Time Period Independent of Amplitude

Q.

A mass is attached to a spring and it executes simple harmonic motion. The mass is also attached to a sensor such that it makes a beeping sound whenever it is in its mean position. The mass is pulled to a length d from the mean position and the frequency f of the beep is noted Now the mass is pulled to a length d' (d' $>$ d) and the frequency of beep was noted to be f'. After this the mass was pushed to d'' (d'' $<$ d) and the frequency of beep was noted f''. Given d, d', d'' are all close to the mean position.

1. we find $f>f^{\prime }>f^{\prime \prime }$

2. we find $f^{\prime }<f^{\prime \prime }<f$

3. we find $f^{\prime }>f>f^{\prime \prime }$

4. we find $f=f^{\prime }=f^{\prime \prime }$

Q. A particle is performing S.H.M with time period $T$. Find the shortest time taken by the particle to go from the mean position to an extreme position.

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

Q. A particle is executing SHM along a straight line. Its velocities at distances $x_1$ and $x_2$ from the mean position are $v_1$ and $v_2$ respectively. Its time period of oscillation is

1. $2\pi \sqrt{\frac{x_2^2-x_1^2}{v_1^2-v_2^2}}$
2. $2\pi \sqrt{\frac{v_1^2+v_2^2}{x_1^2+x_2^2}}$
3. $2\pi \sqrt{\frac{v_1^2-v_2^2}{x_1^2-x_2^2}}$
4. $2\pi \sqrt{\frac{x_1^2+x_2^2}{v_1^2+v_2^2}}$

Q. A spring block system is performing $SHM$ with a time period of $4\text{s}$. If at any instant, the kinetic energy of the system is represented as $K_E$ and potential energy as $K_P$, then what is the time period of oscillation of $K_E-K_P$?

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

Q.

A mass is attached to a spring and it executes simple harmonic motion. The mass is also attached to a sensor such that it makes a beeping sound whenever it is in its mean position. The mass is pulled to a length d from the mean position and the frequency f of the beep is noted Now the mass is pulled to a length d' (d' $>$ d) and the frequency of beep was noted to be f'. After this the mass was pushed to d'' (d'' $<$ d) and the frequency of beep was noted f''. Given d, d', d'' are all close to the mean position.

1. we find $f>f^{\prime }>f^{\prime \prime }$

2. we find $f^{\prime }<f^{\prime \prime }<f$

3. we find $f^{\prime }>f>f^{\prime \prime }$

4. we find $f=f^{\prime }=f^{\prime \prime }$

Q. A block of mass $m$ compresses a spring of stiffness constant $k$ and natural length $l$, through a distance $\frac{l}{2}$ as shown in the figure. If the block is not fixed with the spring, the period of motion of the block is
[Assume collision to be elastic with the wall and horizontal surface to be smooth]

1. $2\pi \sqrt{\frac{m}{K}}$
2. $(\pi +4)\sqrt{\frac{m}{K}}$
3. $(1+\pi )\sqrt{\frac{m}{k}}$
4. None of these

Q. A brass cube of side $a$ and density $\sigma$ is floating in mercury of density $\rho$. If the cube is slightly displaced vertically, it executes $SHM$. Its time period will be

1. $2\pi \sqrt{\frac{\sigma a}{\rho g}}$
2. $2\pi \sqrt{\frac{\rho a}{\sigma g}}$
3. $2\pi \sqrt{\frac{\rho g}{\sigma a}}$
4. $2\pi \sqrt{\frac{\sigma g}{\rho a}}$

Q. A brass cube of side $a$ and density $\sigma$ is floating in mercury of density $\rho$. If the cube is slightly displaced vertically, it executes $SHM$. Its time period will be

1. $2\pi \sqrt{\frac{\sigma a}{\rho g}}$
2. $2\pi \sqrt{\frac{\rho a}{\sigma g}}$
3. $2\pi \sqrt{\frac{\rho g}{\sigma a}}$
4. $2\pi \sqrt{\frac{\sigma g}{\rho a}}$

Q.

A particle executing SHM while moving from one extremity is found to be at distances $x_1,x_{2}and{x}_3$ from the mean position at the end of three successive seconds. The time period of oscillation is ($\theta$ used in the following choices is given by $cos\theta =\frac{x_1+x_3}{2x_2}$

1. $\frac{2\pi }{\theta }$

2. $\frac{\pi }{\theta }$

3. $\theta$

4. $\frac{\pi }{2\theta }$

Q. A cylindrical block of density $d$ stays fully immersed in a beaker filled with two immiscible liquids of different densities $d_1$ and $d_2$. The block is in equilibrium with half of it in liquid $1$ and the other half in liquid $2$ as shown in the figure. If the block is given a displacement downwards and released, then neglecting friction, which of the following statement(s) is/are correct?

1. Block executes simple harmonic motion.
2. The block's motion is periodic but not simple harmonic.
3. The frequency of oscillation is independent of the size of the block.
4. The motion of the block is symmetric about its equilibrium position.

Q. A cylindrical block of density $d$ stays fully immersed in a beaker filled with two immiscible liquids of different densities $d_1$ and $d_2$. The block is in equilibrium with half of it in liquid $1$ and the other half in liquid $2$ as shown in the figure. If the block is given a displacement downwards and released, then neglecting friction, which of the following statement(s) is/are correct?

1. Block executes simple harmonic motion.
2. The block's motion is periodic but not simple harmonic.
3. The frequency of oscillation is independent of the size of the block.
4. The motion of the block is symmetric about its equilibrium position.

Q. A hollow cylindrical shell of radius $R=0.1\text{m}$ has mass $M=6\text{kg}$. It is filled with water having mass $m=3\text{kg}$. It is placed on an inclined plane connected to a spring of spring constant $k=15\text{N/m}$ as shown in the figure. When it is disturbed, it performs oscillations without slipping on the inclined plane. The frequency of the resulting oscillations is
[Assume that the liquid does not rotate inside the cylindrical shell]

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

Q. A cylindrical conducting piston of mass $M$ slides smoothly inside a long cylinder closed at one end, enclosing a certain mass of a gas. The cylinder is kept with its axis horizontal. If the piston is disturbed from its equilibrium position, it oscillates simple harmonically. The period of oscillation will be (process is isothermal)

1. $2\pi \sqrt{\frac{Mh}{PA}}$
2. $2\pi \sqrt{\frac{MA}{Ph}}$
3. $2\pi \sqrt{\frac{M}{PAh}}$
4. $2\pi \sqrt{MPhA}$

Q. A particle executes SHM with a time period of $4\text{s}$. Find the time taken by the particle to go directly from its mean position to half of its amplitude.

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

Q. A spring block system is performing $SHM$ with a time period of $4\text{s}$. If at any instant, the kinetic energy of the system is represented as $K_E$ and potential energy as $K_P$, then what is the time period of oscillation of $K_E-K_P$?

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

Q. The equation of motion of a particle executing simple harmonic motion is $a+16{\pi }^{2}x=0$. In this equation, $a$ is linear acceleration in ${\text{m/s}}^2$ of the particle at displacement $x$ in $\text{m}$. The time period of this simple harmonic motion is

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