Analytical Picture of Forced Oscillation

Q.

A student performs an experiment for determination of $g=\frac{4{\pi }^{2}l}{T^2}$ The error in length $l$ is $\Delta l$ and in time $T$ is $\Delta T$ and $n$ is number of times the reading is taken. The measurement of $g$ is most accurate for:

1. $\Delta l=0.5mm,T=0.2sec,n=10$
2. $\Delta l=0.5mm,T=0.2sec,n=20$
3. $\Delta l=0.5mm,T=0.1sec,n=10$
4. $\Delta l=0.5mm,T=0.2sec,n=50$

Q. The amplitude of a damped oscillator decreases to $0.9$ times its original magnitude in $5\text{s}$. In another $10\text{s}$, it will decrease to $\alpha$ times its original magnitude, where $\alpha$ equals

1. $0.7$
2. $0.81$
3. $0.729$
4. $0.6$

Q.

A particle is moving unidirectionally on a horizontal plane under the action of a constant power supplying energy source. The displacement $\left(s\right)$ – time $\left(t\right)$ graph that describes the motion of the particle is (graphs are drawn schematically and are not to scale):

1. 

2. 

3. 

4. 

Q. What will happen if the earth stops the rotation about its axis?

Q. A damped harmonic oscillator has a frequency of $5$ oscillations per second. The amplitude drops to half its value for every $10$ oscillations. The time it will take to drop to $\frac{1}{1000}$ of the original amplitude, is close to,

1. $10\text{sec.}$
2. $50\text{sec.}$
3. $100\text{sec.}$
4. $20\text{sec.}$

Q. The energy of the particle executing damped oscillations decreases with time, because work is done against

1. elastic tension
2. frictional force
3. both restoring force and friction
4. restoring force

Q.

Find the dimensions of

(a) angular speed $\omega$

(b) angular acceleration $\alpha$

(c) torque $\Gamma$ and

(d) moment of interia I.

Some of the equations involving these quantities are

$\omega =\frac{{\theta }_2-{\theta }_1}{t_2-t_1}$,

$\alpha =\frac{{\omega }_2-{\omega }_1}{t_2-t_1}$

$\Gamma =Fj$ and $I=m{r}^2$.

The symbols have standard meanings.

Q. Column 1 Column 2
a) A linear S.H.M e) $\frac{d^2\Theta }{d{t}^2}=\frac{c}{\Theta }$
b) Angular S.H.M f)$\frac{d^{2}x}{d{t}^2}+\frac{R}{m}\frac{dx}{Mt}+x{w}^2=\frac{F}{m}cos\theta$
c) Damped harmonic

1. a-e, b-h, c-g, d-f
2. a-g, b-h, c-e, d-f
3. a-f, b-g, c-e, d-h
4. a-g, b-e, c-h, d-f

Q. A student performs an experiment for determination of $⟮g=\frac{4{\pi }^{2}l}{T^2}⟯$, l = 1m, and he commits an error of $\Delta l$ For T he takes the time of n oscillations with the stop watch of least count $\Delta T$ and he commits a human error of 0.1 s. For which of the following data, the measurement of g will be most accurate?

1. $\Delta L=0.5,\Delta T=0.1,n=20$
2. $\Delta L=0.5,\Delta T=0.1,n=50$
3. $\Delta L=0.5,\Delta T=0.01,n=20$
4. $\Delta L=0.5,\Delta T=0.05,n=50$

Q. Give one example of damped vibrations.

Q. A simple harmonic oscillator of angular frequency $2rad{s}^{-1}$ is acted upon by an external force $F=\sin tN$. If the oscillator is at rest in its equilibrium position at $t=0$, its position at later times is proportional to

1. $\sin t+\frac{1}{2}\sin 2t$
2. $\sin t+\frac{1}{2}\cos 2t$
3. $\cos t-\frac{1}{2}\sin 2t$
4. $\sin t-\frac{1}{2}\sin 2t$

Q. Which of the following statement(s) is/are correct regarding the amplitude vs frequency curve for a driven system?

1. Smaller the damping, taller and narrower the resonance peak
2. The amplitude tends to infinity when it equals to natural frequency results in zero damping
3. Small damping results in driving frequency almost equal to natural frequency
4. The resonant amplitude decreases with increasing damping

Q. A particle executes simple harmonic oscillation. Its amplitude is $a$. The period of oscillation is $T$. The minimum time taken by the particle to travel half of the amplitude from the equilibrium position is

1. $\frac{T}{8}$
2. $\frac{T}{12}$
3. $\frac{T}{2}$
4. $\frac{T}{4}$

Q. The phase of particle in SHM is found to increase by $14\pi$ in 3.5 sec. Its frequency of oscillation is

1. $2Hz$
2. $1/2Hz$
3. $1Hz$
4. $2\pi Hz$

Q. A vibrating tuning fork is first held in the hand and then its end is brought in contact with a table. Which of the following statement(s) is/are correct in respect of this situation?

1. The sound is louder when the tuning fork is held in hand.
2. The sound is louder when the tuning fork is in contact with table.
3. The sound dies away sooner when tuning fork is brought in contact with the table.
4. The sound remains for a longer duration when turning fork is held in hand.

Q. Stand on the end of a diving board and bounce to set it into oscillation. When you bounce at frequency $f$, the response is maximum in terms of the amplitude of oscillation at the end of the board. When you move to the middle of the board and repeat the experiment, the resonance frequency for forced oscillations at this point is:

1. Higher
2. Lower
3. Same as $f$
4. None of these

Q. A ball of mass $m$ can perform damped harmonic oscillations about the point $x=0$ with natural frequency ${\omega }_0$. At the moment $t=0$, when the ball was in equilibrium, a force $F_x=F_0\cos \omega t$ coinciding with the $x$ axis was applied to it. Find the law of forced oscillation $x\left(t\right)$ fro that ball.

Q. A particle suspended from a fixed point, by a light inextensible thread of length L is projected horizontally from its lowest position with velocity $\sqrt{\frac{7gL}{2}}$. The thread will slack after swinging through an angle $\theta$, such that $\theta$ equal.

1. ${120}^o$
2. ${135}^o$
3. ${150}^o$
4. ${30}^o$

Q. Assertion : An earthquake will not cause uniform damage to all building in an affected area, even if they are built with the same strength and materials.
Reason : The one with its natural frequency close to the frequency of seismic wave is likely to be damaged less.

1. If both assertion and reason are true and reason is the correct explanation of assertion.
2. If both assertion and reason are true and reason is not the correct explanation of assertion.
3. If assertion is true but reason is false.
4. If both assertion and reason are false.

Q. A sphere of radius r is kept on a concave mirror of radius of curvature R. The arrangement is kept on a horizontal table (the surface of concave mirror is frictionless and sliding not rolling). If the sphere is displaced from its equilibrium position and left, then it executes S.H.M. The period of oscillation will be

1. $\pi \times \left(\frac{(R-r)1.4}{g}\right)$
2. $2\pi \times \left(\frac{R-r}{g}\right)$
3. $\sqrt[2\pi ]{2\pi \left(\frac{rR}{g}\right)}$
4. $\left(\frac{R}{gr}\right)$

Q. A point particle of mass $0.1kg$ executing SHM with amplitude of $0.1m$. When the particle passes through the mean position is KE is $8\times {10}^{-3}J$. Obtain the equation of motion of this particle if the initial phase of oscillation is ${45}^o$

Q. A damped harmonic oscillator has a frequency of 5 oscillations per second. The amplitude drops to half its value for every 10 oscillations. The time it will take to drop to $\frac{1}{1000}$ of the original amplitude is close to :-

1. $100s$
2. $20s$
3. $50s$
4. $10s$