Mechanics PYQ — Page 7

A small block of mass $m$ slides down from the top of a frictionless inclined surface, while the inclined plane is moving towards left with constant acceleration $a_{0}$. The angle between the inclined plane and ground is $\theta$ and its base length is $L$. Assuming that initially the small block is at the top of the inclined plane, the time it takes to reach the lowest point of the inclined plane is $\_\_\_\_$. 

A metal string $A$ is suspended from a rigid support and its free end is attached to a block of mass $M$. Second block having mass $2M$ is suspended at the bottom of the first block using a string $B$. The area of cross sections of strings $A$ and $B$ are same. The ratio of lengths of strings of $A$ to $B$ is $2$ and the ratio of their Young's moduli $(Y_A/Y_B)$ is $0.5$. The ratio of elongations in $A$ to $B$ is ______.

A particle starts moving from time $t=0$ and its coordinate is given as $x(t)=4 t^{3}-3 t$ A. The particle returns to its original position (origin) 0.866 units later B. The particle is 1 unit away from origin at its turning point C. Acceleration of the particle is non-negative D. The particle is 0.5 units away from origin at its turning point E. Particle never turns back as acceleration is non-negative Choose the correct answer from the options given below :

When one moves from a point $16$ km below the earth's surface to a point $16$ km above the earth's surface. The change in $g$ is approximately $\alpha$ %. The value of $\alpha$ is _____. (Take radius of the earth $= 6400$ km.)

A cylindrical vessel of $40$ cm radius is completely filled with water and its capacity is $528$ dm$^3$ (dm : decimeter). The vessel is placed on a solid block of exactly same height as vessel. If a small hole is made at $70$ cm below the top of water level, then horizontal range of water falling on the ground in the beginning is __________ cm.

An object of uniform density rolls up the curved path with the initial velocity $v_0$ as shown in the figure. If the maximum height attained by an object is $\dfrac{7v_0^2}{10g}$ ($g$ = acceleration due to gravity), the object is a _______. 

A paratrooper jumps from an aeroplane and opens a parachute after 2 s of free fall and starts deaccelerating with $3 \mathrm{~m} / \mathrm{s}^{2}$. At 10 m height from ground, while descending with the help of parachute, the speed of paratrooper is $5 \mathrm{~m} / \mathrm{s}$. The initial height of the airplane is $\_\_\_\_$ m. $\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^{2}\right)$

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 ________.

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$^2$) 

An air bubble of radius 0.1 cm lies at a depth of 20 cm below the free surface of a liquid of density $1000 \mathrm{~kg} / \mathrm{m}^3$. If the pressure inside the bubble is $2100 \mathrm{~N} / \mathrm{m}^2$ greater than the atmospheric pressure, then the surface tension of the liquid in SI unit is (use $\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2$ )

Two slabs with square cross section of different materials $(1,2)$ with equal sides $(l)$ and thickness $\mathrm{d}_1$ and $\mathrm{d}_2$ such that $\mathrm{d}_2=2 \mathrm{~d}_1$ and $l \gt \mathrm{d}_2$. Considering lower edges of these slabs are fixed to the floor, we apply equal shearing force on the narrow faces. The angle of deformation is $\theta_2=2 \theta_1$. If the shear moduli of material 1 is $4 \times 10^9 \mathrm{~N} / \mathrm{m}^2$, then shear moduli of material 2 is $\mathrm{x} \times 10^9 \mathrm{~N} / \mathrm{m}^2$, where value of $x$ is ________.

For the determination of refractive index of glass slab, a travelling microscope is used whose main scale contains 300 equal divisions equals to 15 cm. The vernier scale attached to the microscope has 25 divisions equals to 24 divisions of main scale. The least count (LC) of the travelling microscope is (in cm) :

If $\mu_0$ and $\varepsilon_0$ are the permeability and permittivity of free space, respectively, then the dimension of $\left(\frac{1}{\mu_0 \varepsilon_0}\right)$ is :

A tube of length 1 m is filled completely with an ideal liquid of mass 2 M , and closed at both ends. The tube is rotated uniformly in horizontal plane about one of its ends. If the force exerted by the liquid at the other end is F then angular velocity of the tube is $\sqrt{\frac{\mathrm{F}}{\alpha \mathrm{M}}}$ in SI unit. The value of $\alpha$ is __________.

The position vectors of two 1 kg particles, (A) and (B), are given by $\overrightarrow{\mathrm{r}}_{\mathrm{A}}=\left(\alpha_1 \mathrm{t}^2 \hat{i}+\alpha_2 \mathrm{t} \hat{j}+\alpha_3 \mathrm{t} \hat{k}\right) \mathrm{m}$ and $\overrightarrow{\mathrm{r}}_{\mathrm{B}}=\left(\beta_1 \mathrm{t} \hat{i}+\beta_2 \mathrm{t}^2 \hat{j}+\beta_3 \mathrm{t} \hat{k}\right) \mathrm{m}$, respectively; $\left(\alpha_1=1 \mathrm{~m} / \mathrm{s}^2, \alpha_2=3 \mathrm{n} \mathrm{m} / \mathrm{s}, \alpha_3=2 \mathrm{~m} / \mathrm{s}, \beta_1=2 \mathrm{~m} / \mathrm{s}, \beta_2=-1 \mathrm{~m} / \mathrm{s}^2, \beta_3=4 \mathrm{pm} / \mathrm{s}\right)$, where t is time, n and p are constants. At $t=1 \mathrm{~s},\left|\overrightarrow{V_A}\right|=\left|\vec{V}_B\right|$ and velocities $\vec{V}_A$ and $\vec{V}_B$ of the particles are orthogonal to each other. At $t=1 \mathrm{~s}$, the magnitude of angular momentum of particle (A) with respect to the position of particle (B) is $\sqrt{\mathrm{L}} \mathrm{kgm}^2 \mathrm{~s}^{-1}$. The value of L is _______ .

In an electromagnetic system, a quantity defined as the ratio of electric dipole moment and magnetic dipole moment has dimension of $\left[M^P L^{\circ} \mathrm{T}^{\mathrm{R}} \mathrm{A}^{\mathrm{S}}\right]$. The value of P and Q are :

The center of mass of a thin rectangular plate (fig - x ) with sides of length $a$ and $b$, whose mass per unit area $(\sigma)$ varies as $\sigma=\frac{\sigma_0 x}{a b}$ (where $\sigma_0$ is a constant), would be $\qquad$ 

A ball of mass 100 g is projected with velocity $20 \mathrm{~m} / \mathrm{s}$ at $60^{\circ}$ with horizontal. The decrease in kinetic energy of the ball during the motion from point of projection to highest point is

The energy of a system is given as $\mathrm{E}(\mathrm{t})=\alpha^3 \mathrm{e}^{-\beta t}$, where t is the time and $\beta=0.3 \mathrm{~s}^{-1}$. The errors in the measurement of $\alpha$ and $t$ are $1.2 \%$ and $1.6 \%$, respectively. At $t=5 \mathrm{~s}$, maximum percentage error in the energy is :

A person measures mass of 3 different particles as $435.42 \mathrm{~g}, 226.3 \mathrm{~g}$ and 0.125 g. According to the rules for arithmetic operations with significant figures, the additions of the masses of 3 particles will be.

A uniform solid cylinder of mass ' $m$ ' and radius ' $r$ ' rolls along an inclined rough plane of inclination $45^{\circ}$. If it starts to roll from rest from the top of the plane then the linear acceleration of the cylinder's axis will be

A bead of mass ' $m$ ' slides without friction on the wall of a vertical circular hoop of radius ' $R$ ' as shown in figure. The bead moves under the combined action of gravity and a massless spring ( k ) attached to the bottom of the hoop. The equilibrium length of the spring is ' $R$ '. If the bead is released from top of the hoop with (negligible) zero initial speed, velocity of bead, when the length of spring becomes ' R ', would be (spring constant is ' k ', g is accleration due to gravity) 

Given below are two statements : Statement I : In a vernier callipers, one vernier scale division is always smaller than one main scale division. Statement II : The vernier constant is given by one main scale division multiplied by the number of vernier scale divisions. In the light of the above statements, choose the correct answer from the options given below.

$\mathrm{A}, \mathrm{B}$ and C are disc, solid sphere and spherical shell respectively with same radii and masses. These masses are placed as shown in figure.  The moment of inertia of the given system about $P Q$ is $\frac{X}{15} I$, where $I$ is the moment of inertia of the disc about its diameter. The value of x is _______.