1

### GATE ME 2004

A uniform stiff rod of length $300$ $mm$ and having a weight of $300$ $N$ is pivoted at one end and connected to a spring at the other end. For keeping the rod vertical in a stable position the minimum value of spring constant $K$ needed is
A
$300$ $N/m$
B
$400$ $N/m$
C
$500$ $N/m$
D
$1000$ $N/m$
2

### GATE ME 2004

A mass M, of 20 kg is attached to the free end of a steel cantilever beam of length 1000 mm having a cross-section of 25 $\, \times \,$ 25 mm. Assume the mass of the cantilever to be negligible and Esteel = 200 Gpa. If the lateral vibration of this system is critically damped using a viscous damper, the damping constant of the damper is
A
1250 Ns/m
B
625 Ns/m
C
312.50 Ns/m
D
156.25 Ns/m
3

### GATE ME 2003

A flexible rotor-shaft system comprises of a $10kg$ rotor disc placed in the middle of a mass-less shaft of diameter $30$ $mm$ and length $500mm$ between bearings (shaft is being taken mass-less as the equivalent mass of the shaft is included in the rotor mass) mounted at the ends. The bearings are assumed to simulate simply supported boundary conditions. The shaft is made of steel for which the value of $E$ is $\,2.1\, \times \,{10^{11}}\,$. Pa. What is the critical speed of rotation of the shaft?
A
$60$ $Hz$
B
$90$ $Hz$
C
$135$ $Hz$
D
$180$ $Hz$
4

### GATE ME 2003

A uniform rigid slender bar of mass $10$ $kg.$ is hinged at the left end is suspended with the help of spring and damper arrangement as shown in the figure where $K = 2kN/m,$ $C =500Ns/m$ and the stiffness of the torsional spring ${K_\theta }$ is 1 kN/m/rad. Ignore the hinge dimensions.

The un-damped natural frequency of oscillations of the bar about the hinge point is

A
$42.43$ rad/s
B
$30$ rad/s
C
$17.32$ rad/s
D
$14.14$ rad/s

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