Mechanics PYQ

In an experiment the values of two spring constants were measured as $k_{1}=(10 \pm 0.2) \mathrm{N} / \mathrm{m}$ and $k_{2}=(20 \pm 0.3) \mathrm{N} / \mathrm{m}$. If these springs are connected in parallel, then the percentage error in equivalent spring constant is :

A river of width 200 m is flowing from west to east with a speed of $18 \mathrm{~km} / \mathrm{h}$. A boat, moving with speed of $36 \mathrm{~km} / \mathrm{h}$ in still water, is made to travel one-round trip (bank to bank of the river). Minimum time taken by the boat for this journey and also the displacement along the river bank are $\_\_\_\_$ and $\_\_\_\_$ respectively.

Two masses $m$ and 2 m are connected by a light string going over a pulley (disc) of mass 30 m with radius $r=0.1 \mathrm{~m}$. The pulley is mounted in a vertical plane and it is free to rotate about its axis. The 2 m mass is released from rest and its speed when it has descended through a height of 3.6 m is $\_\_\_\_$ $\mathrm{m} / \mathrm{s}$. (Assume string does not slip and $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$)

When both jaws of vernier callipers touch each other, zero mark of the vernier scale is right to zero mark of main scale, $4^{\text {th }}$ mark on vernier scale coincides with certain mark on the main scale. While measuring the length of a cylinder, observer observes 15 divisions on main scale and $5^{\text {th }}$ division of vernier scale coincides with a main scale division. Measured length of cylinder is $\_\_\_\_$ mm. (Least count of Vernier calliper $=0.1 \mathrm{~mm}$)

A $0.5$ kg mass is in contact against the inner wall of a cylindrical drum of radius $4$ m rotating about its vertical axis. The minimum rotational speed of the drum to enable the mass to remain stuck to the wall (without falling) is $5$ rad/s. The coefficient of friction between the drum's inner wall surface and mass is _______. (Take $g = 10$ m/s$^2$)

A solid sphere of radius $4$ cm and mass $5$ kg is rotating (rotation axis is passing through the centre of the sphere) with an angular velocity of $1200$ rpm. It is brought to rest in $10$ s by applying a constant torque. The torque applied and the number of rotations it made before it comes to rest are _______ and _______ respectively.

Two circular discs of radius each 10 cm are joined at their centres by a rod of length 30 cm and mass 600 gm as shown in figure. If the mass of each disc is 600 gm and applied torque between two discs is $43 \times 10^{5}$ dyne. cm, the angular acceleration of the discs about the given axis $A B$ is $\_\_\_\_$ $\mathrm{rad} / \mathrm{s}^{2}$. 

An air bubble of volume $2.9 \mathrm{~cm}^{3}$ rises from the bottom of a swimming pool of 5 m deep. At the bottom of the pool water temperature is $17^{\circ} \mathrm{C}$. The volume of the bubble when it reaches the surface, where the water temperature is $27^{\circ} \mathrm{C}$, is $\_\_\_\_$ $\mathrm{cm}^{3}$. $\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right.$, density of water $=10^{3} \mathrm{~kg} / \mathrm{m}^{3}$, and 1 atm pressure is $\left.10^{5} \mathrm{~Pa}\right)$

A spherical body of radius $r$ and density $\sigma$ falls freely through a viscous liquid having density $\rho$ and viscosity $\eta$ and attains a terminal velocity $v_{0}$. Estimated maximum error in the quantity $\eta$ is : (Ignore errors associated with $\sigma, \rho$ and $g$, gravitational acceleration)

The rain drop of mass $1$ g, starts with zero velocity from a height of $1$ km. It hits the ground with a speed of $5$ m/s. The work done by the unknown resistive force is _______ J. (take $g = 10$ m/s$^2$)

Two identical bodies $A$ and $B$ of equal masses have initial velocities $\vec{v_1} = 4\hat{i}$ m/s and $\vec{v_2} = 4\hat{j}$ m/s respectively. The body $A$ has acceleration $\vec{a_1} = 6\hat{i} + 6\hat{j}$ m/s$^2$ while the acceleration of the other body $B$ is zero. The centre of mass of the two bodies moves in __________ path.

Match List-I with List-II. <table class="pyq-table"><tbody><tr><th>List-I</th><th>List-II</th></tr><tr><td>A. Meter (L)</td><td>I. $\sqrt{\dfrac{hc}{G}}$</td></tr><tr><td>B. Second (S)</td><td>II. $\sqrt{\dfrac{Gh}{c^5}}$</td></tr><tr><td>C. Kilogram (M)</td><td>III. $\sqrt{\dfrac{K^2L^2c^3}{Gh}}$</td></tr><tr><td>D. Kelvin (K)</td><td>IV. $\sqrt{\dfrac{Gh}{c^3}}$</td></tr></tbody></table> where $h$ (Planck's constant), $G$ (gravitational constant) and $c$ (speed of light in vacuum) as fundamental units. Choose the correct answer from the options given below :

A cubical block of density $\rho_{b}=600 \mathrm{~kg} / \mathrm{m}^{3}$ floats in a liquid of density $\rho_{\mathrm{e}}=900 \mathrm{kg} / \mathrm{m}^{3}$. If the height of block is $H=8.0 \mathrm{~cm}$ then height of the submerged part is $\_\_\_\_$ cm.

A water spray gun is attached to a hose of cross sectional area $30$ cm$^2$. The gun comprises of $10$ perforations each of cross sectional area $15$ mm$^2$. If the water flows in the hose with the speed of $50$ cm/s, calculate the speed at which the water flows out from each perforation. (Neglect any edge effects)

Two small balls with masses $m$ and 2 m are attached to both ends of a rigid rod of length $d$ and negligible mass. If angular momentum of this system is $L$ about an axis ($A$) passing through its centre of mass and perpendicular to the rod then angular velocity of the system about $A$ is :

Two blocks of masses $2$ kg and $1$ kg respectively, are tied to the ends of a string which passes over a light frictionless pulley as shown in the figure below. The masses are held at rest at the same horizontal level and then released. The distance traversed by the centre of mass in $2$ s is _______ m. (Take $g = 10$ m/s$^2$) 

When the position vector $\vec{r}=x \hat{i}+y \hat{j}+z \hat{k}$ changes sign as $-\vec{r}$, which one of the following vector will not flip under sign change ?

A body of mass $2$ kg begins to move under the influence of time dependent force $\vec{F} = (2t\hat{i} + 6t^2\hat{j})$ N, where $\hat{i}$ and $\hat{j}$ are unit vectors along $x$ and $y$-axis respectively. The power produced by the force at $t = 2$ s is _____ W.

The position of an object having mass $0.1$ kg as a function of time $t$ is given as $\vec{r} = \left(10t^2\hat{i} + 5t^3\hat{j}\right)$ m. At $t = 1$ s, which of the following statements are correct? A. The linear momentum $\vec{p} = \left(2\hat{i} + 1.5\hat{j}\right)$ kg·m/s. B. The force acting on the object $\vec{F} = \left(2\hat{i} + 3\hat{j}\right)$ N. C. The angular momentum of the object about its origin $\vec{L} = 15\hat{k}$ J·s. D. The torque acting on the object about its origin $\vec{\tau} = 20\hat{k}$ N·m. Choose the correct answer from the options given below :

Suppose there is a uniform circular disc of mass $M \mathrm{~kg}$ and radius $r \mathrm{~m}$ shown in figure. The shaded regions are cut out from the disc. The moment of inertia of the remainder about the axis $A$ of the disc is given by $\frac{x}{256} M r^{2}$. The value of $x$ is $\_\_\_\_$. 

Potential energy ($V$) versus distance ($x$) is given by the graph. Rank various regions as per the magnitudes of the force (F) acting on a particle from high to low. 

Two identical bodies, projected with the same speed at two different angles cover the same horizontal range $R$. If the time of flight of these bodies are $5$ s and $10$ s, respectively, then the value of $R$ is ________ m. (Take $g=10$ m/s$^2$)