A cylindrical piston of mass M slides smoothly inside a long cylinder closed at one end, enclosing a certain mass of gas. The cylinder is kept with its axis horizontal. If the piston is disturbed from its equilibrium position, it oscillates simple harmonically. The period of oscillation will be

 

(1) T=2πMhPA

(2) T=2πMAPh

(3) T=2πMPAh

(4) T=2πMPhA

Subtopic:  Simple Harmonic Motion |
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The metallic bob of a simple pendulum has the relative density ρ. The time period of this pendulum is T. If the metallic bob is immersed in water, then the new time period is given by

(1) Tρ-1ρ                 

(2)     Tρρ-1

(3) Tρ-1ρ               

(4) Tρρ-1

Subtopic:  Simple Harmonic Motion |
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One end of a long metallic wire of length L is tied to the ceiling. The other end is tied to massless spring of spring constant K. A mass m hangs freely from the free end of the spring. The area of cross-section and Young's modulus of the wire is A and Y respectively. If the mass is slightly pulled down and released, it will oscillate with a time period T equal to -

(1)    2πmK                   

(2)     2πYA+KLmYAK1/2

(3)    2πmYAKL               

(4)      2πmLYA

Subtopic:  Combination of Springs |
 72%
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On a smooth inclined plane, a body of mass M is attached between two springs. The other ends of the springs are fixed to firm supports. If each spring has force constant K, the period of oscillation of the body (assuming the springs as massless) is

(a) 2πM2K1/2           (b) 2π2MK1/2

(c) 2πMg sinθ2K           (d) 2π2MgK1/2

                

Subtopic:  Combination of Springs |
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An ideal spring with spring-constant K is hung from the ceiling and a block of mass M is attached to its lower end. The mass is released with the spring initially unstretched. Then the maximum extension in the spring is -

(1) 4 Mg/K         

(2) 2 Mg/K

(3) Mg/K             

(4) Mg/2K

Subtopic:  Combination of Springs |
 56%
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A particle of mass m is attached to three identical springs A, B and C each of force constant k a shown in figure. If the particle of mass m is pushed slightly against the spring A and released then the time period of oscillations is -

(a) 2π2mk          (b) 2πm2k

(c) 2πmk            (d) 2πm3k

              

                 

Subtopic:  Combination of Springs |
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The graph shows the variation of displacement of a particle executing SHM with time. We infer from this graph that:

   
 

1. the force is zero at the time \(T/8\).
2. the velocity is maximum at the time \(T/4\).
3. the acceleration is maximum at the time \(T\).
4. the P.E. is equal to the total energy at the time \(T/4\).

Subtopic:  Energy of SHM |
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For a particle executing SHM the displacement \(x \) is given by, \(A\cos \omega t.\)  Identify the graph which represents the variation of potential energy (P.E.) as a function of time \(t\) and displacement \(x.\)

   
1. I, III
2. II, IV
3. II, III
4. I, IV

Subtopic:  Energy of SHM |
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The velocity-time diagram of a harmonic oscillator is shown in the adjoining figure. The frequency of oscillation is

                                     

(1) 25 Hz           

(2) 50 Hz

(3) 12.25 Hz       

(4) 33.3 Hz

Subtopic:  Simple Harmonic Motion |
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The variation of potential energy of harmonic oscillator is as shown in figure. The spring constant is

                                                      

(1) 1 ×102 N/m               

(2) 150 N/m

(3) 0.667 × 102 N/m       

(4) 3 × 102 N/m

Subtopic:  Energy of SHM |
 63%
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