Oscillations

Q.

A block of mass 2$M$ is attached to a massless spring with spring constant $k$. This block is connected to two other blocks of masses $M$ and $2M$ using two massless pulleys and strings. The accelerations of the blocks are $a_1,a_2$ and $a_3$ as shown in the figure. The system is released from rest with the spring in its unstretched state. The maximum extension of the spring is $x_0$. Which of the following option(s) is/are correct?
[$g$ is the acceleration due to gravity. Neglect friction]

1. At an extension of $\frac{x_0}{4}$ of the spring. The magnitude of acceleration of the block connected to the spring is $\frac{3g}{10}$

2. $x_0=\frac{4Mg}{k}$

3. When spring achieves an extension of $\frac{x_0}{2}$ for the first time, the speed of the block connected to the spring is $3g\sqrt{\frac{M}{5k}}$

4. $a_2-a_1=a_1-a_3$

Q. A particle executes a SHM of time period T. The time taken by the particle to go directly from its mean position to half the amplitude.

1. 
2. 
3. 
4. 

Q.

The dimension of stopping potential ${𝑉}_0$ in photoelectric effect in units of Planck’s constant$\left(h\right)$, speed of light$\left(c\right)$, and gravitational constant $\left(G\right)$ and Ampere $\left(A\right)$ is

1. $h^{2/3}{c}^{5/3}{G}^{1/3}{A}^{-1}$

2. $h^2{c}^{1/3}{G}^{3/2}{A}^{-1}$

3. $h^0{G}^{-1}{c}^5{A}^{-1}$

4. $h^{-2/3}{c}^{-1/3}{G}^{4/3}{A}^{-1}$

Q. A particle undergoes SHM of period T starting from mean position. The time taken in ${\frac{3}{8}}^{th}$ oscillation is

1. 
2. 
3. 
4. 

Q. A particle is making simple harmonic motion along the $x-\text{axis}$. If at a distance $x_1$ and $x_2$ from the mean position the velocities of the particle are $v_1$ and $v_2$ respectively. The time period of its oscillation is given as :

1. $T=2\pi \sqrt{\frac{x_2^2-x_1^2}{v_1^2+v_2^2}}$
2. $T=2\pi \sqrt{\frac{x_2^2-x_1^2}{v_1^2-v_2^2}}$
3. $T=2\pi \sqrt{\frac{x_2^2+x_1^2}{v_1^2-v_2^2}}$
4. $T=2\pi \sqrt{\frac{x_2^2+x_1^2}{v_1^2+v_2^2}}$

Q.

In a resonance tube experiment when a tuning fork of frequency 500 Hz is sounded over it, the first and second resonances are obtained at 17 cm and 52cm. the velocity of sound is

Ans is 350 m/s.

Please give detailed explanation as I won't be able to understand concise answers.

Q.

A particle executes simple harmonic motion with an amplitude of 10 cm and time period 6 s. At t = 0 it is at position x = 5 cm going towards positive x-direction. Write the equation for the displacement x at time t. Find the magnitude of the acceleration of the particle at t = 4 s.

Q. A swimmer can swim in still water with speed $v$ and the river flowing with velocity $\frac{v}{2}$. To cross the river in shortest distance, he should swim making an angle $\theta$ with the upstream. What is the ratio of the time taken to swim across in the shortest time to that in swimming across over shortest distance ?

1. $\sin \theta$
2. $\cos \theta$
3. $\tan \theta$
4. $\cot \theta$

Q.

Which is a necessary and sufficient condition for simple harmonic motion?

Q. Explain briefly about phasor in angular simple harmonic motion

Q. Spring of spring constant $k$ is cut into $n$ equal parts, out of which are parts $r$ placed in parallel and connected with mass $M$ as shown in figure. The time period of oscillatory motion of mass $M$ is:

1. $T=2\pi \sqrt{\frac{rM}{nK}}$
   
2. $T=2\pi \sqrt{\frac{nM}{rK}}$
3. $T=2\pi \sqrt{\frac{nrM}{K}}$
   
4. $T=2\pi \sqrt{\frac{M}{nrK}}$

Q. A uniform chain of length $l$ and mass $m$ overhangs a smooth table with two third part lying on the table. Find the kinetic energy of the chain as it completely slips off the table.

1. $\frac{5}{9}mgl$
2. $\frac{6}{9}mgl$
3. $\frac{4}{9}mgl$
4. $\frac{8}{9}mgl$

Q.

$1000kHz$ carrier wave is amplitude modulated by the signal frequency$200Hz-4000Hz$. The channel width of this case is

1. $8kHz$

2. $7.6kHz$

3. $3.8kHz$

4. $400kHz$

5. $4kHz$

Q. A stretched wire fixed at both ends is oscillating in the third overtone mode. Equation of the transverse stationary wave produced in this wire is $y=A\sin \left(6\pi x\right)\sin \left(20\pi t\right)$ ($x$ is in $\text{m}$ and $t$ in $\text{seconds}$). Find the length of the wire.

1. $\frac{2}{3}\text{m}$
2. $\frac{3}{2}\text{m}$
3. $\frac{4}{3}\text{m}$
4. $\frac{3}{4}\text{m}$

Q.

Suppose the block of the previous problem is pusshed down the incline with a force of 4 N. How far will the block move in the first two seconds after starting from rest ? The mass of the block is 4 kg.

Q. Two simple pendulum of lengths 1. 44m and 1m start swinging at the Same time . After how time they will be (1)out of phase and (2)in phase again? Take g=10

Q.

Give two examples, where a body possesses potential as well as kinetic energy.

Q.

Position of a body with acceleration a is given by $x=k{a}^m{t}^n$ . Here, $t$ is time. Find the values of $m$ and $n$

1. $m=1,n=1$

2. $m=1,n=2$

3. $m=2,n=1$

4. $m=2,n=2$

Q. Which of the following expressions represents an oscillatory motion ?

1. $F=-2(x-2{)}^3$
2. $F=-2(x-2{)}^2$
3. $F=2(x-2{)}^3$
4. $F=2(x-2{)}^2$

Q. Which of the following statements is incorrect for a particle performing linear SHM?

1. Acceleration is proportional to the displacement from mean position.
2. Acceleration is always directed opposite to the direction of displacement.
3. Velocity of the particle is constant.
4. The net force on the particle is zero at mean position.

Q. A particle executes S.H.M. period of 8 seconds. After what time of its passing through the mean position will the energy be half kinetic and half potential?

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

Q.

Two waves, travelling in the same direction through the same region, have equal frequencies, wavelengths and amplitudes. If the amplitude of each wave is 4 mm and the phase difference between the waves is ${90}^{\circ }$, what is the resultant amplitude ?

Q. A particle starts from mean position $x=0$ and starts moving towards negative extreme position. Find the initial phase constant $\varphi$ of the particle.

1. $\varphi =\frac{\pi }{2}$
2. $\varphi =\pi$
3. $\varphi =0^{\circ }$
4. $\varphi =\frac{\pi }{4}$

Q.

The acceleration of a body moving with initial velocity $u$ changes with distance $x$ as $a=k^2\sqrt{x}$where $k$ is a positive constant. The distance travelled by the body when its velocity becomes $2u$ is

1. ${\left(\frac{3u}{2k}\right)}^{\frac{3}{4}}$

2. ${\left(\frac{3u}{2k}\right)}^{\frac{4}{3}}$

3. ${\left(\frac{3u}{2k}\right)}^{\frac{3}{2}}$

4. ${\left(\frac{3u^2}{2k^2}\right)}^{\frac{2}{3}}$

Q. Which of the following expressions represents an oscillatory motion ?

1. $F=-2(x-2{)}^3$
2. $F=-2(x-2{)}^2$
3. $F=-2x^3$
4. $F=2x^3$

Q. The radii of circular paths of 2 particles of same mass are in ratio 6:8 then what will be velocities ratio if they have a "constant centripetal force"? in the solution of this q why F1 =F2 taken does constant centripetal force mean by this??

Q. Write two differences between centripetal force and centrifugal force.

Q.

Two simple harmonic motions of an angular frequencies 100 and 100radS^-1 have the same amplitude.Find the ratio of their maximum acceleration.

Q. Define Oscillatory motion.

Q.

Choose the correct statement:

1. In amplitude modulation the frequency of the high frequency carrier wave is made to vary in proportion to the amplitude of the audio signal.

2. In amplitude modulation the amplitude of the high frequency carrier wave is made to vary in proportion to the amplitude of the audio signal.

3. In frequency modulation the amplitude of the high frequency carrier wave is made to vary in proportion to the amplitude of the audio signal.

4. In frequency modulation the amplitude of the high frequency carrier wave is made to vary in proportion to the frequency of the audio signal.