A particle moves according to the law \(x= r\cos\left(\frac{\pi t}{2}\right)\). The distance covered by it in the time interval between \(t=0\) to \(t=3~\text{s}\) will be:

1.	\(r\)	2.	\(2r\)
3.	\(3r\)	4.	\(4r\)

The acceleration ${\mathrm{d}}^2\mathrm{x}/{\mathrm{dt}}^2$ of a particle varies with displacement x as $\frac{d^{2}x}{d{t}^2}=-kx$

where k is a constant of the motion. The time period T of the motion is equal to:

1. $2\mathrm{\pi k}$

2. $2\mathrm{\pi }\sqrt{\mathrm{k}}$

3. $2\mathrm{\pi }/\sqrt{\mathrm{k}}$

4. $2\mathrm{\pi }/\mathrm{k}$

The acceleration of a particle performing SHM is $12 {\mathrm{cm}}^{-2}$ at a distance of 3cm from the mean position. Its time period is:

1. 2.0s

2. 3.14s

3. 0.5s

4. 1.0s

A particle is performing simple harmonic motion along the x-axis with amplitude 4cm and time period 1.2 s. The minimum time taken by the particle to move from x=+2 to x=4cm and back again is given by:

1. 0.4s

2. 0.3s

3. 0.2s

4. 0.6s

This time period of a particle undergoing SHM is 16s. It starts motion from the mean position. After 2s, velocity is $0.4 {\mathrm{ms}}^{-1}$. The amplitude is:

1. 1.44m

2. 0.72m

3. 2.88m

4. 0.36m

Motion of an oscillating liquid column in a U-tube is:

1. periodic but not simple harmonic

2. non-periodic

3. simple harmonic and time period is independent of the density of the liquid

4. simple harmonic and time-period is directly proportional to the density of the liquid

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/$s^2$

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

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

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