What is the moment of inertia of a rod about an axis passing through the centre and perpendicular to its central axis? Given that mass of rod is 1kg, length = 10cm.

A hollow rod has frictionless inner walls. Inside the rod are two small spheres that are kept on either side of the centre of rod. The rod is rotated about its central axis which is perpendicular to the plane of rotation. If the rod had been rotated by an impulsive torque which gave it an instantaneous angular velocity of 5rad/s, what will be the angular velocity after some finite but long time ‘t’? Assume no external forces act on rod after the impulse. Also state what will happen to the spheres in the centre of the rod. The rod has a mass =2kg & length =10cm. The two spheres have a mass =1kg each.

A

2.5 rad/s, spheres will stay where they are as there is no friction to move them

B

0 rad/s, spheres will move outwards and decrease velocity of rod to zero

C

1.25 rad/s, spheres will move to opposite ends of rod

D

2.5 rad/s, spheres will move to opposite ends of rod

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.

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 particle of mass m is attached to a thin uniform rod of length a and mass 4m. The distance of the particle from the centre of mass of the rod is $$\frac{a}{2}.$$

The moment of inertia of the combination about an axis passing through a normal to the rod is

A

$$\frac{{64}}{{48}}m{a^2}$$

B

$$\frac{{91}}{{48}}m{a^2}$$

C

$$\frac{{27}}{{48}}m{a^2}$$

D

$$\frac{{51}}{{48}}m{a^2}$$

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

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

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

For a steam engine, the following data is given: Piston diameter = 0.24 m, length of stroke = 0.6 m, length of connecting rod = 1.5 m, mass of reciprocating parts = 300 kg, mass of connecting rod = 250 kg; speed of rotation = 125 r.p.m ; centre of gravity of connecting rod from crank pin = 0.5 m ; Kg of the connecting rod about an axis through the centre of gravity = 0.65 m calculate inertia force at θ=30 degrees from IDC.

A

19000 N

B

19064 N

C

19032 N

D

20064 N

Piston diameter = 0.24 m, length of stroke = 0.6 m, length of connecting rod = 1.5 m, mass of reciprocating parts = 300 kg, mass of connecting rod = 250 kg; speed of rotation = 125 r.p.m; centre of gravity of connecting rod from crank pin = 0.5 m ; Kg of the connecting rod about an axis through the centre of gravity = 0.65 m Find the equivalent length L of a simple pendulum swung about an axis.

A

1.35 m

B

1.42 m

C

1.48 m

D

1.50 m

A small connecting rod of mass 1.5 kg is suspended in a horizontal plane by two wires 1.25 m long. The wires are attached to the rod at points 120 mm on either side of the centre of gravity. If the rod makes 20 oscillations in 40 seconds, find the mass moment of inertia of the rod about a vertical axis through the centre of gravity.

A

0.014 kg-m2

B

0.015 kg-m2

C

0.016 kg-m2

D

0.017 kg-m2

Piston diameter = 0.24 m, length of stroke = 0.6 m, length of connecting rod = 1.5 m, mass of reciprocating parts = 300 kg, mass of connecting rod = 250 kg; speed of rotation = 125 r.p.m; centre of gravity of connecting rod from crank pin = 0.5 m ; Kg of the connecting rod about an axis through the centre of gravity = 0.65 m Find angular acceleration of connecting rod in rad/s2.

A

16.782

B

17.824

C

15.142

D

17.161

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

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