Mechanics PYQ — Page 5

The surface tension of a soap solution is $3.5 \times 10^{-2}$ N/m. The work required to increase the radius of a soap bubble from $1$ cm to $2$ cm is $\alpha \times 10^{-6}$ J. The value of $\alpha$ is _____. ($\pi = 22/7$)

A uniform wire of length $l$ of weight $w$ is suspended from the roof with a weight of $W$ at the other end. The stress in the wire at $\dfrac{l}{3}$ distance from the top is $\left(\dfrac{W}{A} + \dfrac{2}{\gamma}\dfrac{w}{A}\right)$, where, $A$ is the cross sectional area of the wire. The value of $\gamma$ is _______.

In a screw gauge when the circular scale is given five complete rotations it moves linearly by $2.5$ mm. If the circular scale has $100$ divisions, the least count of screw gauge is _____ mm.

Given below are two statements: Statement I: Pressure of a fluid is exerted only on a solid surface in contact as the fluid-pressure does not exist everywhere in a still fluid. Statement II: Excess potential energy of the molecules on the surface of a liquid, when compared to interior, results in surface tension. In the light of the above statements, choose the correct answer from the options given below

A spherical liquid drop of radius $R$ acquires the terminal velocity $v_1$ when falls through a gas of viscosity $\eta$. Now the drop is broken into $64$ identical droplets and each droplet acquires terminal velocity $v_2$ falling through the same gas. The ratio of terminal velocities $v_1/v_2$ is ________.

The diameter of a wire measured by a screw gauge of least count $0.001$ cm is $0.08$ cm. The length measured by a scale of least count $0.1$ cm is $150$ cm. When a weight of $100$ N is applied to the wire, the extension in length is $0.5$ cm, measured by a micrometer of least count $0.001$ cm. The error in the measured Young's modulus is $\alpha \times 10^9$ N/m$^2$. The value of $\alpha$ is _______. (Ignore the contribution of the load to Young's modulus error calculation)

A tub is filled with water and a wooden cube $10$ cm $\times$ $10$ cm $\times$ $10$ cm is placed in the water. The wooden cube is found to float on the water with a part of it submerged in water. When a metal coin is placed on the wooden cube, the submerged part is increased by $3.87$ cm. The mass of the metal coin is _______ gram. (Take water density as $1$ g/cm$^3$ and density of wood as $0.4$ g/cm$^3$)

A ball of radius $r$ and density $\rho$ dropped through a viscous liquid of density $\sigma$ and viscosity $\eta$ attains its terminal velocity at time $t$, given by $t=A \rho^{a} r^{b} \eta^{\mathrm{c}} \sigma^{d}$, where $A$ is a constant and $a, b, c$ and $d$ are integers. The value of $\frac{b+c}{a+d}$ is $\_\_\_\_$.

A solid sphere ($A$) of mass $5m$ and a spherical shell ($B$) of mass $m$, both having same radius, are placed on a rough surface. When a force of same magnitude is applied tangentially at the highest points of $A$ and $B$, they start rolling without slipping with an acceleration of $a_A$ and $a_B$, respectively. The ratio of $a_A$ and $a_B$ is __________.

Water flows through a horizontal tube as shown in the figure. The difference in height between the water columns in vertical tubes is 5 cm and the area of cross-sections at $A$ and $B$ are $6 \mathrm{~cm}^{2}$ and $3 \mathrm{~cm}^{2}$ respectively. The rate of flow will be $\_\_\_\_$ $\mathrm{cm}^{3} / \mathrm{s}$. (take $g=10 \mathrm{~m} / \mathrm{s}^{2}$) 

The pulley shown in figure is made using a thin rim and two rods of length equal to diameter of the rim. The rim and each rod have a mass of $M$. Two blocks of mass of $M$ and $m$ are attached to two ends of a light string passing over the pulley, which is hinged to rotate freely in vertical plane about its center. The magnitudes of the acceleration experienced by the blocks is $\_\_\_\_$ (assume no slipping of string on pulley). 

Four persons measure the length of a rod as $20.00 \mathrm{~cm}, 19.75 \mathrm{~cm}, 17.01 \mathrm{~cm}$ and 18.25 cm. The relative error in the measurement of average length of the rod is :

A flexible chain of mass $m$ hangs between two fixed points at the same level. The inclination of the chain with the horizontal at the two points of support is $30^{\circ}$. Considering the equilibrium of each half of the chain, the tension of the chain at the lowest point is $\_\_\_\_$.

A large drum having radius $R$ is spinning around its axis with angular velocity $\omega$, as shown in figure. The minimum value of $\omega$ so that a body of mass $M$ remains stuck to the inner wall of the drum, taking the coefficient of friction between the drum surface and mass $M$ as $\mu$, is : 

The velocity at which $6$ kg mass (shown in figure) strikes the ground when it is released from a height of $6$ m above the ground is __________ m/s. Assume pulley is massless and string is light and inextensible. (Take g $= 10$ m/s$^2$) 

A body of mass $1$ kg moves along a straight line with a velocity $v = 2x^2$. The work done by the body during displacement from $x = 0$ to $5$ m is __________ J.

A circular disc has radius $R_{1}$ and thickness $T_{1}$. Another circular disc made of the same material has radius $R_{2}$ and thickness $T_{2}$. If the moment of inertia of both discs are same and $\frac{R_{1}}{R_{2}}=2$ then $\frac{T_{1}}{T_{2}}=\frac{1}{\alpha}$. The value of $\alpha$ is $\_\_\_\_$.

The velocity $(v)$ - Distance $(x)$ graph is shown in figure. Which graph represents acceleration $(a)$ versus distance $(x)$ variation of this system? 

A bead $P$ sliding on a frictionless semi-circular string $(A C B)$ and it is at point $S$ at $t =0$ and at this instant the horizontal component of its velocity is $v$. Another bead $Q$ of the same mass as $P$ is ejected from point $A$ at $t=0$ along the horizontal string $A B$, with the speed $v$, friction between the beads and the respective strings may be neglected in both cases. Let $t_{P}$ and $t_{Q}$ be the respective times taken by beads $P$ and $Q$ to reach the point $B$, then the relation between $t_{P}$ and $t_{Q}$ is 

The escape velocity from a spherical planet $A$ is $10 \mathrm{~km} / \mathrm{s}$. The escape velocity from another planet $B$ whose density and radius are $10 \%$ of those of planet $A$, is $\_\_\_\_$ $\mathrm{m} / \mathrm{s}$.

Dimensions of universal gravitational constant $(G)$ in terms of Planck's constant $(h)$, distance $(L)$, mass $(M)$ and time $(T)$ are _______.

If an air bubble of diameter $2$ mm rises steadily through a liquid of density $2000$ kg/m$^3$ at a rate of $0.5$ cm/s, then the coefficient of viscosity of liquid is _______ Poise. (Take $g = 10$ m/s$^2$)

The velocity of a particle is given as $\vec{v} = -x\hat{i} + 2y\hat{j} - z\hat{k}$ m/s. The magnitude of acceleration at point $(1, 2, 4)$ is _______ m/s$^2$.

In an experiment, a set of reading are obtained as follows - $1.24 \mathrm{~mm}, 1.25 \mathrm{~mm}, 1.23 \mathrm{~mm}$, 1.21 mm. The expected least count of the instrument used in recording these readings is $\_\_\_\_$ mm.