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. 

Match List - I with List - II.<table class="pyq-table"><tbody><tr><th>List - I</th><th>List - II</th></tr><tr><td>A. Boltzmann constant</td><td>I. $[M^{-1}L^3T^{-2}]$</td></tr><tr><td>B. Stefan's constant</td><td>II. $[ML^2T^{-1}]$</td></tr><tr><td>C. Planck's constant</td><td>III. $[ML^2T^{-2}K^{-1}]$</td></tr><tr><td>D. Gravitational constant</td><td>IV. $[ML^0T^{-3}K^{-4}]$</td></tr></tbody></table>Choose the correct answer from the options given below :

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 _______. 

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 ________.

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$^2$)

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$^2$) 

Three masses $m_1 = 4$ kg, $m_2 = 4$ kg and $m_3 = 6$ kg are suspended from a fixed smooth frictionless pulley as shown in the figure below. The value of $T_1/T_2$ is _____. (take $g = 10$ m/s$^2$) 

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 ______.

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 $\_\_\_\_$ $\mathrm{J}.\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$

Eight mercury drops, each of radius $r$, coalesce to form a bigger drop. The surface energy released in this process is _______. ($S$ is the surface tension of mercury).

Two projectiles are projected with the same initial velocities at the $15^\circ$ and $30^\circ$ with respect to the horizontal. The ratio of their ranges is $1:x$. The value of $x$ is:

Net gravitational force at the center of a square is found to be $F_{1}$ when four particles having mass $M, 2 M, 3 M$ and $4 M$ are placed at the four corners of the square as shown in figure and it is $F_{2}$ when the positions of $3 M$ and $4 M$ are interchanged. The ratio $\frac{F_{1}}{F_{2}}$ is $\frac{\alpha}{\sqrt{5}}$. The value of $\alpha$ is $\_\_\_\_$. 

The time taken by a block of mass $m$ to slide down from the highest point to the lowest point on a rough inclined plane is $50\%$ more compared to the time taken by the same block on identical inclined smooth plane. Both inclined planes are at $45^\circ$ with the horizontal. The coefficient of kinetic friction between the rough inclined surface and block is _____.

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.

A $1$ kg block subjected to two simultaneous forces $(2\hat{i} + 3\hat{j} + 4\hat{k})$ N and $(3\hat{i} - \hat{j} - 2\hat{k})$ N is moved a distance of $25$ m along $(3\hat{i} - 4\hat{j})$ direction. The work done in this process is _____ J.

The percentage error in the calculated volume of a sphere, if there is $2\%$ error in its diameter measurement, is __________.

Match the LIST-I with LIST-II \(\begin{array}{||c|l||c|l||} \hline & \text{List-I} & & \text{List-II} \\ \hline \text{A.} & \text{Magnetic induction} & \text{I.} & \mathrm{M\,L\,T^{-2}\,A^{-2}} \\ \hline \text{B.} & \text{Magnetic flux} & \text{II.} & \mathrm{M\,L^{2}\,T^{-2}\,A^{-2}} \\ \hline \text{C.} & \text{Magnetic permeability} & \text{III.} & \mathrm{M\,L^{0}\,T^{-2}\,A^{-1}} \\ \hline \text{D.} & \text{Self inductance} & \text{IV.} & \mathrm{M\,L^{2}\,T^{-2}\,A^{-1}} \\ \hline \end{array}\) Choose the correct answer from the options given below:

A particle is rotating in a circular path and at any instant its motion can be described as $\theta = \dfrac{5t^4}{40} - \dfrac{t^3}{3}$. The angular acceleration of the particle after $10$ seconds is _______ rad/s$^2$.

A uniform bar of length 12 cm and mass $20 m$ lies on a smooth horizontal table. Two point masses $m$ and $2 m$ are moving in opposite directions with same speed of $v$ and in the same plane as the bar, as shown in figure. These masses strike the bar simultaneously and get stuck to it. After collision the entire system is rotating with angular frequency $\omega$. The ratio of $v$ and $\omega$ is : 

In a screw gauge, the zero of the circular scale lies 3 divisions above the horizontal pitch line when their metallic studs are brought in contact. Using this instrument thickness of a sheet is measured. If pitch scale reading is 1 mm and the circular scale reading is 51 then the correct thickness of the sheet is $\_\_\_\_$ mm. [Assume least count is 0.01 mm ]

A wheel initially at rest is subjected to a uniform angular acceleration about its axis. In the first $2\text{ s}$ it rotates through an angle $\theta_1$ and in the next $2\text{ s}$ it rotates through an angle $\theta_2$. The ratio $\dfrac{\theta_2}{\theta_1}$ is _______.

Two cars $A$ and $B$ each of mass $10^{3} \mathrm{~kg}$ are moving on parallel tracks separated by a distance of 10 m, in same direction with speeds $72 \mathrm{~km} / \mathrm{h}$ and $36 \mathrm{~km} / \mathrm{h}$. The magnitude of angular momentum of $\operatorname{car} A$ with respect to $\operatorname{car} B$ is $\_\_\_\_$ J.s.

The height in terms of radius of the earth $(R)$, at which the acceleration due to gravity becomes $\dfrac{g}{9}$, where $g$ is acceleration due to gravity on earth's surface, is ______.