JEE Main 2013 — Mechanics

A ball projected from ground at an angle of $45^{\circ}$ just clears a wall in front. If point of projection is $4 \mathrm{~m}$ from the foot of wall and ball strikes the ground at a distance of $6 \mathrm{~m}$ on the other side of the wall, the height of the wall is :

A block is placed on a rough horizontal plane. A time dependent horizontal force $\mathrm{F}=\mathrm{kt}$ acts on the block, where $\mathrm{k}$ is a positive constant. The acceleration - time graph of the block is :

A body of mass ' $m$ ' is tied to one end of a spring and whirled round in a horizontal plane with a constant angular velocity. The elongation in the spring is $1 \mathrm{~cm}$. If the angular velocity is doubled, the elongation in the spring is $5 \mathrm{~cm}$. The original length of the spring is :

A body starts from rest on a long inclined plane of slope $45^{\circ}$. The coefficient of friction between the body and the plane varies as $\mu=0.3 x$, where $x$ is distance travelled down the plane. The body will have maximum speed (for $g=10 \mathrm{~m} / \mathrm{s}^2$ ) when $x=$

A boy of mass $20 \mathrm{~kg}$ is standing on a $80 \mathrm{~kg}$ free to move long cart. There is negligible friction between cart and ground. Initially, the boy is standing $25 \mathrm{~m}$ from a wall. If he walks $10 \mathrm{~m}$ on the cart towards the wall, then the final distance of the boy from the wall will be

A bullet of mass $10 \mathrm{~g}$ and speed $500 \mathrm{~m} / \mathrm{s}$ is fired into a door and gets embedded exactly at the centre of the door. The door is $1.0 \mathrm{~m}$ wide and weighs $12 \mathrm{~kg}$. It is hinged at one end and rotates about a vertical axis practically without friction. The angular speed of the door just after the bullet embeds into it will be :

A copper wire of length $1.0 \mathrm{~m}$ and a steel wire of length $0.5 \mathrm{~m}$ having equal cross-sectional areas are joined end to end. The composite wire is stretched by a certain load which stretches the copper wire by $1 \mathrm{~mm}$. If the Young's modulii of copper and steel are respectively $1.0 \times 10^{11}$ $\mathrm{Nm}^{-2}$ and $2.0 \times 10^{11} \mathrm{Nm}^{-2}$, the total extension of the composite wire is :

A hoop of radius $\text{r}$ and mass $\text{m}$ rotating with an angular velocity ${\omega }_{0}$ is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?

A $70 \mathrm{~kg}$ man leaps vertically into the air from a crouching position. To take the leap the man pushes the ground with a constant force $F$ to raise himself. The center of gravity rises by $0.5 \mathrm{~m}$ before he leaps. After the leap the c.g. rises by another $1 \mathrm{~m}$. The maximum power delivered by the muscles is : (Take $\mathrm{g}=10 \mathrm{~ms}^{-2}$ )

A particle of mass $2 \mathrm{~kg}$ is moving such that at time $t$, its position, in meter, is given by $\vec{r}(t)=5 \hat{i}-2 t^2 \hat{j}$. The angular momentum of the particle at $t=2 s$ about the origin in $\mathrm{kg} \mathrm{m}^{-2} \mathrm{~s}^{-1}$ is :

A projectile is given an initial velocity of $(\hat{i}+2\hat{j})m{s}^{-1}$, where $\hat{\text{i}}$ is along the ground and $\hat{\text{j}}$ is along the vertical upward. If $g=10m{s}^{-2}$, the equation of its trajectory is :

A projectile of mass $M$ is fired so that the horizontal range is $4 \mathrm{~km}$. At the highest point the projectile explodes in two parts of masses $M / 4$ and $3 M / 4$ respectively and the heavier part starts falling down vertically with zero initial speed. The horizontal range (distance from point of firing) of the lighter part is :

A ring of mass $M$ and radius $R$ is rotating about its axis with angular velocity $\omega$. Two identical bodies each of mass $m$ are now gently attached at the two ends of a diameter of the ring. Because of this, the kinetic energy loss will be :

A tennis ball (treated as hollow spherical shell) starting from $\mathrm{O}$ rolls down a hill. At point $\mathrm{A}$ the ball becomes air borne leaving at an angle of $30^{\circ}$ with the horizontal. The ball strikes the ground at $\mathrm{B}$. What is the value of the distance $\mathrm{AB}$ ? (Moment of inertia of a spherical shell of mass $m$ and radius $R$ about its diameter $=\frac{2}{3} m R^2$ ) 

A thin tube sealed at both ends is $100 \mathrm{~cm}$ long. It lies horizontally, the middle $20 \mathrm{~cm}$ containing mercury and two equal ends containing air at standard atmospheric pressure. If the tube is now turned to a vertical position, by what amount will the mercury be displaced ?  (Given : cross-section of the tube can be assumed to be uniform)

A uniform cylinder of length $\text{L}$ and mass $\text{M}$ having cross-sectional area $\text{A}$ is suspended, with its length vertical, from a fixed point by a massless spring, such that it is half submerged in a liquid of density $\sigma$ at equilibrium position. The extension ${\text{x}}_{0}$ of the spring when it is in equilibrium is :

A uniform sphere of weight $W$ and radius $5 \mathrm{~cm}$ is being held by a string as shown in the figure. The tension in the string will be : 

A uniform wire (Young's modulus $2 \times 10^{11} \mathrm{Nm}^{-2}$ ) is subjected to longitudinal tensile stress of $5 \times 10^7 \mathrm{Nm}^{-2}$. If the overall volume change in the wire is $0.02 \%$, the fractional decrease in the radius of the wire is close to:

A wind-powered generator converts wind energy into electrical energy. Assume that the generator converts a fixed fraction of the wind energy intercepted by its blades into electrical energy. For wind speed $v$, the electrical power output will be most likely proportional to

Air of density $1.2 \mathrm{~kg} \mathrm{~m}^{-3}$ is blowing across the horizontal wings of an aeroplane in such a way that its speeds above and below the wings are $150 \mathrm{~ms}^{-1}$ and $100 \mathrm{~ms}^{-1}$, respectively. The pressure difference between the upper and lower sides of the wings, is :

Assume that a drop of a liquid evaporates by a decrease in its surface energy so that its temperature remains unchanged. The minimum radius of the drop for this to be possible is. (The surface tension is $T$, the density of the liquid is$\rho$ and $L$ is its latent heat of vaporisation.)

Correct set up to verify Ohm's law is :

From the following, the quantity (constructed from the basic constants of nature), that has the dimensions, as well as correct order of magnitude, vis-a-vis typical atomic size, is:

If the ratio of lengths, radii and Young's moduli of steel and brass wires in the figure are $a, b$ and $c$ respectively, then the corresponding ratio of increase in their lengths is : 

If the time period $t$ of the oscillation of a drop of liquid of density $d$, radius $r$, vibrating under surface tension $s$ is given by the formula $t=\sqrt{r^{2 b} s^c d^{a / 2}}$. It is observed that the time period is directly proportional to $\sqrt{\frac{d}{s}}$. The value of $b$ should therefore be :

In an experiment, a small steel ball falls through a liquid at a constant speed of $10 \mathrm{~cm} / \mathrm{s}$. If the steel ball is pulled upward with a force equal to twice its effective weight, how fast will it move upward ?

Let $[ {\in }_{0} ]$ denote the dimensional formula of the permittivity of vacuum. If M = mass, L = length, T = time and A = electric current, then :

The change in the value of acceleration of earth towards sun, when the moon comes from the position of solar eclipse to the position on the other side of earth in line with sun is: (mass of the moon $=7.36 \times 10^{22} \mathrm{~kg}$, radius of the moon's orbit $=3.8 \times 10^8 \mathrm{~m}$ ).

The dimensions of angular momentum, latent heat and capacitance are, respectively.

The gravitational field, due to the 'left over part' of a uniform sphere (from which a part as shown, has been 'removed out'), at a very far off point, $\mathrm{P}$, located as shown, would be (nearly) : 

The maximum range of a bullet fired from a toy pistol mounted on a car at rest is $R_0=40 \mathrm{~m}$. What will be the acute angle of inclination of the pistol for maximum range when the car is moving in the direction of firing with uniform velocity $\mathrm{v}=20 \mathrm{~m} / \mathrm{s}$ on a horizontal surface? $\left(\mathrm{g}=10 \mathrm{~m} / \mathrm{s}^2\right)$

This question has $Statement - I$ and $Statement - II$ of the four choices given after the Statements, choose the one that best describes the two Statements. $Statement - I:$ A point particle of mass $m$ moving with speed $\nu$ collides with stationary point particle of mass $M$. If the maximum energy loss possible is given as $f(\frac{1}{2}m{\nu }^{2})$ then $f=(\frac{m}{M+m})$. $Statement - II:$ Maximum energy loss occurs when the particles get stuck together as a result of the collision.

This question has Statement-1 and Statement- 2 . Of the four choices given after the Statements, choose the one that best describes the two Statetnents. Statement-1: A capillary is dipped in a liquid and liquid rises to a height $h$ in it. As the temperature of the liquid is raised, the height $h$ increases (if the density of the liquid and the angle of contact remain the same). Statement-2: Surface tension of a liquid decreases with the rise in its temperature.

Two blocks of mass $M_1=20 \mathrm{~kg}$ and $M_2=12 \mathrm{~kg}$ are connected by a metal rod of mass $8 \mathrm{~kg}$. The system is pulled vertically up by applying a force of $480 \mathrm{~N}$ as shown. The tension at the mid-point of the rod is: 

Two blocks of masses $\mathrm{m}$ and $\mathrm{M}$ are connected by means of a metal wire of cross-sectional area A passing over a frictionless fixed pulley as shown in the figure. The system is then released. If $\mathrm{M}=2 \mathrm{~m}$, then the stress produced in the wire is: 

Two springs of force constants $300 \mathrm{~N} / \mathrm{m}$ (Spring A) and $400 \mathrm{~N} / \mathrm{m}$ (Spring B) are joined together in series. The combination is compressed by $8.75 \mathrm{~cm}$. The ratio of energy stored in $\mathrm{A}$ and $\mathrm{B}$ is $\frac{E_A}{E_B}$. Then $\frac{E_A}{E_B}$ is equal to:

Wax is coated on the inner wall of a capillary tube and the tube is then dipped in water. Then, compared to the unwaxed capillary, the angle of contact $\theta$ and the height $h$ upto which water rises change. These changes are :

What is the minimum energy required to launch a satellite of mass $\text{m}$ from the surface of a planet of mass $\text{M}$ and radius $\text{R}$ in a circular orbit at an altitude of $\text{2R}$?