Simple Harmonic Oscillation

Q.

Convert 1 dyne into newton.

Q. Two charges each of $1\mu \text{C}$ but opposite in sign are $1\text{cm}$ apart. The electric field at a distance $10\text{cm}$ from the mid point on the axial line of this dipole is

1. $18\times {10}^4\text{N/C}$
2. $20\times {10}^4\text{N/C}$
3. $36\times {10}^4\text{N/C}$
4. $40\times {10}^4\text{N/C}$

Q. When a particle of mass $m$ is attached to a vertical spring of spring constant $k$ and released, its motion is described by $y\left(t\right)=y_0{\sin }^2\omega t,$ where ${}^{\prime }{y}^{\prime }$ is measured from the lower end of unstretched spring. Then $\omega$ is :

1. $\frac{1}{2}\sqrt{\frac{g}{y_0}}$
2. $\sqrt{\frac{g}{y_0}}$
3. $\sqrt{\frac{g}{2y_0}}$
4. $\sqrt{\frac{2g}{y_0}}$

Q.

A linear harmonic oscillator of force constant $2\times {10}^6$ N/m and amplitude 0.01 $m$ has a total mechanical energy 160 $J$. Its

1. Maximum potential energy is 100 $J$
2. Maximum K.E. is 100 $J$
3. Maximum P.E is 160 $J$
4. Minimum P.E. is zero

Q.

The percentage errors in the measurements of the length of a simple pendulum and its time period are 3% and 5% respectively. The maximum error in the value of the acceleration due to gravity obtained from these measurements is

1. 2%
2. 15%
3. 8%
4. 13%

Q.

What does tangential force produce?

Q. A dipole consists of two particles one with charge $+1\mu C$ and mass $1\text{kg}$ and the other with charge $-1\mu C$ and mass $2\text{kg}$ separated by a distance of 3 m. For small oscillations about its equilibrium position, the angular frequency, when placed in a uniform electric field of $20\text{kV/m}$ is

1. $1.1\text{rad/s}$
2. $10\text{rad/s}$
3. $0.1\text{rad/s}$
4. $2.5\text{rad/s}$

Q. Every SHM is a periodic motion. But all periodic motions are not SHM why? Explain with an example

Q.

Define force constant. Give its unit and dimensional formula.

Q. A spring has natural length of $50\text{cm}$ and a force constant of $2.0\times {10}^3{\text{Nm}}^{-1}$ . A body of mass $10\text{kg}$ is suspended from it in equilibrium position. If the body is pulled down, such that length of spring becomes $58\text{cm}$ and released. If it executes simple harmonic motion, what is the net force on the body, when it is at it's lowermost position of oscillation ?
$(\text{Take g}=10{\text{ms}}^{-2})$.

1. $20\text{N}$
2. $40\text{N}$
3. $60\text{N}$
4. $80\text{N}$

Q. A block of mass $m$, attached to a spring of spring constant $k$, oscillates on a smooth horizontal table. The other end of the spring is fixed to a wall. The block has a speed $v,$ when the spring is at its natural length. How far will the block move on the table before coming to an instantaneous rest?

1. $x=2v\sqrt{\frac{m}{k}}$
2. $x=\frac{1}{v}\sqrt{\frac{m}{k}}$
3. $x=3\sqrt{\frac{mv}{k}}$
4. $x=v\sqrt{\frac{m}{k}}$

Q.

Motion of a child on a swing is ______________.

Q. A circular spring of natural length $l$ is cut and welded with two beads of masses $m_1$ and $m_2$ such that the ratio of the lengths of the springs between the beads is $4:1$. If the stiffness of the original spring is $k$, then find the angular frequency of oscillation of the beads in a smooth horizontal rigid tube.
[Assume $m_1=m$ and $m_2=3m$].

1. $\omega =\sqrt{\frac{25k}{3m}}$
2. $\omega =\sqrt{\frac{5k}{3m}}$
3. $\omega =\sqrt{\frac{25k}{4m}}$
4. $\omega =\sqrt{\frac{5k}{4m}}$

Q.

A body of mass 2 kg suspended through a vertical spring executes simple harmonic motion of period 4 s. If the oscillations are stopped and the body hangs in equilibrium, find the potential energy stored in the spring.

Q. The speed $v$ of a particle moving along a straight line, when it is at a distance $x$ from a fixed point on the line, is given by $v^2=144-9x^2$.
Choose the wrong option.

1. $\text{Displacement of the particle}\le \text{distance moved by it}$
2. $\text{The magnitude of acceleration at a distance 3 units from the fixed point is 27 units}$
3. $\text{The maximum displacement from the fixed point is 4 units}$
4. $\text{The motion is simple harmonic with}T=\frac{\pi }{3}\text{units}$

Q. In the reported figure, two bodies \(A\) and \(B\) of masses \(200~\text {gms}\) and \(800~\text {gms}\) are attached with the system of springs. Springs are kept in a stretched position with some extension, when the system is released. The horizontal surface is assumed to be frictionless. The angular frequency will be \(\text{rad/s}\) when \(k = 20~ \text {N/m}\)

Q. A street car moves rectilinearly from station A (here car stops) to the next station B (here also car stops ) with an acceleration varying according to the law $f=a-bx$, where $a$ and $b$ are positive constants and $x$ is the distance from station A. The distance between the two stations & the maximum velocity are respectively.

1. $x=\frac{2a}{b};v_{max}=\frac{a}{\sqrt{b}}$
2. $x=\frac{a}{b};v_{max}=\frac{a}{2\sqrt{b}}$
3. $x=\frac{2a}{b};v_{max}=\frac{2a}{\sqrt{b}}$
4. $x=\frac{a}{2b};v_{max}=\frac{a}{\sqrt{b}}$

Q.

The two velocities of a train with constant acceleration are $u$and $v$ respectively. What is the velocity at which the middle point of the train passes through the same point?

Q.

If the distance ‘$\mathrm{s}$’ traveled by a particle in time $\mathrm{t}$ is $\mathrm{s}=\mathrm{a}\sin \mathrm{t}+\mathrm{b}\cos 2\mathrm{t}$, then the acceleration at $\mathrm{t}=0$ is

1. $\mathrm{a}$

2. $-\mathrm{a}$

3. $4\mathrm{b}$

4. $-4\mathrm{b}$

Q.

How do you find the maximum speed of a pendulum?

Q.

How do I understand Neutral Equilibrium?

I have understood what it means in terms of Energy diagram.

But I have difficulty in understanding the examples of it and differentiating from Stable Equilibrium.

Q. The displacement of a damped harmonic oscillator is given by $x\left(t\right)=e^{-0.1t}\cos (10\pi t+\phi ).$ 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. $13\text{s}$
2. $7\text{s}$
3. $4\text{s}$
4. $27\text{s}$

Q. The period of oscillation of a simple pendulum of length l is given by $T=2\pi \sqrt{l/g}$. The length l is about 10 cm and is known to 1 mm accuracy. The period of oscillation is about 0.5 s. The time of 100 oscillations has been measured with a stop watch of 1 s resolution. Find the percentage error in the determination of g.

Q. Two masses $m_1=1\text{kg}$ and $m_2=0.5\text{kg}$ are suspended together by a massless spring of spring constant $12.5\text{N/m}$. When masses are in equilibrium, $m_1$ is removed without disturbing the system. New amplitude of oscillation will be

1. $30\text{cm}$
2. $50\text{cm}$
3. $80\text{cm}$
4. $60\text{cm}$

Q. Consider a situation as shown in figure. Find the minimum value of $m$ so that the blocks do not move.

1. $2kg$
2. $8kg$
3. $10kg$
4. $1kg$

Q.

The average resisting force that must act on a $5kg$ mass to reduce its speed from $65\raisebox{1ex}{$cm$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ to $0.15\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$ in 0.2s is:

1. 12.5 N

2. 25 N

3. 50 N

4. 100 N

Q. The position-time $(x-t)$ equation of a particle moving along $x-$ axis is given by $x=A+A(1-\cos \omega t)$. Select the correct statement(s):

(Consider S.I units everywhere)

1. The particle oscillates simple harmonically between points $x=2A$ and $x=A$
2. Velocity of the particle is maximum at $x=2A$
3. Minimum time taken by the particle in travelling from $x=A$ to $x=3A$ is $\frac{\pi }{\omega }$
4. Minimum time taken by the particle in travelling from $x=A$ to $x=2A$ is $\frac{\pi }{2\omega }$

Q. The potential energy of a particle of mass $0.1\text{kg}$, moving along the $x-$axis, is given by $U=5x(x-4)\text{J}$, where $x$ is in meters.
Choose the wrong option.

1. The speed of the particle is maximum at $x=2\text{m}$.
2. The particle executes simple harmonic motion.
3. The period of oscillation of the particle is $\frac{\pi }{5}s$.
4. The particle does not execute simple harmonic motion.

Q.

A moving mass of $8kg$ collides elastically with a stationary mass of $2kg$.If KE be the initial kinetic energy of the moving mass, what will be the kinetic energy left with it after the collision?

Q. The resultant of two rectangular simple harmonic motions of the same frequency and unequal amplitudes but differing in phase by $\frac{\pi }{2}$ is

1. Circular
2. Elliptical
3. Simple harmonic
4. Parabolic