centre of gravity is (where, M = Mass of the body and S = Sum of the squares of the two semi-axes.)

A

$$\frac{{{\text{MS}}}}{3}$$

B

$$\frac{{{\text{MS}}}}{4}$$

C

$$\frac{{{\text{MS}}}}{5}$$

D

None of these

In order to have a connecting rod equally strong in buckling about X-axis and Y-axis, the moment of inertia about X-axis should be __________ the moment of inertia about Y-axis.

A

Equal to

B

Twice

C

Three times

D

Four times

The parallel axis theorem can add any angle varied moment of inertias to give the perpendicular moment of inertia and it can be used in the determination of the moment of inertia about inclined axis.

A

True

B

False

Mohan traveled five cities (A, B, C, D and E) starting from home by a car which has a mileage such that the car can travel 10 km in 1 liter petrol. Distance between city A and B is twice the distance between home to city A. Distance between city B and C is thrice the distance between home to city A. Distance between city C and D is 100. Distance between city D and E is 1.5 times than the distance between B and C. Average distance traveled by Mohan was 41 km between five cities. Find how much petrol was needed by Mohan to travel from city B to D.

A

12

B

13

C

10

D

15

Two solid spheres of radius R and mass M each are connected by a thin rigid rod of negligible mass. The distance between the centre is 4R. The moment of inertia about an axis passing through the centre of symmetry and perpendicular to the line joining the sphere is

A

$$\frac{{11}}{5}M{R^2}$$

B

$$\frac{{22}}{5}M{R^2}$$

C

$$\frac{{44}}{5}M{R^2}$$

D

$$\frac{{88}}{5}M{R^2}$$

Three consecutive absorption lines at 64.275 cm^-1, 77.130 cm^-12 and 89.985 cm^-1 have been observed in a microwave spectrum for a linear rigid diatomic molecule. The moments of inertia $${I_A}$$ and $${I_B}$$ are ($${I_A}$$ is with respect to the bond axis passing through the centre of mass and $${I_B}$$ is with respect to an axis passing through the centre of mass and perpendicular to bond axis)

A

both equal to $$\frac{{{\hbar ^2}}}{{12.855\,hc}}$$ g-cm²

B

zero and $$\frac{{{\hbar ^2}}}{{12.855\,hc}}$$ g-cm²

C

both equal to $$\frac{{{\hbar ^2}}}{{6.427\,hc}}$$ g-cm²

D

zero and $$\frac{{{\hbar ^2}}}{{6.427\,hc}}$$ g-cm²

Let I1 be the moment of inertia about the centre of mass of a thick asymmetrical body. Let I2 be the moment of inertia about an axis parallel to I1. The distance between the two axes is ‘a’ & the mass of the body is ‘m’. Find the relation between I1 & I2.

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