Phase and the General Equation

Q. Two particles execute SHM of the same amplitude and frequency along the same straight line. They pass one another when going in opposite directions each time their displacement is half their amplitude. What is the phase difference between them?

1. ${60}^{\circ }$
2. ${30}^{\circ }$
3. ${90}^{\circ }$
4. ${120}^{\circ }$

Q. Two particles $P$ and $Q$ describe $\text{SHM}$ of same amplitude $A$, same frequency $f$ along the same straight line. The maximum distance between the two particles is $A\sqrt{2}$. The phase difference between the particle is

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

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 equilibrium position is

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

Q. A particle executes simple harmonic motion between x = -A and x = +A. The time taken for it to go from 0 to A/2 is $T_1$ and to go from 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. A particle executing S.H.M. of amplitude 4 cm and T=4 sec. The time taken by it to move from positive extreme position(+a) to half the amplitude is

1. 1 sec
2. 1/3 sec
3. 2/3 sec
4. √3/2 sec

Q.

Two bodies performing $\text{S.H.M}$. have same amplitude and frequency. Their phases at a certain instant are as shown in the figure. The phase difference between them is

1. $\pi$
2. $\frac{\pi }{3}$
3. $\frac{3}{5}\pi$
4. $\frac{11}{6}\pi$

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

Q. A particle is performing simple harmonic motion along $x-$ axis with amplitude $4\text{cm}$ and time period $1.2\text{sec}$. The minimum time taken by the particle to move from $x=+2\text{cm}$ to $x=+4\text{cm}$ and back again is given by

1. $0.2\text{sec}$
2. $0.4\text{sec}$
3. $0.3\text{sec}$
4. $0.6\text{sec}$

Q.

A particle performs simple harmonic motion with a period of $2s$. The time taken by the particle to cover a displacement equal to half of its amplitude from the mean position is $\left(\frac{1}{a}\right)s$. The value of ‘$a$’ to the nearest integer is _____.

Q. A particle is executing simple harmonic motion, Its displacement is given by x=5 sin $\pi$t, where x is in cm and t in seconds. How long will the particle take to move from the position of equilibrium to the position of maximum displacement?

1. 0.5 s
2. 1.0 s
3. 1.5 s
4. 2.0 s

Q. Starting from rest, a particle rotates in a circle of radius $R=\sqrt{2}m$ with an angular acceleration $\alpha =\frac{\pi }{4}rad/s^2$. Average velocity $\left(\text{in}m/s\right)$ of the particle over the time it rotates quarter circle.

Q. The amplitude of a particle executing S.H.M. with frequency of 60 Hz is 0.01 m. The maximum value of the acceleration of the particle is

1. 
2. 
3. 
4. 

Q. A particle moves on the X-axis according to the equation $x=A+Bsin\omega t$. The motion is simple harmonic with amplitude.

1. A

2. B

3. A+B

4. 

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.

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. 

3. 

4. 

Q. Let $T_1$ and $T_2$ be the time period of oscillation of two springs $A$ and $B$ when a mass $m$ is suspended from them separately. Now both the springs are connected in parallel and the same mass $m$ is suspended with them. If $T$ is time period of the parallel system, then

1. $T=T_1+T_2$
2. $T=\frac{T_1{T}_2}{T_1+T_2}$
3. $T^2=T_1^2+T_2^2$
4. $\frac{1}{T^2}=\frac{1}{T_1^2}+\frac{1}{T_2^2}$

Q.

A bullet when fired at a target has its velocity decreased to $50\%$ after penetrating $30cm$ into it. Then the additional thickness it will penetrate before resting is

1. $10cm$

2. $30cm$

3. $40cm$

4. $60cm$

Q. The displacement of a particle is represented by the equation $y=3\text{cos}(\frac{\pi }{4}-2\omega t)$. The motion of the particle is

1. Simple harmonic with period $\frac{2\pi }{\omega }$
2. Periodic but not simple harmonic
3. Simple harmonic with period $\frac{\pi }{\omega }$
4. Non-periodic

Q. 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. $T^{-1}=t_1^{-1}+t_2^{-1}$

2. $T^2=t_1^2+t_2^2$

3. $T=t_1+t_2$

4. $T^{-2}=t_1^{-2}+t_2^{-2}$

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.

Consider a particle moving in simple harmonic motion according to the equation $x=2.0cos(50\Pi t+ta{n}^{-1}0.75)$

where x is in centimeter and t in second. The motion is started at t = 0. (a) When does the particle come to rest for the second time?

1. sec

2. sec

3. sec

4. sec

Q.

A particle is moving in a straight line with SHM. Its velocity has the values $3m/s$ and $2m/s$ when its distances from mean position are $1m$ and $2m$ respectively. Find the length of its path and period of its motion.

1. $5.06m$, $6.03s$

2. $5.06m$, $4.87s$

3. $2.53m$, $4.87s$

4. None of these

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

1. Zero
2. 
3. 
4. 

Q. Find out the phase constant $\varphi$ [in $\text{rad}$] of a particle performing SHM, if the initial conditions are as shown in the figure.

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

Q. In the given figure, the block is attached with a system of three ideal springs $A,B$ and $C$. The block is displaced by a small distance $x$ from its equlibrium position vertically downwards and released. $T$ represents the time period of small vertical oscillations of the block (pulleys are ideal). Choose the correct alternative.

1. $T=2\pi \sqrt{\frac{11m}{2k}}$
2. the deformation of the spring $A$ is $\frac{2}{11}$ times the displacement of the block.
3. the deformation of the spring $C$ is $\frac{1}{11}$ times the displacement of the block.
4. the deformation of the spring $B$ is $\frac{4}{11}$ times the displacement of the block.

Q.

A block of mass 5 kg executes simple harmonic motion under the restoring force of a spring. The amplitude and the time period of the motion are 0.1 m and 3.14 s respectively. Find the maximum force exerted by the spring on the block.

1. 2N

2. 1N

3. 0.5N

4. 20 N

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 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. 
2. 
3. 
4. 

Q. A particle executes $\text{SHM}$ of amplitude $A$ and time period $T$. Find the distance travelled by the particle in the duration its phase changes by $\frac{\pi }{12}$ to $\frac{5\pi }{12}$.

1. $\frac{A}{\sqrt{2}}$
2. $\sqrt{\frac{2}{3}}A$
3. $\sqrt{\frac{3}{2}}A$
4. $\frac{2}{\sqrt{3}}A$

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