Body in Water

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. 
2. 
3. 
4. 

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. 
2. 
3. 
4. 

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 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 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 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. $T\sqrt{\frac{\rho }{\rho -1}}$

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 cylindrical piece of cork of density of base area A and height h floats in a liquid of density p₁. The cork is depressed slightly and then released. Show that the cork oscillates up and down simple harmonically with a period T= 2 π √( hp/p₁/g) where r is the density of cork. (Ignore damping due to viscosity of the liquid).

Q. Two solid cylinders connected with a short light rod about common axis have radius R and total mass M rest on a horizontal table top connected to a spring of spring constant k as shown. The cylinders are pulled to the left by x and released. There is sufficient friction for the cylinders to roll. Find time period of oscillation

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

Q. A uniform rod of length l and mass M is provided at the centre. Its two ends are attached to two springs of equal spring constant k. The springs are fixed to rigid support as shown in figure and the rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle $\theta$ in one direction and released. The frequency of oscillation is

1. $\frac{1}{2\pi }\sqrt{\frac{2k}{6M}}$
2. $\frac{1}{2\pi }\sqrt{\frac{k}{M}}$
3. $\frac{1}{2\pi }\sqrt{\frac{6k}{M}}$
4. $\frac{1}{2\pi }\sqrt{\frac{24k}{M}}$