Oscillations (SHM) · JEE Main previous year questions

A box weight 196 N on a spring balance at the north pole. Its weight recorded on the same balance if it is shifted to the equator is close to (Take $\text{g}=10{\text{ ms}}^{-2}$ at the north pole and the radius of the earth $=6400\text{ km}$) :

1. A.194.32 N
2. B.195.66 N
3. C.195.32 N
4. D.194.66 N

A spring whose unstretched length is l has a force constant k. The spring is cut into two pieces of unstretched lengths $l_1$ and $l_2$ where, $l_1=n{l}_2$ and n is an integer. The ratio $k_1/k_2$ of the corresponding force constant, $k_1$ and $k_2$ will be :

1. A.$\frac{1}{n^2}$
2. B.$\frac{1}{n}$
3. C.$n^2$
4. D.n

The displacement of a damped harmonic oscillator is given by $x(t)=e^{-0.1t}\cos (10\pi t+f)$. Here t is in seconds. The time taken for its amplitude of vibration to drop to half of its initial value is close to :

1. A.27 s
2. B.13 s
3. C.7 s
4. D.4 s

A simple pendulum oscillating in air has period T. The bob of the pendulum is completely immersed in a non-viscous liquid. The density of the liquid is 1/16 th of the material of the bob. If the bob is inside liquid all the time, its period of oscillation in this liquid is :

1. A.$2T\sqrt{\frac{1}{10}}$
2. B.$4T\sqrt{\frac{1}{14}}$
3. C.$4T\sqrt{\frac{1}{15}}$
4. D.$2T\sqrt{\frac{1}{14}}$

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 1/1000 of the original amplitude is close to :-

1. A.100 s
2. B.10 s
3. C.20 s
4. D.50 s

A simple pendulum, made of a string of length $\ell$ and a bob of mass m, is released from a small angle ${\theta }_0$. It strikes a block of mass M, kept on a horizontal surface at its lowest point of oscillations, elastically. It bounces back and goes up to an angle ${\theta }_1$. Then M is given by :

1. A.$\frac{m}{2}\left(\frac{{\theta }_0+{\theta }_1}{{\theta }_0-{\theta }_1}\right)$
2. B.$\frac{m}{2}\left(\frac{{\theta }_0-{\theta }_1}{{\theta }_0+{\theta }_1}\right)$
3. C.$m\left(\frac{{\theta }_0+{\theta }_1}{{\theta }_0-{\theta }_1}\right)$
4. D.$m\left(\frac{{\theta }_0-{\theta }_1}{{\theta }_0+{\theta }_1}\right)$

A simple harmonic motion is represented by : $y=5(\sin 3\pi t+\sqrt{3}\cos 3\pi t)\mathrm{cm}$ The amplitude and time period of the motion are :

1. A.$10\mathrm{cm},\frac{3}{2}\mathrm{s}$
2. B.$5\mathrm{cm},\frac{2}{3}\mathrm{s}$
3. C.$5\mathrm{cm},\frac{3}{2}\mathrm{s}$
4. D.$10\mathrm{cm},\frac{2}{3}\mathrm{s}$

A pendulum is executing simple harmonic motion and its maximum kinetic energy is $K_1$. If the length of the pendulum is doubled and it performs simple harmonic motion with the same amplitude as in the first case, its maximum kinetic energy is $K_2$. Then :

1. A.$K_2=\frac{K_1}{2}$
2. B.$K_2=2K_1$
3. C.$K_2=K_1$
4. D.$K_2=\frac{K_1}{4}$

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

1. A.$\frac{\sqrt{3}}{2}\text{ s}$
2. B.$\frac{3}{2}\text{ s}$
3. C.$\frac{2}{\sqrt{3}}\text{ s}$
4. D.$2\sqrt{3}\text{ s}$

A simple pendulum of length $1\text{ m}$ is oscillating with an angular frequency $10\text{ rad/s}$. The support of the pendulum starts oscillating up and down with a small angular frequency of $1\text{ rad/s}$ and an amplitude of $10^{-2}\text{ m}$. The relative change in the angular frequency of the pendulum is best given by :

1. A.$1\text{ rad/s}$
2. B.$10^{-3}\text{ rad/s}$
3. C.$10^{-1}\text{ rad/s}$
4. D.$10^{-5}\text{ rad/s}$

A particle executes simple harmonic motion and is located at x = a, b and c at times ${\text{t}}_0,2{\text{t}}_0$ and $3{\text{t}}_0$ respectively. The frequency of the oscillation is :

1. A.$\frac{1}{2\pi t_0}{\cos }^{-1}\left(\frac{a+c}{2b}\right)$
2. B.$\frac{1}{2\pi t_0}{\cos }^{-1}\left(\frac{a+b}{2c}\right)$
3. C.$\frac{1}{2\pi t_0}{\cos }^{-1}\left(\frac{2a+3c}{b}\right)$
4. D.$\frac{1}{2\pi t_0}{\cos }^{-1}\left(\frac{a+2b}{3c}\right)$

The end correction of a resonance column is 1 cm. If the shortest length resonating with the tuning fork is 10 cm, the next resonating length should be :

1. A.28 cm
2. B.32 cm
3. C.36 cm
4. D.40 cm

A silver atom in a solid oscillates in simple harmonic motion in some direction with a frequency of $10^{12}/sec$. What is the force constant of the bonds connecting one atom with the other? (Mole wt. of silver = 108 and Avogadro number = $6.02\times 10^{23}\mathrm{gm}\mathrm{mol}{\mathrm{e}}^{-1}$)

1. A.$5.5N/m$
2. B.$6.4N/m$
3. C.$7.1N/m$
4. D.$2.2N/m$

For an RLC circuit driven with voltage of amplitude vm and frequency ${\omega }_0=\frac{1}{\sqrt{LC}}$ the current exhibits resonance. The quality factor, Q is given by :

1. A.$\frac{CR}{{\omega }_0}$
2. B.$\frac{{\omega }_{0}L}{R}$
3. C.$\frac{{\omega }_{0}R}{L}$
4. D.$\frac{R}{({\omega }_{0}C)}$

In an experiment to determine the period of a simple pendulum of length 1 m, it is attached to different spherical bobs of radii $r_1$ and $r_2$ . The two spherical bobs have uniform mass distribution. If the relative difference in the periods, is found to be $5\times 10^{-4}\mathrm{s}$, the difference in radii, $|r_1-r_2|$ is best given by :

1. A.$1\mathrm{cm}$
2. B.$0.1\mathrm{cm}$
3. C.$0.5\mathrm{cm}$
4. D.$0.01\mathrm{cm}$

1. A.$0.4m/s$
2. B.$4m/s$
3. C.$0.2m/s$
4. D.$2m/s$

A block of mass 0.1 kg is connected to an elastic spring of spring constant $640\mathrm{N}{\mathrm{m}}^{-1}$ and oscillates in a damping medium of damping constant $10^{-2}\mathrm{kg}{\mathrm{s}}^{-1}$ . The system dissipates its energy gradually. The time taken for its mechanical energy of vibration to drop to half of its initial value, is closest to :

1. A.$2\mathrm{s}$
2. B.$3.5\mathrm{s}$
3. C.$5\mathrm{s}$
4. D.$7\mathrm{s}$

The ratio of maximum acceleration to maximum velocity in a simple harmonic motion is $10{\mathrm{s}}^{-1}$. At, $\mathrm{t}=0$ the displacement is 5 m. What is the maximum acceleration ? The initial phase is $\frac{\pi }{4}$.

1. A.$500m/{\mathrm{s}}^2$
2. B.$500\sqrt{2}m/{\mathrm{s}}^2$
3. C.$750m/{\mathrm{s}}^2$
4. D.$750\sqrt{2}m/{\mathrm{s}}^2$

A student measures the time period of 100 oscillations of a simple pendulum four times. The data set is 90 s, 91 s, 95 s and 92 s. If the minimum division in the measuring clock is 1 s, then the reported mean time should be :

1. A.$92\pm 1.8\mathrm{s}$
2. B.$92\pm 3\mathrm{s}$
3. C.$92\pm 2\mathrm{s}$
4. D.$92\pm 5.0\mathrm{s}$

A pendulum clock loses $12\mathrm{s}$ a day if the temperature is $40^{\circ }\mathrm{C}$ and gains $4\mathrm{s}$ a day if the temperature is $20^{\circ }\mathrm{C}$. The temperature at which the clock will show correct time, and the co-efficient of linear expansion ($\alpha$) of the metal of the pendulum shaft are respectively :

1. A.$30^{\circ }\mathrm{C}$; $\alpha =1.85\times 10^{-3}{/}^{\circ }\mathrm{C}$
2. B.$55^{\circ }\mathrm{C}$; $\alpha =1.85\times 10^{-2}{/}^{\circ }\mathrm{C}$
3. C.$25^{\circ }\mathrm{C}$; $\alpha =1.85\times 10^{-5}{/}^{\circ }\mathrm{C}$
4. D.$60^{\circ }\mathrm{C}$; $\alpha =1.85\times 10^{-4}{/}^{\circ }\mathrm{C}$

A particle performs simple harmonic motion with amplitude $A$. Its speed is trebled at the instant that it is at a distance $\frac{2A}{3}$ from equilibrium position. The new amplitude of the motion is:

1. A.$A\sqrt{3}$
2. B.$\frac{7A}{3}$
3. C.$\frac{A}{3}\sqrt{41}$
4. D.$3A$

The period of oscillation of a simple pendulum is $T=2\pi \sqrt{\frac{L}{g}}$. Measured value of L is $20.0\mathrm{cm}$ known to $1\mathrm{mm}$ accuracy and time for 100 oscillations of the pendulum is found to be $90\mathrm{s}$ using wrist watch of $1\mathrm{s}$ resolution. The accuracy in the determination of g is:

1. A.$1\%$
2. B.$5\%$
3. C.$2\%$
4. D.$3\%$

A pendulum made of a uniform wire of cross sectional area $A$ has time period $T$. When an additional mass $M$ is added to its bob, the time period changes to $T_M$. If the Young's modulus of the material of the wire is $Y$ then $\frac{1}{Y}$ is equal to : ($g=\text{gravitational acceleration}$)

1. A.$\left[1-{\left(\frac{T_M}{T}\right)}^2\right]\frac{A}{Mg}$
2. B.$\left[1-{\left(\frac{T}{T_M}\right)}^2\right]\frac{A}{Mg}$
3. C.$\left[{\left(\frac{T_M}{T}\right)}^2-1\right]\frac{A}{Mg}$
4. D.$\left[{\left(\frac{T_M}{T}\right)}^2-1\right]\frac{Mg}{A}$

1. A.
2. B.
3. C.
4. D.

1. A.
2. B.
3. C.
4. D.