Mechanics PYQ — Page 2

A small bob $A$ of mass $m$ is attached to a massless rigid rod of length 1 m pivoted at point $P$ and kept at an angle of $60^{\circ}$ with vertical as shown in figure. At distance of 1 m below point $P$, an identical bob $B$ is kept at rest on a smooth horizontal surface that extends to a circular track of radius $R$ as shown in figure. If bob $B$ just manages to complete the circular path of radius $R$ upto a point $Q$ after being hit elastically by $\operatorname{bob} A$, then radius $R$ is $\_\_\_\_$ m. 

In a screw gauge the zero of main scale reference line coincides with the fifth division of the circular scale when two studs are in contact. There are $100$ divisions in circular scale and pitch of screw gauge is $0.1$ mm. When diameter of a sphere is measured, the reading of main scale is $5$ mm and $50^{\text{th}}$ division of circular scale coincides with the reference line of main scale. The diameter of sphere is _______ mm.

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 $\_\_\_\_$ $\mathrm{rad} / \mathrm{s}^{2}$. 

A lift of mass $1600$ kg is supported by thick iron wire. If the maximum stress which the wire can withstand is $4 \times 10^8$ N/m$^2$ and its radius is $4$ mm, then maximum acceleration the lift can take is _______ m/s$^2$. (take $g = 10$ m/s$^2$ and $\pi = 3.14$)

Initially a satellite of 100 kg is in a circular orbit of radius $1.5 \mathrm{R}_{\mathrm{E}}$. This satellite can be moved to a circular orbit of radius $3 R_{E}$ by supplying $\alpha \times 10^{6} \mathrm{~J}$ of energy. The value of $\alpha$ is $\_\_\_\_$. (Take Radius of Earth $R_{E}=6 \times 10^{6} \mathrm{~m}$ and $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}$)

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 :

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)

At $t=0$, a body of mass $100$ g starts moving under the influence of a force $(5\hat{i}+10\hat{j})$ N. After $2$ s its position is $(2x\hat{i}+5y\hat{j})$ m. The ratio $x:y$ 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 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 particle of mass 2 kg is projected vertically upward with a speed of 30 m/s. The maximum height reached by the particle is (g = 10 m/s²):

Water drops fall from a tap on the floor, 5 m below, at regular intervals of time, the first drop strikes the floor when the sixth drop begins to fall. The height at which the fourth drop will be from ground, at the instant when the first drop strikes the ground is $\_\_\_\_$ m. $\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$

Two masses of $3.4$ kg and $2.5$ kg are accelerated from an initial speed of $5$ m/s and $12$ m/s, respectively. The distances traversed by the masses in the $5^\text{th}$ second are $104$ m and $129$ m, respectively. The ratio of their momenta after $10$ s is $\dfrac{x}{8}$. The value of $x$ is ________.

The velocity ($v$) versus time ($t$) plot of a particle is shown in the figure, for a time interval of $40\text{ s}$. The total distance travelled by the particle and the average velocity during this period are, respectively _______. 

A car moving with a speed of $54$ km/h takes a turn of radius $20$ m. A simple pendulum is suspended from the ceiling of the car. Determine the angle made by the string of the pendulum with the vertical during the turning. (Take $g=10$ m/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). 

A wedge $Y$ with mass of $10$ kg and all frictionless surfaces and the inclined surface making $37°$ with horizontal. A block $X$ with mass $2$ kg is placed at the highest point of the wedge as shown in figure is at rest. At $t = 0$ wedge $(Y)$ is pulled toward right with constant force $(f)$ of $24$ N. Taking the block $X$ at rest at $t = 0$, the time taken by it to slide down $8.8$ m on the slope, while $Y$ is on the move, is _____ s. (take $\tan(37°) = 3/4$ and $g = 10$ m/s$^2$) 

A fly wheel having mass 3 kg and radius 5 m is free to rotate about a horizontal axis. A string having negligible mass is wound around the wheel and the loose end of the string is connected to 3 kg mass. The mass is kept at rest initially and released. Kinetic energy of the wheel when the mass descends by 3 m is $\_\_\_\_$ $\mathrm{J}.\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$

Keeping the significant figures in view, the sum of the physical quantities $52.01 \mathrm{~m}, 153.2 \mathrm{~m}$ and 0.123 m is :

A body of mass 2 kg is moving along $x$-direction such that its displacement as function of time is given by $x(t)=\alpha t^{2}+\beta t+\gamma \mathrm{m}$, where $\alpha=1 \mathrm{~m} / \mathrm{s}^{2}, \beta=1 \mathrm{~m} / \mathrm{s}$ and $\gamma=1 \mathrm{~m}$. The work done on the body during the time interval $t=2 \mathrm{~s}$ to $t=3 \mathrm{~s}$, is $\_\_\_\_$ J.