Phase and the General Equation

Q.

The maximum acceleration of a body moving in SHM is $a_0$ and maximum velocity is $v_0$. The amplitude of the oscillation is given by,

1. $\frac{v_0^2}{a_0}$

2. $a_0{v}_0$

3. $\frac{a_0^2}{v_0}$

4. $\frac{1}{a_0{v}_0}$

Q.

A particle executes harmonic motion with an angular velocity and maximum acceleration of 3.5 rad/sec and 7.5 m/$s^2$ respectively. The amplitude of oscillation is

[AIIMS 1999; Pb. PET 1999]

1. 0.28 m

2. 0.53 m

3. 0.61 m

4. 0.36 m

Q. Which of the following functions of time represent (a) simple harmonic, (b) periodic but not simple harmonic, and (c) non-periodic motion? Give period for each case of periodic motion (w is any positive constant): (a) sin ωt – cos ωt (b) sin³ ωt (c) 3 cos (p/4 – 2ωt) (d) cos ωt + cos ³ωt + cos ⁵ωt (e) exp (–ω²t²) (f) 1 + ωt + ω²t²

Q.

The equation of a transverse wave is given by

$y=10sin\pi (0.01x-2t)$

where x and y are in cm and t is in second. Its frequency is

[MP PET 1990; MNR 1986; RPET 2003]

1. 

2. 

3. 

4. 

Q. The motion of a particle executing simple harmonic motion is described by the displacement function, x(t) = A cos (ωt + ∅ ). If the initial (t = 0) position of the particle is 1 cm and its initial velocity is ω cm/s, what are its amplitude and initial phase angle ? The angular frequency of the particle is p s–1. If instead of the cosine function, we choose the sine function to describe the SHM : x = B sin (ωt + α), what are the amplitude and initial phase of the particle with the above initial conditions.

Q. A mass attached to a spring is free to oscillate, with angular velocity w, in a horizontal plane without friction or damping. It is pulled to a distance x0 and pushed towards the centre with a velocity υₒ at time t = 0. Determine the amplitude of the resulting oscillations in terms of the parameters ω, xₒ and υₒ. [Hint : Start with the equation x = α cos (ωt+θ) and note that the initial velocity is negative.]

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. 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.

A horizontal platform with an object placed on it is executing S.H.M. in the vertical direction. The amplitude of oscillation is 3.92x${10}^{-3}$. What must be the least period of these oscillations, so that the object is not detached from the platform

1. 0.1256 sec

2. 0.1356 sec

3. 0.1556 sec

4. 0.1456 sec

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. 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. $\sqrt{A^2+B^2}$

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

Q.

The maximum acceleration of a body moving in SHM is $a_0$ and maximum velocity is $v_0$. The amplitude of the oscillation is given by,

1. $\frac{v_0^2}{a_0}$

2. $a_0{v}_0$

3. $\frac{a_0^2}{v_0}$

4. $\frac{1}{a_0{v}_0}$

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 100 gm block is connected to a horizontal massless spring of force constant 25.6 N/m and is free to oscillate on a horizontal frictionless surface. The block is displaced on a horizontal frictionless surface. The block is displaced by 3 cm along positive x - direction and released with an initial velocity of $v_i=-16\sqrt{3}$ cm.

Express position of block as function of time.

1. $x=3cos(16t+\frac{\pi }{2})cm$

2. $x=3cos(16t+\frac{\pi }{3})cm$
3. $x=3.5cos(16t+\frac{\pi }{6})cm$
4. $x=3.2cos(16t+\frac{\pi }{4})cm$

Q. A particle is executing SHM. Find the phase constants of the particle, when it starts from the points $\text{A},\text{B}$ and $\text{C}$ respectively as shown below.
(Phase constant $\varphi$ is the range of $[0,2\pi ]$)

1. ${\varphi }_A=\frac{3\pi }{2},{\varphi }_B=0,{\varphi }_C=\frac{\pi }{2}$
2. ${\varphi }_A=\frac{\pi }{2},{\varphi }_B=\pi ,{\varphi }_C=\frac{\pi }{2}$
3. ${\varphi }_A=\frac{3\pi }{2},{\varphi }_B=\pi ,{\varphi }_C=\frac{\pi }{2}$
4. ${\varphi }_A=\frac{\pi }{2},{\varphi }_B=\pi ,{\varphi }_C=\frac{3\pi }{2}$

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.

The position, velocity and acceleration of a particle executing simple harmonic motion are found to have magnitudes $2cm,1m{s}^{-1}and10m{s}^{-2}$ at a certain instant. Find the amplitude and the time period of the motion.

1. 4.9 cm, 0.28s

2. 9.8 cm, 1s

3. 4.9 cm, 2s

4. 9.8 cm, 0.28s

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. Two simple harmonic motions are represented by equations $y_1=4\sin (10t+\varphi )$ and $y_2=5\cos \left(10t\right)$. What is the phase difference between their velocities?

1. $\varphi$
2. $-\varphi$
3. $(\varphi +\frac{\pi }{2})$
4. $(\varphi -\frac{\pi }{2})$

Q. The displacement x (in metres) of a particle is simple harmonic motion is related to time t (in seconds) as x = $0.01cos(\pi t+\frac{\pi }{4})$ What is the frequency of the motion?

1. 0.5 Hz
2. 1.0 Hz
3. $\frac{\pi }{2}$HZ
4. $\pi HZ$

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

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. 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 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. If position of the particle varies as X=kt^3 then the velocity versus time graph is (where k is positive constant)

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.

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. The displacement x (in metres) of a particle is simple harmonic motion is related to time t (in seconds) as x = $0.01cos(\pi t+\frac{\pi }{4})$ What is the frequency of the motion?

1. 0.5 Hz
2. 1.0 Hz
3. $\frac{\pi }{2}$HZ
4. $\pi HZ$

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. $3\text{ms}$