Velocity Displacement Relationship

Q. A particle executes linear simple harmonic motion with an amplitude of $3\text{cm}$. When the particle is at $2\text{cm}$ from the mean position, the magnitude of its velocity is equal to that of its acceleration. Find the time period in seconds.

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

Q. A particle performs SHM with a period $T$ and amplitude $A$. The mean velocity of the particle over the time interval during which it travels a distance $\frac{A}{2}$ from the extreme position is

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

Q. Two particles are projected from the same point with the same speed $u$ such that they have the same range $R$, but different maximum heights, $h_1$ and $h_2$. Which of the following is correct ?

1. $R^2=4h_1{h}_2$
2. $R^2=16h_1{h}_2$
3. $R^2=2h_1{h}_2$
4. $R^2=h_1{h}_2$

Q.

The average Kinetic energy of the molecule of a gas is directly proportional to the _______________.

Q. A body executing linear simple harmonic motion has a velocity of $3cm/s$, when its displacement is $4cm$ and a velocity of $4cm/s$, when its displacement is $3cm$. What is the amplitude of oscillation?

1. $7.5cm$
2. $10cm$
3. $12.5cm$
4. $5cm$

Q. The displacement of a damped harmonic oscillator is given by $x\left(t\right)=e^{-0.1t}\cos (10\pi t+\varphi )$, where $t$ is in $\text{seconds}$. The time taken for its amplitude of vibration to drop to half of its initial value is close to:
$[\ln 2=0.7]$

1. $27\text{s}$
2. $13\text{s}$
3. $4\text{s}$
4. $7\text{s}$

Q.

What are the most probable velocity and the average velocity for a system that follows the Maxwell-Boltzmann distribution?

Q. A particle is performing SHM with an amplitude $A$ and angular frequency $\omega$. Find the displacement of the particle from the mean position where the speed of the particle becomes half of the maximum speed.

1. $x=\frac{A}{\sqrt{2}}$
2. $x=\pm \frac{\sqrt{3}A}{2}$
3. $x=\frac{-A}{\sqrt{2}}$
4. $x=\pm \frac{A}{2}$

Q.

A particle on a stretched string supporting a travelling wave, takes 5.0 ms to move from its mean position to the extreme position. The distance between two consecutive particles, which are at their mean positions, is 2.0 cm. Find the frequency, the wavelength and the wave speed.

Q.

Find the acceleration of the 500 g block in figure.

Q.

The maximum speed and acceleration of a particle executing simple harmonic motion are $10cm{s}^{-1}and50cm{s}^{-2}$. Find the position(s) of the particle when the speed is $8c{m}^{-1}$.

1. 

2. 

3. 

4. 

Q. A particle executing SHM oscillates between two fixed points separated by $20\text{cm}$. If its maximum velocity is $30\text{cm/s}$, find its velocity when the displacement is $5\text{cm}$ from the mean position.

1. $15\sqrt{3}\text{cm/s}$
2. $20\sqrt{3}\text{cm/s}$
3. $10\sqrt{3}\text{cm/s}$
4. $5\sqrt{3}\text{cm/s}$

Q. The particle is executing $\text{SHM}$ on a line $4\text{cm}$ long. If its velocity at mean position is $12\text{cm/s}$, its frequency in $\text{Hertz}$ will be :

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

Q. The time period of an object performing SHM is $16\text{s}$. It starts its motion from the equilibrium position. After $2\text{s}$ its velocity is $\pi \text{m/s}$. What is its displacement amplitude?

1. $\sqrt{2}\text{m}$
2. $4\sqrt{2}\text{m}$
3. $8\sqrt{2}\text{m}$
4. $2\sqrt{2}\text{m}$

Q.

prove that the phase difference between displacement and velocity of an object in shm is pi/2.

Q. A particle performing SHM has a velocity $v_0$ at mean position. Find the velocity of the particle, when it reaches a point which is at a distance $\frac{A}{2}$ from the mean position.
[$A=$ amplitude of SHM]

1. $\frac{v_0}{4}$
2. $\frac{\sqrt{3}}{2}{v}_0$
3. $\frac{3v_0}{2}$
4. $\frac{v_0}{\sqrt{2}}$

Q.

The angular frequency of a simple pendulum is $\omega \mathrm{rad}/\sec$. Now, if the length is made one-fourth of the original length, the angular frequency becomes

1. $\frac{\omega }{2}$

2. $2\omega$

3. $4\omega$

4. $\omega$

Q.

The motion of a particle executing S.H.M. is given by x = 0.01 sin 100 $\pi$ ( t + 0.05) , where x is in metres and time is in seconds. The time period is

1. 0.01 sec

2. 0.02 sec

3. 0.1 sec

4. 0.2 sec

Q. A particle moves along the positive branch of the curve $y=\frac{x^2}{2}$ where $x=\frac{t^2}{2}$, $x$ and $y$ are measured in metre and t in second. At $t=2s$, the velocity of the particle is

1. $(4\hat{i}-2\hat{j})m/s$
2. $(2\hat{i}+4\hat{j})m/s$
3. $(2\hat{i}-4\hat{j})m/s$
4. $(2\hat{i}+2\hat{j})m/s$

Q. A particle executing SHM oscillates between two fixed points separated by $20\text{cm}$. If its maximum velocity is $30\text{cm/s}$, find its velocity when the displacement is $5\text{cm}$ from the mean position.

1. $15\sqrt{3}\text{cm/s}$
2. $20\sqrt{3}\text{cm/s}$
3. $10\sqrt{3}\text{cm/s}$
4. $5\sqrt{3}\text{cm/s}$

Q.

The acceleration of a particle starting from rest varies with time according to relation $a=\alpha t+\beta$. What will be the velocity of the particle after time $t$?

Q. In an a.c. circuit the voltage applied is $E=E_{0}sin\omega t$. The resulting current in the circuit is $I=I_{0}sin(\omega T-\frac{\pi }{2})$ The power consumption in the circuit is given by

1. $P=\sqrt{2}{E}_0{I}_0$
2. $P=\frac{E_0{I}_0}{\sqrt{2}}$
3. $P=Zero$
4. $P=\frac{E_0{I}_0}{2}$

Q. A block with a mass of $1\text{kg}$ is connected to a massless spring with a force constant of $100\text{N/m}$ and is placed on a smooth horizontal surface. At $t=0$, a constant force of $F=10\text{N}$ is applied on the block, when the spring is in its natural length. The speed of the block, when it is displaced by $8\text{cm}$ from its mean position will be

1. $80\text{cm/s}$
2. $16\text{cm/s}$
3. $25\text{cm/s}$
4. $60\text{cm/s}$

Q.

A 660 Hz tuning fork sets up vibration in a string clamped at both ends. The wave speed for a transverse wave on this string is $220m{s}^{-1}$ and the string vibrates in three loops. (a) Find the length of the string. (b) If the maximum amplitude of a particle is 0.5 cm, write a suitable equation describing the motion.

Q. A particle executes SHM, with an amplitude of $10\text{cm}$. When the particle is at $6\text{cm}$ from the mean position, the magnitude of its velocity is equal to twice of its acceleration. Find the time period (in seconds)

1. $4.1\text{s}$
2. $9.42\text{s}$
3. $14.8\text{s}$
4. $1.17\text{s}$

Q.

A cyclist goes uphill at a speed of 8 km/h and downhill at a speed of 32 km/h. If uphill and downhill journeys involve the same distance, what is the average speed during the whole journey?

Q. Two particles P and Q start from origin and execute Simple Harmonic Motion along X-axis with same amplitude but with periods 3 seconds and 6 seconds respectively. The ratio of the velocities of P and Q when they meet at mean position is

1. 1 : 2
2. 2 : 1
3. 3 : 2
4. 2 : 3

Q.

Does amplitude affect period$?$

Q. Two blocks A and B of masses 10 kg and 15 kg are placed in contact with each other rest on a rough horizontal surface as shown in the figure. The coefficient of friction between the blocks and surface is 0.2. A horizontal force of 200 N is applied to block A. The acceleration of the system is: (Take $g=10m{s}^{-2}$)

1. $6m{s}^{-2}$
2. $4m{s}^{-2}$
3. $8m{s}^{-2}$
4. $10m{s}^{-2}$

Q.

A body covers 26, 28, 30, 32 meters in 10th, 11th, 12th and 13th seconds respectively. The body starts

1. from rest and moves with uniform velocity

2. from rest and moves with uniform acceleration

3. with an initial velocity and moves with uniform acceleration

4. with an initial velocity and moves with uniform velocity