The amplitude of velocity of a particle is given by, ${\mathrm{V}}_{\mathrm{m}}={\mathrm{V}}_0/\left({\mathrm{a\omega }}^2-\mathrm{b\omega }+\mathrm{c}\right)$ where ${\mathrm{V}}_0,\mathrm{a},\mathrm{b},\mathrm{c}$ are positive:

The condition for a single resonant frequency is 

1. ${\mathrm{b}}^2<4\mathrm{ac}$

2. ${\mathrm{b}}^2=4\mathrm{ac}$

3. ${\mathrm{b}}^2=5\mathrm{ac}$

4. ${\mathrm{b}}^2=7\mathrm{ac}$

Two pendulums have time periods T and 5T/4. They start SHM at the same time from the mean position. What will be the phase difference between them after the bigger pendulum completed one oscillation?

1. $45°$

2. $90°$

3. $60°$

4. $30°$

A simple harmonic oscillator has amplitude 'a' and time period T. The time required by it to travel from x=a to $\mathrm{x}=\frac{\mathrm{a}}{2}$ is

1. $\frac{\mathrm{T}}{6}$

2. $\frac{\mathrm{T}}{4}$

3. $\frac{\mathrm{T}}{3}$

4. $\frac{\mathrm{T}}{2}$

The piston in the cylinder head of a locomotive has a stroke (twice the amplitude) of 1.0 m. If the piston moves with simple harmonic motion with an angular frequency of 200 rad/min, what is the maximum speed?

1. 100 m/min

2. 200 m/min

3. 300 m/min

4. 50 m/min

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 more than its true weight on the plank when the latter moves fastest

A body has a time period ${\mathrm{T}}_1$ under the action of one force and ${\mathrm{T}}_2$ under the action of another force, the square of the time period when both the forces are acting simultaneously in the same direction is

1. ${\mathrm{T}}_1^2{\mathrm{T}}_2^2$

2. $2{\mathrm{T}}_1^2{\mathrm{T}}_2^2$

3. ${\mathrm{T}}_1^2+{\mathrm{T}}_2^2$

4. ${\mathrm{T}}_1^2{\mathrm{T}}_2^2/({\mathrm{T}}_1^2+{\mathrm{T}}_2^2)$

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

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

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

1. 0.4 sec

2. 0.3 sec

3. 0.2 sec

4. 0.6 sec

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

1. 2.0s

2. 3.14s

3. 0.5s

4. 1.0s