Body in Water

Q. The time period of a spring-mass system is $T$ in air. When the mass is partially immersed in water, the time period of oscillation is $T^{\prime }$. Then:


[Assume the mass is cubical]

1. $T^{\prime }=T$
2. $T^{\prime }<T$
3. $T^{\prime }>T$
4. $T^{\prime }\le T$

Q. A test tube of cross-sectional area a has some lead shots in it. The total mass is m. It floats upright in liquid of density d. when pushed down little and released, it oscillates up and down with a period T. Use dimensional considerations and choose the correct relationship from the following.

1. $T=2\pi \sqrt{\frac{ma}{gd}}$
2. $T=2\pi \sqrt{\frac{mg}{ad}}$
3. $T=2\pi \sqrt{\frac{md}{ad}}$
4. $T=2\pi \sqrt{\frac{m}{adg}}$

Q. In the figure shown, the mass of the plank is m and that of the solid cylinder is $8\text{m}$. Springs are light. The plank is slightly displaced from equilibrium and then released. There is no slipping at any contact point. The ratio of mass of the plank and stiffness of the spring i.e. $\frac{m}{k}=\frac{2}{{\pi }^2}$ . The period of small oscillations of the plank (in seconds) is

Q. A vertical U – tube of uniform cross – sectional area A contains a liquid of density $\rho$ The total length of the liquid column in the tube is L. the liquid column is disturbed by gently blowing into the tube. If viscous effects are neglected, the time period of the resulting oscillation of the liquid column is given by

1. $T=2\pi \sqrt{\frac{L\rho }{gA}}$
2. $T=2\pi \sqrt{\frac{L}{\rho gA}}$
3. $T=2\pi \sqrt{\frac{L}{g}}$
4. $T=2\pi \sqrt{\frac{L}{2g}}$

Q. A rectangular block of mass $m$ and area of cross-section $A$ floats in a liquid of density $\rho$. If it is given a small vertical displacement from equilibrium it undergoes oscillation with a time period $T$. Then,

1. $T\propto \sqrt{\rho }$
2. $T\propto {\rho }^0$
3. $T\propto \frac{1}{\rho }$
4. $T\propto \frac{1}{\sqrt{\rho }}$

Q. A test tube of cross-sectional area a has some lead shots in it. The total mass is m. It floats upright in liquid of density d. when pushed down little and released, it oscillates up and down with a period T. Use dimensional considerations and choose the correct relationship from the following.

1. $T=2\pi \sqrt{\frac{ma}{gd}}$
2. $T=2\pi \sqrt{\frac{mg}{ad}}$
3. $T=2\pi \sqrt{\frac{md}{ad}}$
4. $T=2\pi \sqrt{\frac{m}{adg}}$

Q. A vertical U – tube of uniform cross – sectional area A contains a liquid of density $\rho$ The total length of the liquid column in the tube is L. the liquid column is disturbed by gently blowing into the tube. If viscous effects are neglected, the time period of the resulting oscillation of the liquid column is given by

1. $T=2\pi \sqrt{\frac{L\rho }{gA}}$
2. $T=2\pi \sqrt{\frac{L}{\rho gA}}$
3. $T=2\pi \sqrt{\frac{L}{g}}$
4. $T=2\pi \sqrt{\frac{L}{2g}}$

Q. A solid cube floats in water half immersed and has small vertical oscillations of time period $\frac{\pi }{5}\text{s}$. Its mass $\left(\text{in kg}\right)$ is:
$(\text{Take}g=10{\text{ms}}^{-2})$

1. $4$
2. $2$
3. $1$
4. $0.5$

Q. The time period of a spring-mass system is $T$ in air. When the mass is partially immersed in water, the time period of oscillation is $T^{\prime }$. Then:


[Assume the mass is cubical]

1. $T^{\prime }=T$
2. $T^{\prime }<T$
3. $T^{\prime }>T$
4. $T^{\prime }\le T$

Q. The metallic bob of a simple pendulum has the relative density $\rho$. The time period of this pendulum is $T$. If the metallic bob is immersed completely in water, then the new time period is given by:

1. $T\left(\frac{\rho -1}{\rho }\right)$
2. $T\left(\frac{\rho }{\rho -1}\right)$
3. $T\sqrt{\frac{\rho -1}{\rho }}$
4. $T\sqrt{\frac{\rho }{\rho -1}}$