SHM expression

Q. Out of the following functions representing motion of a particle which represent S.H.M?
$I.y=sin\omega t-cos\omega t$
$II.y=si{n}^3\omega t$
$III.y=5cos(\frac{3\pi }{4}-3\omega t)$
$IV.y=1+{\omega }^2{t}^2+\omega t$

1. Only $\left(IV\right)$ does not represent S.H.M.
2. $\left(I\right)$ and $\left(III\right)$
3. $\left(I\right)$ and $\left(II\right)$
4. Only $\left(I\right)$

Q. The displacement $y$ of a particle varies with time $t$ according to the relation $y=a\sin \omega t+b\cos \omega t$. Which of the following statement(s) is/are correct regarding the motion?

1. The motion is SHM.
2. The motion is SHM with amplitude $a+b$.
3. The motion is SHM with amplitude $a^2+b^2$.
4. The motion is SHM with amplitude $\sqrt{a^2+b^2}$.

Q. A block of mass $100\text{g}$ attached to a spring of spring constant $100\text{N/m}$. The block is moved to compress the spring by $10\text{cm}$ and then released on a frictionless floor as shown below. If the collisions with the wall in the front are elastic in nature, then the time period of the motion is

1. $0.2\text{s}$
2. $0.1\text{s}$
3. $0.15\text{s}$
4. $0.132\text{s}$

Q. Two pendulums of time periods $3\text{s}$ and $7\text{s}$ respectively start oscillating simultaneously from two opposite extreme positions. After how much time they will be in same phase?

1. $\frac{21}{8}\text{s}$
2. $\frac{21}{4}\text{s}$
3. $\frac{21}{2}\text{s}$
4. $\frac{21}{10}\text{s}$

Q. A point moves along x-axis according to the equation $x=Asi{n}^2(\omega t-\frac{\pi }{4}).$ Find the amplitude, period and velocity as a function of x.

1. $\frac{A}{2},\frac{\pi }{\omega },2\omega \sqrt{(A-x)x}$
2. $3\frac{A}{2},\frac{\pi }{\omega },\omega \sqrt{(A-x)x}$
3. $\frac{3A}{2},\frac{\pi }{\omega },3\omega \sqrt{(A-x)x}$
4. $\frac{A}{2},2\frac{\pi }{\omega },2\omega \sqrt{(A-x)x}$

Q. In order that the resultant path on superimposing two mutually perpendicular SHM be a circle, the conditions are that:

1. the amplitudes on both SHM should be equal and they should have a phase difference of $\frac{\pi }{2}$
2. the amplitudes should be in the ratio $1:2$ and the phase difference should be zero
3. the amplitudes should be in the ratio $1:2$ and the phase difference should be $\frac{\pi }{2}$
4. the amplitudes should be equal and the phase difference should be zero

Q. The displacement - time $(x-t)$ graph of a particle executing simple harmonic motion is shown in figure. The correct variation of net force $F$ acting on the particle as a function of time is

1. 
2. 
3. 
4. 

Q. A particle executing SHM has maximum acceleration of $2\pi {\text{m/s}}^2$ and maximum velocity of $6\text{m/s}$. Find its time period

1. $\frac{{\pi }^2}{2}\text{s}$
2. $\frac{2{\pi }^2}{3}\text{s}$
3. $6\text{s}$
4. $\frac{\pi }{3}\text{s}$

Q. A particle performs SHM in a straight line. In the first second, starting from rest, it travels a distance $a$ and in the next second, it travels a distance $b$ on the same side of the mean position. The amplitude of the SHM is

1. $a-b$
2. $\frac{2a-b}{3}$
3. $\frac{2a^2}{3a-b}$
4. None of these

Q. Two particles are executing S.H.M of the same amplitude of $20\text{cm}$ with the same period along the same line about the same equilibrium position. The maximum distance between the two is $20\text{cm}$. Their phase difference (in radians) is equal to

1. $\frac{\pi }{3}$
2. $\frac{\pi }{2}$
3. $\frac{2\pi }{3}$
4. $\frac{4\pi }{5}$

Q. If $x,v$ and $a$ denotes 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. $a^2{T}^2+4{\pi }^2{v}^2$
2. $aT/x$
3. $aT+2\pi v$
4. $aT/v$

Q. Two SHM’s are represented by the equations, $y_1=0.1sin(100\pi t+\frac{\pi }{3})$ and $y_2=0.1cos100\pi t$. The phase difference between the speeds of the two particles is,

1. $\frac{\pi }{3}$
2. $\frac{\pi }{8}$
3. $\frac{\pi }{6}$
4. $\frac{\pi }{4}$

Q. Two particles are executing SHM, one with angular frequency $\omega$ and the other with ${}^{\prime }2{\omega }^{\prime }$. Which of the following acceleration $\left(a\right)$ vs position $\left(x\right)$ graphs correctly represent SHM's of two particles.

1. 
2. 
3. 
4. 

Q.

The co-ordinates of a moving particle are $\mathrm{x}={\mathrm{at}}^2$ and $\mathrm{y}={\mathrm{bt}}^2$, where $\mathrm{a}$ and $\mathrm{b}$ are constants. the velocity of a particle at any instant is

1. $2\mathrm{t}\sqrt{{\mathrm{a}}^2}++{\mathrm{b}}^2$

2. $2\mathrm{t}\left(\mathrm{a}+\mathrm{b}\right)$

3. $2\mathrm{t}\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}$

4. $\sqrt{{\mathrm{a}}^2+{\mathrm{b}}^2}$

Q.

Which of the following equations can be that of an SHM? Here, x represents displacement from mean position.

t represents time

k is a positive constant

v represents velocity

1. $\frac{m{d}^{2}x}{d{t}^2}-kx=0$

2. $\frac{m{d}^{2}x}{d{t}^2}+kx=0$

3. $\frac{mdv}{dt}+kx=0$

4. $K{\int }_0^{t}vdt=-m\frac{d^{2}x}{d{t}^2}$ .

Q. If $x,v$ and $a$ denotes 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. $a^2{T}^2+4{\pi }^2{v}^2$
2. $aT/x$
3. $aT+2\pi v$
4. $aT/v$

Q. A transverse sinusoidal wave moves along a string in the positive $x-$direction at a speed of $10\text{cm/s}$. The wavelength of the wave is $0.5\text{m}$ and its amplitude is $10\text{cm}$. At a particular time $t$, the snapshot of the wave is shown in the figure. The velocity of a particle at $P$ when its displacement is $5\text{cm}$ along $y-$axis is

1. $\frac{\sqrt{3}\pi }{50}\hat{j}\text{m/s}$
2. $-\frac{\sqrt{3}\pi }{50}\hat{j}\text{m/s}$
3. $\frac{\sqrt{3}\pi }{50}\hat{i}\text{m/s}$
4. $-\frac{\sqrt{3}\pi }{50}\hat{i}\text{m/s}$

Q. In a horizontal spring mass system, mass $m$ is released after being displaced towards right by some distance at $t=0$ on a frictionless surface. The phase angle of the motion in radian when it passes through the equilibrium position for the first time is

1. $\frac{\pi }{2}$
2. $\pi$
3. $\frac{3\pi }{2}$
4. $0$

Q.

A particle executing simple harmonic motion along y-axis has its motion described by the equation. $y=Asin\left(\omega t\right)+B$. The amplitude of the simple harmonic motion is

1. A

2. B

3. A+B

4. $\sqrt{A+B}$

Q. Which of the following expressions represent simple harmonic motion?

1. $x=asin(\omega t+\varphi )$
2. $x=acos(\omega t+\delta )$
3. $x=asin\omega t+bcos\omega t$
4. $x=asin\omega tcos\omega t$

Q. A particle executes linear $\text{SHM}$ with a time period of $10\text{s}$. Find the time taken by the particle to go directly from its mean position (at $t=0)$ to $\frac{1}{2}$ of its amplitude.

1. $\frac{5}{6}\text{s}$
2. $\frac{5\pi }{6}\text{s}$
3. $1.2\text{s}$
4. $\frac{5}{3}\text{s}$

Q. A particle ís executing simple harmonic motion with amplitude of $0.1\text{m}$. At a certain instant when its displacement is $0.02\text{m}$, its acceleration is $0.5{\text{ms}}^{-2}$. The maximum velocity of the particle is (in ${\text{ms}}^{-1}$)

1. $0.01$
2. $0.05$
3. $0.5$
4. $0.25$

Q. The displacement $y$ of a particle varies with time $t$ according to the relation $y=a\sin \omega t+b\cos \omega t$. Which of the following statement(s) is/are correct regarding the motion?

1. The motion is SHM.
2. The motion is SHM with amplitude $a+b$.
3. The motion is SHM with amplitude $a^2+b^2$.
4. The motion is SHM with amplitude $\sqrt{a^2+b^2}$.

Q. A particle is placed at the lowest point of a smooth wire frame in the shape of a parabola, lying in the vertical $xy-$ plane having equation $x^2=5y$, where $(x,y)$ are in $\text{metres}$. After slight displacement, the particle is set free to move. Find the angular frequency of oscillation $\text{in rad/s}$.
[Take $g=10{\text{m/s}}^2)$]

1. $2\text{rad/s}$
2. $4\text{rad/s}$
3. $6\text{rad/s}$
4. $8\text{rad/s}$

Q.

A test tube of cross-sectional area $a$ has some lead shots in it. The total mass is $m$. It floats upright in a liquid of density $d$. When pushed down a little and released, it oscillates up and down with a period $T$. Choose the correct relationship from the following.

1. $T=2\pi \sqrt{\frac{ma}{gd}}$

2. $T=2\pi \sqrt{\frac{mg}{ad}}$

3. $T=2\pi \sqrt{\frac{md}{ag}}$

4. $T=2\pi \sqrt{\frac{m}{agd}}$

Q. Two blocks $A\left(5\text{kg}\right)\text{and B}\left(2\text{kg}\right)$ attached to the ends of a spring of spring constant 1120 N/m are placed on a smooth horizontal plane with spring undeformed. Simultaneously velocities of $3\text{m/s and 10 m/s}$ along the line of the spring in the same direction are imparted to $A$ and $B$ then

1. the amplitude of oscillation of $5\text{kg}$ block is $0.18\text{m}$
2. the first maximum compression occurs after start at $\frac{3\pi }{56}\text{s}$
3. the maximum extension of the spring is $0.50\text{m}$
4. time period of oscillation is $\frac{\pi }{14}\text{s}$

Q. If the period of S.H.M. is 12 seconds and amplitude 10 cm. What is the displacement 14 Seconds after passage of the particle through its extreme positive displacement?

1. 5 cm
2. 4 cm
3. 2 cm
4. 10 cm

Q. A large horizontal surface is in $SHM$ with an amplitude of $1\text{cm}$. A mass of $10\text{kg}$ is kept gently on it when it is at mean position. For the mass to always remain in contact with the surface, the maximum frequency of $SHM$ will be

1. $5\text{Hz}$
2. $6\text{Hz}$
3. $7\text{Hz}$
4. $8\text{Hz}$

Q. Choose the correct statement (s) from the following in which k is a real, positive constant.

1. Function f(t) = sin kt + coskt is simple harmonic having a period $\frac{2\pi }{k}$
2. Function f(t) = $sin\pi t+2cos2\pi +3sin3\pi t$ is periodic but not simple harmonic having a period of 2s
3. Function f(t) = $coskt+2si{n}^{2}kt$ is simple harmonic having a period $\frac{2\pi }{k}$.
4. Function f(t) = $e^{-kt}$ is not periodic.

Q. A particle is executing SHM according to the equation, $x=A\cos \omega t$. Average speed of the particle during the interval $0\le t\le \frac{\pi }{6\omega }$ is

1. $\frac{\sqrt{3}A\omega }{2}$
2. $\frac{\sqrt{3}A\omega }{4}$
3. $\frac{3A\omega }{\pi }$
4. $\frac{3A\omega }{\pi }(2-\sqrt{3})$