Body in Water

Q. A horizontal rod of mass m and length L is pivoted smoothly at one end. The rod’s other end is supported by a spring of force constant k. The rod is rotated (in the vertical plane) by a small angle from its horizontal equilibrium position and released. The angular frequency of the subsequent simple harmonic motion is

1. 
2. 
3. 
4. 

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 bob of mass $m$ suspended by a length $l$ undergoes simple harmonic oscillations with time period $T$. If the bob is immersed in a liquid that has density $\frac{1}{4}$ times that of the bob and the length of the threat is increased by $(1/3{)}^{rd}$ of the original length, then the time period of the simple harmonic oscillation will be

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

Q. Two stones are projected with the same magnitude of velocity, but making different angles with horizontal. The angle of projection of one is $\frac{\pi }{3}$ and its maximum height is Y, the maximum height attained by the other stone with as $\frac{\pi }{6}$ angle of projection is

1. 
2. 3 Y
3. 2 Y
4. Y

Q.

A metal ball immersed in alcohol weighs $W_1$at 0°C and$W_2$ at 59°C. The coefficient of cubical expansion of the metal is less than that of alcohol. Assuming that the density of metal is large compared to that of alcohol, it can be shown that

1. 

2. 

3. 

4. 

Q. A uniform cylinder of length L and mass M having cross – sectional area A is suspended with its vertical length, from a fixed point by a massless spring, such that it is half – submerged in a liquid of density d at equilibrium position. When the cylinder is given a small downward push and released, it starts oscillating vertically with a small amplitude. If the force constant of the spring is K, the frequency of oscillation of the cylinder is.

1. 
2. 
3. 
4. 

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. An incompressible fluid of density $\rho$ flows through a horizontal pipe of radius r and then passes through a constriction of radius $\frac{r}{2}$ . If the fluid has pressure $P_o$ and velocity $V_o$ before the constriction, pressure in the constriction is $P_o-\frac{3\alpha }{2}\rho V_o^2$. Here $\alpha$ is an integer. Find $\alpha$ .

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. The loss of weight of a solid when immersed in a liquid at $0^{\circ }C$ is $W_0$ then the loss of weight W at $t^{\circ }C$ is - where $\alpha$ and $\beta$ are the cubical expansion coefficients of solid and liquid respectively.

1. 
2. 
3. 
4. 

Q. For the arrangement shown, a solid block of mass $M$ is connected to another block of mass $m$ by string. The block of mass $m$ is connected to the spring. The cross-sectional area of the submerged block is $A$. Initially, the submerged block is in equilibrium position and is displaced slightly inside the liquid of density $\rho$. The motion of mass $M$ is a simple harmonic motion. What is its time period?

1. $2\pi \sqrt{\frac{2M}{(2K+\rho Ag)}}$
2. $2\pi \sqrt{\frac{2M}{(K+\rho Ag)}}$
3. $2\pi \sqrt{\frac{M}{(2K+\rho Ag)}}$
4. $2\pi \sqrt{\frac{M}{(K+\rho Ag)}}$

Q. On a planet, a body is found to freely fall through a distance of 8 m in 2 seconds. On that planet what is the time period of a simple pendulum of length 1 m?

Q. On a smooth inclined surface a body of mass M is attached between two springs. The other ends of the springs are fixed to firm supports. If each spring has force constant k, the period of oscillation of the body is

1. 
2. 
3. 
4. 

Q. If the time period ($T$) of vibrating a liquid drop depends on surface tension ($S$), radius ($r$) of the drop and density $\left(\rho \right)$ of the liquid, then the expression of $T$ is :

1. $T=K\sqrt{\frac{\rho r^3}{S^{1/2}}z}$
2. $T=K\sqrt{\frac{\rho {{}^{1/2}r}^3}{S}}$
3. none
4. $T=K\sqrt{\frac{\rho r^3}{S}}$

Q. A pendulum has time period T in air when it is made to oscillate in water, it acquired a time period $T^{\prime }=\sqrt{2}T$. The density of the pendulum bob is equal to (density of water =1) :

1. None of these
2. $\sqrt{2}$
3. $2\sqrt{2}$
4. $2$

Q. $\frac{5\lambda }{ka}$ is the angular divergence between the first minima and third maxima on either side of the central line. Then $k$ is

(Assume $\lambda <<a,a$ is slit width).

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 rod of length l and mass m, pivoted at one end, is held by a spring at its mid-point and a spring at far end. The springs have spring constant k. Find the frequency of small oscillations about the equilibrium position.

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

Q. A disc of mass $m$ and radius $r$ is free to rotate about its centre shown in the figure. A string id wrapped over its rim and a block of mass $m$ is attached to the free end of the string. The system is released from rest. The speed of the block as it descends through a height $h$, is.

1. $\sqrt{2gh}$
2. $\sqrt{\frac{2}{3}gh}$
3. $2\sqrt{\frac{gh}{3}}$
4. $\frac{1}{2}\sqrt{3gh}$

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 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. A student uses a simple pendulum of exactly $1m$ length to determine $g$, the acceleration due to gravity. He uses a stopwatch with the least count of 1 $\sec$ for this and records 40 seconds for 20 oscillations. For this observation, which of the following statements is(are) true?

1. Error $\Delta T$ in measuring $T$, the time period, is 0.05 seconds.
2. Error $\Delta T$ in measuring $T$, the time period, is 1 second.
3. Percentage error in the determination of $g$ is 2.5%.
4. Percentage error in the determination of $g$ is 5%.

Q. A rod of mass m and length l is connected by two spring of spring constants $k_1$ and $k_2$, so that it is horizontal at equilibrium. What is the natural frequency of the system?

1. $\frac{1}{2\pi }\sqrt{\frac{2k_1{b}^2+k_2{l}^2}{m{l}^2}}$
2. $\frac{1}{2\pi }\sqrt{\frac{k_1{b}^2+k_2{l}^2}{2m{l}^2}}$
3. $\frac{1}{2\pi }\sqrt{\frac{3(k_1{b}^2+k_2{l}^2)}{m{l}^2}}$
4. $\frac{1}{2\pi }\sqrt{\frac{k_1{b}^2+k_2{l}^2}{m{l}^2}}$

Q. A pendulum clock loses $12$s a day if the temperature is ${40}^o$C and gains $4$s a day if the temperature is ${20}^o$C. The temperature at which the clock will show correct time, and the co-efficient of linear expansion $\left(\alpha \right)$ of the metal of the pendulum shaft are respectively.

1. ${25}^o$C$;\alpha =1.85\times {10}^{-5}{/}^o$C
2. ${30}^o$C$;\alpha =1.85\times {10}^{-3}{/}^o$C
3. ${60}^o$C $;\alpha =1.85\times {10}^{-4}{/}^o$C
4. ${55}^o$C$;\alpha =1.85\times {10}^{-2}{/}^o$C

Q. A mass $M$ attached to a horizontal spring, executes SHM with amplitude $A_1$. When the mass $M$ passes through its mean position then a smaller mass $m$ is placed over it and both of them move together with amplitude $A_2$. The ratio of $\left(\frac{A_1}{A_2}\right)$ is:

1. $\frac{M}{M+m}$
2. $\frac{M+m}{M}$
3. $\sqrt{\frac{M}{M+m}}$
4. $\sqrt{\frac{M+m}{M}}$

Q. A solid uniform cylinder of mass m performs small oscillations due to the action of two springs of stiffness k each (figure). Find the period of these oscillations in the absence of sliding.

Q. A uniform rod of mass $M$ and length $L$ is pivoted about point $O$ as shown in the figure given. It is slightly rotated from its mean position so that it performs angular simple harmonic motion. For this physical pendulum, determine the time period of oscillation.

1. $2\pi \sqrt{\frac{L}{g}}$
2. $\pi \sqrt{\frac{7L}{3g}}$
3. $2\pi \sqrt{\frac{2L}{3g}}$
4. $noneofthese$

Q. If level of liquid starts decreasing slowly when the level of liquid is at a height $h_1$ above the cylinder, the block starts moving up. Then, value of $h_1$ is :

1. $5h/2$
2. $5h/4$
3. $5h/3$
4. $2h/3$