Forced Oscillations

Q.

What are the characteristics of SHM?

Q. The equation of motion of a particle 𝑖𝑠 \(x=a\cos\left( \alpha t \right)^2.\) The motion is

Q. At what depth from the surface of earth, the time period of a simple pendulum is $0.5\%$ more than that on the surface of the earth?
$($Radius of earth is $6400\text{km})$

1. $32\text{km}$
2. $64\text{km}$
3. $96\text{km}$
4. $128\text{km}$

Q. A simple pendulum has a time period $T_1$ when on the earth's surface and $T_2$ when taken to a height $R$ above the earth's surface, where $R$ is the radius of the earth. The value of $\frac{T_2}{T_1}$ is

1. $1$
2. $\sqrt{2}$
3. $2$
4. $4$

Q. Four pendulums $A,B,C$ and $D$ are suspended from the same elastic support as shown in figure. $A$ and $C$ are of the same length, while $B$ is smaller than $A$ and $D$ is larger than $A$. If $A$ is given a transverse displacement,


[Assume there is no loss of energy]

1. $D$ will vibrate with maximum amplitude
2. $C$ will vibrate with maximum amplitude
3. $B$ will vibrate with maximum amplitude
4. All the four will oscillate with equal amplitude

Q. A particle with restoring force proportional to displacement and resisting force proportional to velocity is subjected to force $Fsin$$\omega$t. If the amplitude of the particle is maximum for $\omega ={\omega }_1$ and the energy of the particle is maximum for $\omega ={\omega }_2$, then (where ${\omega }_0$ natural frequency of oscillationn of particle)

Q. Find the time period of a uniform disc of mass $m$ and radius $r$ suspended through a point at a distance $r/2$ from the centre, oscillating in a plane parallel to its plane.

1. $2\pi \sqrt{\frac{2r}{3g}}$
2. $4\pi \sqrt{\frac{r}{2g}}$
3. $2\pi \sqrt{\frac{3r}{2g}}$
4. None

Q. An organ pipe open at one end is vibrating in first overtone and is in resonance with another pipe open at both ends and vibrating in third harmonic. The ratio of length of two pipes is

1. 1 : 2
2. 4 : 1
3. 8 : 3
4. 3 : 8

Q.

Calculate the natural frequency of a spring-mass system?

Q.

A simple pendulum of length l is suspended from the ceiling of a car moving with a speed v on a circular horizontal road of radius r. (a) Find the tension in the string when it is at rest with respect to the car. (b) Find the time period of small oscillation.

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

Q.

What is forced frequency?

Q. In forced oscillations, a particle oscillates simple harmonically with a frequency equal to

1. Frequency of driving force
2. Natural frequency of body
3. Difference in frequency of driving force and natural frequency
4. Mean of frequency of driving force and natural frequency

Q.

A uniform rod of length l is suspended by an end and is made to undergo small oscillations. Find the length of the simple pendulum having the time period equal to that of the rod.

Q. During the phenomenon of resonance:

1. The amplitude of oscillations becomes large
2. The frequency of oscillations becomes large
3. The time period of oscillation becomes large
4. All of the above

Q. Resonance is a special case of

1. Forced oscillations
2. Damped oscillations
3. Undamped oscillations
4. Coupled oscillations

Q. Shock absorbers in automobiles is an application of damped oscillation.

1. True
2. False

Q.

A 1000 kg car carrying four 82 kg people travels over a "washboard” dirt road with corrugations 4.0 m apart. The car bounces with maximum amplitude when its speed is 16 km/h. when the car stops, and the people get out, by how much does the car body rise on its suspension?

1. 10 cm

2. 3 cm

3. 5 cm

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

Q.

What is the formula for oscillation?

Q. In forced oscillations, a particle oscillates simple harmonically with a frequency equal to

1. Frequency of driving force
2. Natural frequency of body
3. Difference in frequency of driving force and natural frequency
4. Mean of frequency of driving force and natural frequency

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. Which of the following is correct about damped oscillations ?

1. Amplitude of oscillations decreases linearly.
2. Energy of oscillations remains constant.
3. Amplitude of oscillations decreases exponentially.
4. Frequency of oscillations decreases linearly.

Q. The mass and the diameter of a planet are three times the respective values for the Earth. The period of oscillation of a simple pendulum on the Earth is $2\text{Sec.}$ The Period of oscillation of the same pendulum on the planet would be:

1. $\frac{3}{2}\text{sec}$
2. $\frac{2}{\sqrt{3}}\text{sec}$
3. $\frac{\sqrt{3}}{2}\text{sec}$
4. $2\sqrt{3}\text{sec}$

Q. What is the meaning of frequency of small amplitude oscillations ?

Q. Amplitude of vibrations does not remains constant in case of:-

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

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

Q.

How does damping force executing SHM vary?

Q. A linear harmonic oscillator of force constant $2\times {10}^{6}N{m}^{-1}$ and amplitude 0.01 m has a total mechanical energy 160 J. which of the following statements are correct?
(i) Maximum PE is 100 J (ii) Maximum KE is 100 J
(iii) Maximum PE is 160 J (iv) Minimum PE is zero

1. Both (i) and (iv)
2. Both (ii) and (iii)
3. Both (i) and (ii)
4. Both (ii) and (iv)