The natural frequency of free torsional vibrations of a shaft is equal to (where q = Torsional stiffness of the shaft and $$I$$ = Mass moment of inertia of the disc attached at the end of a shaft)

A torsional system with discs of moment of inertia $${I_1}$$ and $${I_2}$$ as shown in the below figure, is gear driven such that the ratio of speed of shaft B to shaft A is 'G'. Neglecting the inertia of gears, the equivalent inertia of disc on shaft B at the speed of shaft A is equal to
Theory of Machine mcq question image

A

$${\text{G}} \times {I_2}$$

B

$${{\text{G}}^2} \times {I_2}$$

C

$$\frac{{{I_2}}}{{\text{G}}}$$

D

$$\frac{{{I_2}}}{{{{\text{G}}^2}}}$$

In a gearing system, pump speed is one third of the motor. Shaft from motor is 6 cm in diameter and 30cm long, the impellar shaft is 10cm diameter and 60cm long. Mass moment of inertia is 1500 Kg-m2, C = 80Gn/m2. Neglecting the inertia of shaft and gears calculate the frequency of free torsional vibrations in Hz.

A

4.7

B

5.7

C

4.5

D

5.5

An undamped spring-mass system with mass m and spring stiffness k is shown in the figure. The natural frequency and natural period of this system are ω rad/s and T s, respectively. If the stiffness of the spring is doubled and the mass is halved, then the natural frequency and the natural period of the modified system, respectively, are

A

2ω rad/s and T/2 s

B

ω/2 rad/s and 2T s

C

4ω rad/s and T/4 s

D

ω rad/s and T s

the given figure, considering left end has one rotor, If the mass moment of inertia of the shaft till node N1 is increased to four times, then what will be the effect on free torsional vibrations of a rotor at left end of N1?

A

Increases 4 times

B

Increases 2 times

C

Decreases 4 times

D

Decreases 2 times

A shaft has an attached disc at the centre of its length. The disc has its centre of gravity located at a distance of 2 mm from the axis of the shaft. When the shaft is allowed to vibrate in its natural bow-shaped mode, it has a frequency of vibration of 10 rad/s. When the shaft is rotated at 300 r.p.m., it will whirl with a radius of

A

2 mm

B

2.22 mm

C

2.50 mm

D

3.0 mm

A flywheel is mounted on a vertical shaft. The both ends of a shaft are fixed and its diameter is 50 mm. The flywheel has a mass of 500 kg, the modulus of rigidity for the shaft material is 80 GN/m2 and its radius of gyration is 0.5 m. Find the natural frequency of torsional vibrations.

A

5.32

B

5.86

C

5.65

D

6.66

In the given figure, considering PN as one shaft, If the mass moment of inertia is increased to four times, then what will be the effect on free torsional vibrations of a rotor at P?

A

Increases 4 times

B

Increases 2 times

C

Decreases 4 times

D

Decreases 2 times

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 $$\frac{a}{2}.$$
Classical Mechanics mcq question image

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

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

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

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

If a disc has moment of inertia I about the axis which is tangential and in plane of the disc. then the moment of inertia about the axis which is tangential and perpendicular to its plane will be

A

$\dfrac{6}{5} I$

B

$\dfrac{3}{4} I$

C

$\dfrac{3}{2} I$

D

$\dfrac{5}{4} I$

The natural frequency of free torsional vibrations of a shaft is equal to (where q = Torsional stiffness of the shaft and $$I$$ = Mass moment of inertia of the disc attached at the end of a shaft)

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