The angular frequency of the damped oscillator is given by, $\omega =\sqrt{\frac{k}{m}-\frac{r^2}{4m^2}}$where k is the spring constant, m is the mass of the oscillator and r is the damping constant. If the ratio $r^2/mk$ is 8%, the change in the time period compared to the undamped oscillator is approximately as follows:

(1) increases by 1%

(2) increases by 8%

(3) decreases by 1%

(4) decreases by 8%

A pendulum with a time period of 1s is losing energy due to damping. At a certain time, its energy is 45 J. If after completing 15 oscillations, its energy has become 15 J, its damping constant $(in s^{-1})$ is:

(1) $\frac{1}{30} \ln 3$

(2) $\frac{1}{15} \ln 3$

(3) 2

(4) $\frac{1}{2}$

Two particles are performing simple harmonic motion in a straight line about the same equilibrium point. The amplitude and time period for both particles are the same and equal to A and T, respectively. At time t = 0 one particle has displacement A while the other one has displacement -A/2 and they are moving towards each other. If they cross each other at time t, then t is:

(1) T/6

(2) 5T/6

(3) T/3

(4) T/4

A block of mass 0.1 kg is connected to an elastic spring constant 640 ${\mathrm{Nm}}^{-1}$ and oscillates in a damping medium of damping constant ${10}^{-2} \mathrm{kg} {\mathrm{s}}^{-1}$. The system dissipates its energy gradually. The time taken for its mechanical energy of vibrations to drop to half of its initial value is closest to:

1. 2s

2. 3.5 s

3. 5 s

4. 7 s

 A damped harmonic oscillator has a frequency of 5 oscillations per second. The amplitude drops to half of its value for every 10 oscillations. The time it will take to drop to $\frac{1}{1000}$ of the original amplitude is close to:

1. 100 s

2. 20 s

3. 10 s

4. 50 s

A simple pendulum oscillating in air has time period T. The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is $\frac{1}{16}th$ of the material of the bob. If the bob is inside liquid all the time, its period of oscillation in this liquid is:

1, $4\mathrm{T}\sqrt{\frac{1}{15}}$

2. $2\mathrm{T}\sqrt{\frac{1}{10}}$

3. $4\mathrm{T}\sqrt{\frac{1}{14}}$

4. $2\mathrm{T}\sqrt{\frac{1}{14}}$

The displacement of a damped harmonic oscillator is given by

$\mathrm{x}\left(\mathrm{t}\right)={\mathrm{e}}^{-0.1\mathrm{t}}\cos (10\mathrm{\pi t}+\mathrm{\varphi })$. Here t is in seconds. The time taken for its amplitude of vibrations to drop to half of its initial value is close to:

1. 13 s

2. 7 s

3. 27 s

4. 4 s

Which of the following expression corresponds to simple harmonic motion along a straight line, where x is the displacement and a, b, c are positive constants?

(1) a + bx - c$x^2$

(2) b$x^2$

(3) a - bx + c$x^2$

(4) -bx

A horizontal plank has a rectangular block placed on it. The plank starts oscillating vertically and simple harmonically with an amplitude of 40 cm. The block just loses contact with the plank when the later is momentarily at rest. Then

1. the period of oscillation is $2\mathrm{\pi }/3$

2. the block weighs double its weight when the plank is at one of the positions of momentary at rest.

3. the block weighs 1.5 times its weights on the plank halfway down

4. the block weighs its true weight on the plank when the latter moves fastest

In an experiment for determining the gravitational acceleration g of a place with the help of a simple pendulum, the square of the measured time period is plotted against the string length of the pendulum in the (Fig. 9.18). What is the value of g at the place?

(1) 9.81 m/$s^2$

(2) 9.87 m/$s^2$

(3) 9.91 m/$s^2$

(4) 10.0 m/$s^2$