Phase and the General Equation

Q.

A simple harmonic motion is represented by $x\left(t\right)={\sin }^2\omega t-2{\cos }^2\omega t$. The angular frequency of oscillation is given by?

Q. A $150\text{g}$ block oscillates in SHM on the end of a spring with $k=1500\text{N/m}$, according to the relation $x=x_0\cos (\omega t+\varphi ).$ How long does the block take to move from the position $+0.800x_0$ to $+0.600x_0$ directly ?

1. $2\text{ms}$
2. $0.3\text{ms}$
3. $4\text{ms}$
4. $3\text{ms}$

Q. A particle executes simple harmonic oscillation with an amplitude $A$. If the time period of oscillation is $T$, then minimum time taken by the particle to travel half of the amplitude from the mean position is

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

Q. Two particles undergo SHM along parallel lines with the same time period $\left(T\right)$ and equal amplitudes. At a particular instant, one particle is at its extreme position while the other is at its mean position. They move in the same direction. They will cross each other after a further time of

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

Q. The equation of motion of a particle started at $t=0$ is given by $x=10\text{sin}(20t+\frac{\pi }{3})$, where $x$ is in $\text{cm}$ and $t$ is in seconds. Find the time when the particle first comes to rest.

1. $\frac{\pi }{60}\text{s}$
2. $\frac{\pi }{120}\text{s}$
3. $\frac{\pi }{30}\text{s}$
4. $\frac{5\pi }{3}\text{s}$

Q. What is the phase difference between the two simple harmonic motions whose displacement – time graphs are shown in fig ?

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

Q. The phase difference between the displacement and acceleration of particle executing $\text{S.H.M.}$ $\left(\text{in radian}\right)$ is

1. $\text{Zero}$
2. $\frac{\pi }{2}$
3. $\pi$
4. $2\pi$

Q.

A particle executing simple harmonic motion has angular frequency $6.28s^{-}1$ and amplitude $10cm$. Find $\left(x\right)$ the speed when the displacement is $6cm$, from the mean position, $\left(y\right)$ the speed at $t=\frac{1}{6}s$ assuming that the motion starts from rest at $t=0$.
(i) $25.1cm{s}^{-1}$
(ii) $54.4cm{s}^{-1}$
(iii) $50.2cm{s}^{-1}$
(iv) $27.2cm{s}^{-1}$

1. $x$ - (ii); $y$ - (iv)

2. $x$ - (iii); $y$ - (iv)

3. $x$ - (i), $y$ - (iii)

4. $x$ - (iii), $y$ - (ii)

Q. Two particles move parallel to $x-$ axis about the origin with the same amplitude and frequency. At a certain instant, they are found at distance $\frac{A}{3}$ from the origin on opposite sides, but their velocities are found to be in the same direction. Find the phase difference between the two particles.

1. $\pi$
2. ${\cos }^{-1}\left(\frac{2}{3}\right)$
3. $\frac{\pi }{3}$
4. ${\cos }^{-1}\left(\frac{7}{9}\right)$

Q. Find the maximum acceleration of a block which is oscillating with frequency 2 Hz and amplitude 50 cm.

1. $78.8m/s^2$

2. $100m/s^2$
3. $50m/s^2$
4. $40m/s^2$

Q. A particle executes $\text{SHM}$ with an angular frequency $4\pi \text{rad/s}$. If it is at its positive extreme position initially, find the first instant when it is at a distance of $\frac{1}{\sqrt{2}}$ times its amplitude from the mean position in positive direction

1. $\frac{1}{24}\text{s}$
2. $\frac{1}{16}\text{s}$
3. $-\frac{1}{16}\text{s}$
4. $\frac{3}{16}\text{s}$

Q. The time taken by a particle executing simple harmonic motion of time period T to move from the mean position to half the maximum displacement is

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

Q. If vectors $\overrightarrow{A}=\cos wt\hat{i}+\sin wt\hat{j}$ and $\overrightarrow{B}=\cos wt/2\hat{i}+\sin wt/2\hat{j}$ are functions of time , then the value of $t$ at which they are orthogonal to each other is

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

Q.

A particle is undergoing circular motion with angular frequency $\omega$ as shown

If the shadow of the particle on the wall is performing SHM such that its acceleration's maximum value is 'a', find the acceleration of the particle and the radius of the circle.

1. $a,\frac{9}{2{\omega }^2}$

2. $2a,\frac{a}{{\omega }^2}$

3. $a,\frac{a}{{\omega }^2}$

4. Data is sufficient

Q. A particle executes simple harmonic motion (amplitude $=A$) between $x=-A$ and $x=+A$. The time taken for it to go from $0$ to $\frac{A}{2}$ is $T_1$ and to go from $\frac{A}{2}$ to $A$ is $T_2$ . Then

1. $T_1<T_2$
2. $T_1>T_2$
3. $T_1=T_2$
4. $T_1=2T_2$

Q.

The equation of a particle executing SHM is $x=\left(5m\right)sin\left[\right(\pi s^{-1})t+\frac{\pi }{3}]$. What is its amplitude, time period, maximum speed and velocity. At t = 1s?

1. $5m,2s,5m{s}^{-1},\frac{-5\pi }{2}m{s}^{-1}$

2. $5m,2s,5\pi m{s}^{-1},\frac{-5\pi }{2}m{s}^{-1}$

3. $5m,\pi s,5m{s}^{-1},\frac{-5\pi }{2}m{s}^{-1}$

4. $5m,2s,5\pi m{s}^{-1},\frac{+5\pi }{2}m{s}^{-1}$

Q. A particle is in linear simple harmonic motion between two points $P$ and $Q$ represented by $x=A\text{sin}(\omega t+\varphi )$. Find the initial phase of the particle when it starts from its extreme position.

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

Q. A particle executes SHM with a period of $T$ second and amplitude $A$ metre. The particle is at its mean position initially. The shortest time it takes to reach point $\frac{A}{\sqrt{2}}$ metre from its mean position in seconds is

1. $T$
2. $\frac{T}{4}$
3. $\frac{T}{8}$
4. $\frac{T}{16}$

Q. The figure shown below represents a particle executing SHM. If the particle starts from the extreme position, $\left(P\right)$, find the correct equation of motion of particle.

1. $x=5\text{cos}\omega t$
2. $x=-5\text{cos}\omega t$
3. $x=5\text{sin}\omega t$
4. $x=10\text{cos}\omega t$

Q.

The phase difference between two SHM

$y_1=10sin(10\pi t+\frac{\pi }{3})and{y}_2=12sin(8\pi t+\frac{\pi }{4})att=0.5sis:$

1. $\frac{11\pi }{12}$

2. $\frac{13\pi }{12}$

3. $\pi$

4. $\frac{17\pi }{12}$

Q. For the given situation, which of the following options correctly represents the equation for SHM?

1. $x=A\sin (\omega t+\frac{5\pi }{6})$
2. $x=A\sin (\omega t+\frac{\pi }{6})$
3. $x=A\sin (\omega t-\frac{5\pi }{6})$
4. $x=A\sin (\omega t-\frac{\pi }{6})$

Q. Two particles move parallel to the $x-$axis about the origin with the same amplitude $A$ and angular frequency $\omega$. At a certain instance they are found at a distance $\frac{A}{3}$ from the origin on opposite sides but their velocities are in the same direction. What is the phase difference between the two?

1. ${\cos }^{-1}\left(\frac{7}{9}\right)$
2. ${\cos }^{-1}\left(\frac{5}{9}\right)$
3. ${\cos }^{-1}\left(\frac{4}{9}\right)$
4. ${\cos }^{-1}\left(\frac{1}{9}\right)$

Q.

A particle is subjected to two simple harmonic motions in the same direction having equal amplitudes and equal frequency. If the resultant amplitude is equal to the amplitude of the individual motions, find the phase difference between the individual motions.

1. 0

2. $\left(\frac{\pi }{3}\right)$

3. $\left(\frac{2\pi }{3}\right)$

4. $\left(\frac{\pi }{2}\right)$

Q.

Corresponding to point A on acceleration time (a-t) curve of a particle executing SHM, there is a point 1 or 2 or 3 or 4 on velocity time (v-t) curve of the motion. The starting value of time on x-axis is different for two curves. Find the point:

1. 1

2. 2

3. 3

4. 4

Q. The following figure shows the displacement versus time graph for two particles $A$ and $B$ executing simple harmonic motions. The ratio of their maximum velocities (respectively) is

1. $3:1$
2. $1:3$
3. $1:9$
4. $9:1$

Q. A horizontal platform is executing simple harmonic motion in the vertical direction of frequency v. A block of mass m is placed on the platform. What is the maximum amplitude of the platform so that the block is not
detached from it ?

1. $\frac{g}{4{\pi }^2{v}^2}$
2. $\frac{mg}{4{\pi }^2{v}^2}$
3. $\frac{g}{2{\pi }^2{v}^2}$
4. $\frac{gm}{2{\pi }^2{v}^2}$

Q.

A spring block system is doing SHM with Amplitude A and angular frequency $\omega$.

At t=T, the particle is at extreme position. Write displacement versus time equation.

1. $x=Asin\omega t$

2. $x=A+Asin\omega t$

3. $x=Acos\omega t$

4. $x=Asin(\omega t+\frac{\Pi }{2})$

Q. The displacement of a particle from its mean position (in metres) is given by $y=0.2\sin (10\pi t+1.5\pi )\cos (10\pi t+1.5\pi )$. The motion of the particle is SHM, with a time period of (in seconds)

Q. An object of mans $m$ is undergoing linear $\text{SHM}$ according to $x=0.2\text{sin}\left(3.0t\right)\text{m}$. Find the velocity and acceleration of the mass when it is $+5\text{cm}$ from its mean position.

1. $\pm 0.58\text{m/s},0.45{\text{m/s}}^2$
2. $\pm 0.8\text{m/s},1.8{\text{m/s}}^2$
3. $\pm 0.58\text{m/s},-0.45{\text{m/s}}^2$
4. $\pm 0.8\text{m/s},0.45{\text{m/s}}^2$

Q. A particle executes SHM with a period of $T$ second and amplitude $A$ metre. The particle is at its mean position initially. The shortest time it takes to reach point $\frac{A}{\sqrt{2}}$ metre from its mean position in seconds is

1. $T$
2. $\frac{T}{4}$
3. $\frac{T}{8}$
4. $\frac{T}{16}$