Forced Oscillations

Q. The frequency of vibration of a string depends on the length L between the nodes, the tension F in the string and its mass per unit length m. Guess the expression for its frequency from dimensional analysis.

Q. The nth harmonic of a closed organ pipe is equal to then mth harmonic of an open pipe. The first overtone frequency of the closed organ pipe is also equal to the first overtone frequency of the open organ pipe. Find the value of n if m=6.

Q.

The amplitude of vibration of a particle is given by $a_m=\frac{\left(a_0\right)}{(a{\omega }^2-b\omega +c)}$ where $a_0,a,b$ and c are positive. The condition for a single resonant frequency is

1. b²=4ac

2. b² >4ac

3. b²=7ac

4. b²=5ac

Q.

If vibrations of a string are to be increased by a factor of two, then tension in

the string must be made

[AIIMS 1999; Pb. PET 2000]

1. Half

2. Twice

3. Eight times

4. Four times

Q.

A mass m is suspended from the two coupled springs connected in series. The force constant for springs are $K_1$ and $K_2$. The time period of the suspended mass will be

1. T = $2\pi \sqrt{\left(\frac{m}{K_1+K_2}\right)}$

2. T = $2\pi \sqrt{\left(\frac{m(K_1+K_2)}{K_1{K}_2}\right)}$

3. T = $2\pi \sqrt{\left(\frac{m}{K_1+K_2}\right)}$

4. T = $2\pi \sqrt{\left(\frac{m{K}_1{K}_2}{K_1+K_2}\right)}$

Q.

One end of a massless spring of spring constant 100 N/m and natural length 0.5 m is fixed and the other end is connected to a particle of mass 0.5 kg lying on a frictionless horizontal table. The spring remains horizontal. If the table is made to rotate at an angular velocity of 2 rad/s, find the elongation of the spring.

1. 50 cm

2. 1 cm

3. 51 cm

4. None of these

Q. Does coswt+ sin2wt+ cos4t represent periodic motion? If yes, then find the period (o is anypositive constant).

Q. Sin^2wt- cos^2wt is shm or periodic motion

Q. A vernier scale with }10 divisions slides over a main }{ scale whose pitch is }0.5mm (pitch }=1MSD) . If a }} bob of diameter }9.75mm is held between the jaws, } determine the M.S.R.R and VC.D if } (a) there is no zero error. } (b) the zero error }=0.35mm.

Q. 34. Is y=asin²wt a periodic motion or a S.H.M. ?

Q.

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

1. Proportionality between restoring force and displacement from equilibrium position

2. Constant speed

3. Proportionality between acceleration and displacement

4. Proportionality between force and displacement from mean position

Q. The Bob of a simple pendulum executes SHM in water with a period t, while the period of oscillation of Bob is 2s in air. Neglecting frictional force of water and given that the density of Bob is 9000/8 kg/m^3, the value of t is…

Q. An electric dipole of dipole moment $p$ is placed in a uniform electric field $E$ in stable equilibrium position. Its moment of inertia about the centroidal axis is $I$. If it is displaced slightly from its mean position, find the period of small oscillations.

1. $2\pi \sqrt{\frac{IE}{p}}$
2. $\pi \sqrt{\frac{I}{pE}}$
3. $\frac{1}{2\pi }\sqrt{\frac{pE}{I}}$
4. $2\pi \sqrt{\frac{I}{pE}}$

Q. pipe has length l .the air in it is vibrating in third overtone with maximum amplitude a. find the amplitude at a distance of l/7 from closed end of the pipe.

Q. Amplitude of vibrations remains constant in case of
(i) Free vibrations
(ii) Damped vibrations
(iii) Maintained vibrations
(iv) Forced vibrations

1. $\left(i\right),\left(iii\right)\text{and}\left(iv\right)$
2. $\left(ii\right)\text{and}\left(iii\right)$
3. $\left(i\right),\left(ii\right)\text{and}\left(iii\right)$
4. $\left(ii\right)\text{and}\left(iv\right)$

Q. 283.Two closed organ pipes A and B have the same length and A is wider than B.They resonate in the fundamental mode at frequencies nA and nB respectively

Q. Two identical springs of spring constant k are attached to a block of mass m and to fixed support. Show that when the mass is displaced from its equilibrium position then either side, it executes a simple harmonic motion. Find the period of oscillation.

Q.

Due to some force $F_1$ a body oscillates with period 4/5 sec and due to other force $F_2$ oscillates with period 3/5 sec. If both forces act simultaneously, the new period will be

1. 0.48 sec

2. 0.36 sec

3. 0.72 sec

4. 0.64 sec

Q. 3 sound waves of equal amplitudes have frequencies f-1 , f+2 , f+4. They superimpose to give beats. The number of beats produced per second will be?

Q. A piano wire A vibrates at a fundamental frequency of 600 Hz. A second identical wire B produces 6 beats per second with it when the tension in A is slightly increased. Find the the ratio of the tension in A to the tension in B.

Q. Amplitude of vibrations remains constant in case of
$\left(i\right)$ free vibrations
$\left(ii\right)$ damped vibrations
$\left(iii\right)$ maintained vibrations
$\left(iv\right)$ forced vibrations

1. $\left(i\right)$, $\left(iii\right)$ and $\left(iv\right)$
2. $\left(ii\right)$ and $\left(iii\right)$
3. $\left(i\right)$, $\left(ii\right)$ and $\left(iii\right)$
4. $\left(ii\right)$ and $\left(iv\right)$

Q. In an oscillatory motion friction effects , changes the harmonic motion into periodic motion but how?

Q. organ pipe has length L The air in it is in second overtone with maximum amplitudeA.The amplitude at a dis†an ce 2(L÷5) from closed end of the pipe is(1) Zero 2)0.5A 3)A 4)0.7071A

Q. A closed organ pipe has lenght l .The air in it is vibrating in 3rd overtone with maximum amplitude a .The amplitude at a dis†an ce of l/7 from closed end of the pe is equal to

Q.

On applying forced vibrations, the resonance wave becomes very fast when

1. Restoring force is big

2. Damping force is small

3. Applied periodic force is more

4. Oscillations are small

Q.

Two sitar strings A and B playing the note ‘Ga’ are slightly out of tune and produce beats of frequency 6 Hz. The tension in the string A is slightly reduced and the beat frequency is found to reduce to 3 Hz. If the original frequency of A is 324 Hz, what is the frequency of B?

Q. 35.What is the value of t if the amplitude is1/8 of initial ampitude in a damped oscillation? It is given that at t=4s the amplitude is half the initial amplitude in it.

Q.

Suppose the amplitude of a forced oscillator is given as $x_m=\frac{F_m}{\left[m^2\right({\omega }_d^2-{\omega }^2)+b^2{\omega }_d^2{]}^{\frac{1}{2}}}$ where Fm is the (constant) amplitude of the external oscillating force. At resonance,

what are the amplitude and velocity amplitude of the oscillating object?

1. $\frac{F_m}{b\omega },\frac{F_m}{b{\omega }^2}$

2. $\frac{F_m}{b{\omega }_d},\frac{F_m}{b}$

3. $\frac{F_m}{b\omega },\frac{F_m}{b}$

4. None of these

Q. A body of mass $600\text{gm}$ is attached to a spring of spring constant $k=100\text{N/m}$ and it is performing damped oscillations. If damping constant is $0.2$ and driving force is $F=F_0\cos \left(\omega t\right)$ where $F_0=20\text{N}$, find the amplitude of oscillations at resonance.

1. $4.1\text{m}$
2. $0.57\text{m}$
3. $7.75\text{m}$
4. $0.98\text{m}$

Q. A uniform horizontal rod of length $0.40\text{m}$ and mass $1.2\text{kg}$ is supported by two identical wires as shown in figure. Where a mass of $4.8\text{kg}$ should be placed from the left end of the rod, so that the same tuning fork may excite the wire on left into its fundamental vibrations and that on right into its first overtone? (Take $g=10{\text{m/s}}^2$)

1. $5\text{cm}$
2. $10\text{cm}$
3. $15\text{cm}$
4. $25\text{cm}$