Oscillations (SHM) · JEE Main previous year questions

A particle moves with simple harmonic motion in a straight line. In first $\tau \mathrm{s}$, after starting from rest it travels a distance $a$, and in next $\tau \mathrm{s}$ it travels $2a$, in same direction, then:

1. A.amplitude of motion is $3a$
2. B.time period of oscillations is $8\tau$
3. C.amplitude of motion is $4a$
4. D.time period of oscillations is $6\tau$

A pipe of length $85\mathrm{cm}$ is closed from one end. Find the number of possible natural oscillations of air column in the pipe whose frequencies lie below $1250\mathrm{Hz}$. The velocity of sound in air is $340m/s$.

1. A.$12$
2. B.$8$
3. C.$6$
4. D.$4$

Following are expressions for four plane simple harmonic waves (i) $y_1=A\cos 2\pi \left(n_{1}t+\frac{x}{{\lambda }_1}\right)$ (ii) $y_2=A\cos \left[2\pi \left(n_{1}t+\frac{x}{{\lambda }_1}\right)+\pi \right]$ (iii) $y_3=A\cos 2\pi \left(n_{2}t+\frac{x}{{\lambda }_2}\right)$ (iv) $y_4=A\cos 2\pi \left(n_{2}t-\frac{x}{{\lambda }_2}\right)$ The pairs of waves which will produce destructive interference and stationary waves respectively in a medium, are

1. A.(iii, iv), (i, ii)
2. B.(i, iii), (ii, iv)
3. C.(i, iv), (ii, iii)
4. D.(i, ii), (iii, iv)

This question has Statement 1 and Statement 2. Of the four choices given after the Statements, choose the one that best describes the two Statements. Statement 1: In the resonance tube experiment, if the tuning fork is replaced by another identical tuning fork but with its arm having been filed, the length of the air column should be increased to obtain resonance again. Statement 2: On filing the arms, the frequency of a tuning fork increases.

1. A.Statement 1 is true, Statement 2 is true, Statement 2 is a correct explanation of Statement 1.
2. B.Statement 1 is true, Statement 2 is false.
3. C.Statement 1 is false, Statement 2 is true.
4. D.Statement 1 is true, Statement 2 is true, Statement 2 is not a correct explanation of Statement 1.

If $x,v$ and $a$ denote the displacement, the velocity and the acceleration of a particle executing simple harmonic motion of time period $T$, then, which of the following does not change with time?

1. A.$aT/x$
2. B.$aT+2\pi v$
3. C.$aT/v$
4. D.$a^2{T}^2+4{\pi }^2{v}^2$

A coin is placed on a horizontal platform which undergoes vertical simple harmonic motion of angular frequency $\omega$. The amplitude of oscillation is gradually increased. The coin will leave contact with the platform for the first time

1. A.at the mean position of the platform
2. B.for an amplitude of $\frac{g}{{\omega }^2}$
3. C.for an amplitude of $\frac{g^2}{{\omega }^2}$
4. D.at the highest position of the platform

The maximum velocity of a particle, executing simple harmonic motion with an amplitude $7mm$, is $4.4m/s$. The period of oscillation is

1. A.$0.01s$
2. B.$10s$
3. C.$0.1s$
4. D.$100s$

Starting from the origin a body oscillates simple harmonically with a period of $2s$. After what time will its kinetic energy be $75\%$ of the total energy?

1. A.$\frac{1}{6}s$
2. B.$\frac{1}{4}s$
3. C.$\frac{1}{3}s$
4. D.$\frac{1}{12}s$

Two simple harmonic motions are represented by the equations $y_1=0.1\sin \left(100\pi t+\frac{\pi }{3}\right)$ and $y_2=0.1\cos \pi t$. The phase difference of the velocity of particle 1 with respect to the velocity of particle 2 is

1. A.$\frac{\pi }{3}$
2. B.$\frac{-\pi }{6}$
3. C.$\frac{\pi }{6}$
4. D.$\frac{-\pi }{3}$

The bob of a simple pendulum is a spherical hollow ball filled with water. A plugged hole near the bottom of the oscillating bob gets suddenly unplugged. During observation, till water is coming out, the time period of oscillation would

1. A.first decrease and then increase to the original value
2. B.first increase and then decrease to the original value
3. C.increase towards a saturation value
4. D.remain unchanged

If a simple harmonic motion is represented by $\frac{d^{2}x}{d{t}^2}+\alpha x=0$. its time period is

1. A.$\frac{2\pi }{\sqrt{\alpha }}$
2. B.$\frac{2\pi }{\alpha }$
3. C.$2\pi \sqrt{\alpha }$
4. D.$2\pi \alpha$

The bob of a simple pendulum executes simple harmonic motion in water with a period $t$, while the period of oscillation of the bob is $t_0$ in air. Neglecting frictional force of water and given that the density of the bob is $(4/3)\times 1000kg/{\mathrm{m}}^3$. What relationship between $t$ and $t_0$ is true

1. A.$t=2t_0$
2. B.$t=t_0/2$
3. C.$t=t_0$
4. D.$t=4t_0$

A particle at the end of a spring executes $S.H.M$ with a period $t_1$. While the corresponding period for another spring is $t_2$. If the period of oscillation with the two springs in series is $T$ then

1. A.$T^{-1}=t_1^{-1}+t_2^{-1}$
2. B.$T^2=t_1^2+t_2^2$
3. C.$T=t_1+t_2$
4. D.$T^{-2}=t_1^{-2}+t_2^{-2}$

The total energy of particle, executing simple harmonic motion is

1. A.independent of $x$
2. B.$\propto x^2$
3. C.$\propto x$
4. D.$\propto x^{1/2}$

A particle of mass $m$ is attached to a spring (of spring constant $k$) and has a natural angular frequency ${\omega }_0$. An external force $F(t)$ proportional to $\cos \omega t(\omega \ne {\omega }_0)$ is applied to the oscillator. The time displacement of the oscillator will be proportional to

1. A.$\frac{1}{m({\omega }_0^2+{\omega }^2)}$
2. B.$\frac{1}{m({\omega }_0^2-{\omega }^2)}$
3. C.$\frac{m}{{\omega }_0^2-{\omega }^2}$
4. D.$\frac{m}{{\omega }_0^2+{\omega }^2}$

The length of a magnet is large compared to its width and breadth. The time period of its oscillation in a vibration magnetometer is $2\mathrm{s}$. The magnet is cut along its length into three equal parts and these parts are then placed on each other with their like poles together. The time period of this combination will be

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

In forced oscillation of a particle the amplitude is maximum for a frequency ${\omega }_1$ of the force while the energy is maximum for a frequency ${\omega }_2$ of the force; then

1. A.${\omega }_1<{\omega }_2$ when damping is small and ${\omega }_1>{\omega }_2$ when damping is large
2. B.${\omega }_1>{\omega }_2$
3. C.${\omega }_1={\omega }_2$
4. D.${\omega }_1<{\omega }_2$

In a simple harmonic oscillator, at the mean position

1. A.kinetic energy is minimum, potential energy is maximum
2. B.both kinetic and potential energies are maximum
3. C.kinetic energy is maximum, potential energy is minimum
4. D.both kinetic and potential energies are minimum.

A child swinging on a swing in sitting position, stands up, then the time period of the swing will

1. A.increase
2. B.decrease
3. C.remains same
4. D.increases of the child is long and decreases if the child is short

If a spring has time period $T$, and is cut into $n$ equal parts, then the time period of each part will be

1. A.$T\sqrt{n}$
2. B.$T/\sqrt{n}$
3. C.$nT$
4. D.$T$

If a current is passed through a spring then the spring will

1. A.expand
2. B.compress
3. C.remain same
4. D.none of these

The mass of product liberated on anode in an electrochemical cell depends on (where $t$ is the time period for which the current is passed).

1. A.$(It{)}^{1/2}$
2. B.$It$
3. C.$I/t$
4. D.$I^{2}t$