Oscillations · Question 3

A cylindrical cork of uniform density floats in a liquid of density 1. If the cork is depressed slightly and released, it oscillates harmonically with time period T. If the same cork floats in another liquid of density 2, then the similar oscillation has time period 2T. The value of 2 / 1 is:

1. 1/4
2. 4
3. 2
4. 1/2

Consider a spring-mass simple harmonic oscillator in one dimension. The mass of the particle is m kg and the spring constant is k Nm-1. At a given instant, the extension of the spring is x-meter and the speed of the particle is v ms-1. On the x - v plane, if the graph of v as a function of x is a circle, then the correct option is:

1. k = m
2. k = 1m
3. k = m
4. k = m2

In an oscillating spring mass system, a spring is connected to a box filled with sand. As the box oscillates, sand leaks slowly out of the box vertically so that the average frequency (t) and average amplitude A(t) of the system change with time t. Which one of the following options schematically depicts these changes correctly?

1. [figure]
2. [figure]
3. [figure]
4. [figure]

Two identical point masses P and Q, suspended from two separate massless springs of spring constants k1 and k2, respectively, oscillate vertically. If their maximum speeds are the same, the ratio (AQ/AP) of the amplitude AQ of mass Q to the amplitude AP of mass P is

1. k2k1
2. k1k2
3. k2k1
4. k1k2

If x = 5 ( t + 3 ) m represents the motion of a particle executing simple harmonic motion, the amplitude and time period of motion, respectively, are

1. 5 cm, 2 s
2. 5 m, 2 s
3. 5 cm, 1 s
4. 5 m, 1 s

If the mass of the bob in a simple pendulum is increased to thrice its original mass and its length is made half its original length, then the new time period of oscillation is x2 times its original time period. Then the value of x is:

1. 3
2. 2
3. 2 3
4. 4

A particle executing simple harmonic motion with amplitude A has the same potential and kinetic energies at the displacement

1. 2 A
2. A2
3. A 2
4. A 2

The two-dimensional motion of a particle, described by r = ( i + 2 j)A t is a/an: Choose the correct answer from the options given below:

1. B, C and D only
2. A, B and C only
3. A, C and D only
4. C and D only

The x-t graph of a particle performing simple harmonic motion is shown in the figure. The acceleration of the particle at t = 2 s is : [figure]

1. - 28 m s-2
2. 216 m s-2
3. - 216 m s-2
4. 28 m s-2

Two pendulums of length 121 cm and 100 cm start vibrating in phase. At some instant, the two are at their mean position in the same phase. The minimum number of vibrations of the shorter pendulum after which the two are again in phase at the mean position is :

1. 11
2. 9
3. 10
4. 8