Mechanics Questions

A particle moves in a straight line with retardation proportional to its displacement. Its loss of kinetic energy for any displacement $x$ is proportional to:

A projectile can have the same range $R$ for two angles of projection. If $T_1$ and $T_2$ be the time of flights in the two cases, then the product of the two time of flights is directly proportional to:

A machine gun fires a bullet of mass $40\text{g}$ with a velocity $1200\text{m/s}$. The man holding it can exert a maximum force of $144\text{N}$ on the gun. How many bullets can he fire per second at the most?

Two masses $m_1=5\text{kg}$ and $m_2=4.8\text{kg}$ tied to a string are hanging over a light frictionless pulley. What is the acceleration of the masses when left free to move? (Take $g=9.8{\text{m/s}}^2$)

A uniform chain of length $2\text{m}$ is kept on a table such that a length of $60\text{cm}$ hangs freely from the edge of the table. The total mass of the chain is $4\text{kg}$. What is the work done in pulling the entire chain on the table?

A particle is acted upon by a force of constant magnitude which is always perpendicular to the velocity of the particle. The motion of the particle takes place in a plane. It follows that:

One solid sphere $A$ and another hollow sphere $B$ are of same mass and same outer radii. Their moment of inertia about their diameters are respectively $I_A$ and $I_B$ such that:

A wire fixed at the upper end stretches by length $l$ by applying a force $F$. The work done in stretching is:

The potential energy of a $1\text{kg}$ particle free to move along the $x$ -axis is given by $V(x)=\left(\frac{x^4}{4}-\frac{x^2}{2}\right)\text{J}$. The total mechanical energy of the particle is $2\text{J}$. Then, the maximum speed (in m/s) is:

A force of $-F\hat{k}$ acts on $O$, the origin of the coordinate system. The torque about the point $(1,-1,0)$ is:

A thin circular ring of mass $m$ and radius $R$ is rotating about its axis with a constant angular velocity $\omega$. Two objects each of mass $M$ are attached gently to the opposite ends of a diameter of the ring. The ring now rotates with an angular velocity ${\omega }^{\prime }$:

Four point masses, each of value $m$, are placed at the corners of a square $ABCD$ of side $\ell$. The moment of inertia through $A$ and parallel to $BD$ is:

A body weighing $13\text{kg}$ is suspended by two strings $5\text{m}$ and $12\text{m}$ long, their other ends being fastened to the extremities of a rod $13\text{m}$ long. If the rod be so held that the body hangs immediately below the middle point. The tensions in the strings are:

A particle just clears a wall of height $b$ at distance $a$ and strikes the ground at a distance $c$ from the point of projection. The angle of projection is:

Consider a two particle system with particles having masses $m_1$ and $m_2$. If the first particle is pushed towards the centre of mass through a distance $d$, by what distance should the second particle be moved, so as to keep the centre of mass at the same position?

A player caught a cricket ball of mass $150\text{g}$ moving at a rate of $20\text{m/s}$. If the catching process is completed in $0.1\text{s}$, the force of the blow exerted by the ball on the hand of the player is equal to:

A mass of $M\text{kg}$ is suspended by a weightless string. The horizontal force $F$ that is required to displace it until the string makes an angle of $45^{\circ }$ with the initial vertical direction is:

$ABC$ is a triangle, right-angled at $A$. The resultant of the forces acting along $AB$ and $AC$ with magnitudes $\frac{1}{AB}$ and $\frac{1}{AC}$ respectively is a force along $AD$, where $D$ is the foot of the perpendicular from $A$ onto $BC$. The magnitude of the resultant is:

A rocket with a lift-off mass $3.5\times 10^4\text{kg}$ is blasted upwards with an initial acceleration of $10{\text{m/s}}^2$. Then the initial thrust of the blast is:

A spring of spring constant $5\times 10^3\text{N/m}$ is stretched initially by $5\text{cm}$ from the unstretched position. Then the work required to stretch it further by another $5\text{cm}$ is:

The co-ordinates of a moving particle at any time $t$ are given by $x=\alpha t^3$ and $y=\beta t^3$. The speed of the particle at time $t$ is given by:

A boy playing on the roof of a $10\text{m}$ high building throws a ball with a speed of $10\text{m/s}$ at an angle of $30^{\circ }$ with the horizontal. How far from the throwing point will the ball be at the height of $10\text{m}$ from the ground? (Take $g=10{\text{m/s}}^2$)

A marble block of mass $2\text{kg}$ lying on ice when given a velocity of $6\text{m/s}$ is stopped by friction in $10\text{s}$. The coefficient of friction is:

A spherical solid ball of volume $V$ is made of a material of density ${\rho }_1$. It is falling through a liquid of density ${\rho }_2$ (${\rho }_2<{\rho }_1$). Assume that the liquid applies a viscous force $F_{\text{viscous}}=-k{v}^2$. The terminal speed of the ball is:

A thin rod of length $L$ is lying along the $x$ -axis from $x=0$ to $x=L$. Its linear density $\lambda$ varies as $\lambda =k(x/L{)}^n$. Which graph best approximates the dependence of the position $x_{CM}$ of the centre of mass on $n$ ($n\ge 0$)?

Consider a uniform square plate of side ‘$a$ ’ and mass ‘$m$ ’. The moment of inertia of this plate about an axis perpendicular to its plane and passing through one of its corners is:

A body of mass $m=3.513\text{kg}$ is moving along the $x$ -axis with a speed of $5.00\text{m/s}$. The magnitude of its momentum is recorded as:

An athlete in the olympic games covers a distance of $100\text{m}$ in $10\text{s}$. His kinetic energy can be estimated to be in the range:

A jar is filled with two non-mixing liquids $1$ and $2$ having densities ${\rho }_1$ and ${\rho }_2$ respectively. A solid ball of density ${\rho }_3$ is dropped and comes to equilibrium such that it floats at the interface, partially in both liquids. Which of the following is true for ${\rho }_1$, ${\rho }_2$ and ${\rho }_3$?

A block of mass $0.50\text{kg}$ moving at $2.00\text{m/s}$ on a smooth surface strikes another mass of $1.00\text{kg}$ and they move together. The energy loss during collision is:

A capillary tube (A) is dipped in water. Another identical tube (B) is dipped in soap water solution. Which of the following shows the relative nature of the liquid columns?

A body starts from rest at $x=0,t=0$ with constant acceleration $a$. Simultaneously another body passes $x=0$ with constant speed $v$. The position of the first is $x_1(t)$ and the second is $x_2(t)$. Which graph describes $(x_1-x_2)$ vs $t$?

Two full turns of a screw gauge cover $1\text{mm}$. Total divisions on circular scale is $50$. Zero error is $-0.03\text{mm}$. While measuring, main scale reading is $3\text{mm}$ and circular scale is $35$. The diameter of the wire is:

Statement-1: Two particles moving in the same direction do not lose all their energy in a completely inelastic collision.Statement-2: Principle of conservation of momentum holds true for all kinds of collisions.

The figure shows the position–time ($x-t$) graph of one-dimensional motion of a body of mass $0.4\text{kg}$. If the graph is a sawtooth wave with peaks at $2\text{m}$ every $2\text{seconds}$, the magnitude of each impulse is:

The potential energy function for the force between two atoms in a diatomic molecule is approximately given by $U(x)=\frac{a}{x^{12}}-\frac{b}{x^6}$, where $a$ and $b$ are constants and $x$ is the distance between the atoms. If the dissociation energy of the molecule is $D=[U(x=\infty )-U_{atequilibrium}]$, $D$ is:

Two fixed frictionless inclined planes making an angle $30^{\circ }$ and $60^{\circ }$ with the vertical are shown. Two blocks $A$ and $B$ are placed on the two planes. What is the relative vertical acceleration of $A$ with respect to $B$?

A small particle of mass $m$ is projected at an angle $\theta$ with the $x$ -axis with an initial velocity $v_0$ in the $x-y$ plane. At a time $t<\frac{v_0\sin \theta }{g}$, the angular momentum of the particle about the origin is:

A wooden wheel of radius $R$ is made of two semicircular parts. The two parts are held together by a ring made of a metal strip of cross sectional area $S$ and length $L$. $L$ is slightly less than $2\pi R$. To fit the ring on the wheel, it is heated so that its temperature rises by $\Delta T$ and it just steps over the wheel. As it cools down to surrounding temperature, it presses the semicircular parts together. If the coefficient of linear expansion of the metal is $\alpha$, and its Young's modulus is $Y$, the force that one part of the wheel applies on the other part is:

The block of mass $M$ moving on the frictionless horizontal surface collides with a spring of spring constant $K$ and compresses it by length $L$. The maximum momentum of the block after collision is:

A particle of mass $10\text{ g}$ is kept on the surface of a uniform sphere of mass $100\text{ kg}$ and radius $10\text{ cm}$. Find the work to be done against the gravitational force between them to take the particle far away from the sphere [$G=6.67\times 10^{-11}{\text{ Nm}}^2/{\text{kg}}^2$ ]:

The change in the value of $g$ at a height ‘$h$ ’ above the surface of the earth is the same as at a depth ‘$d$ ’ below the surface of earth. When both ‘$d$ ’ and ‘$h$ ’ are much smaller than the radius of earth, then which one of the following is correct?

A ‘T’ shaped object with dimensions shown in the figure, is lying on a smooth floor. A force $F$ is applied at the point $P$ parallel to $AB$, such that the object has only the translational motion without rotation. Find the location of $P$ with respect to $C$:

Consider a car moving on a straight road with a speed of $100\text{ m/s}$. The distance at which car can be stopped is [${\mu }_k=0.5$ ]:

A body of mass $m$ is accelerated uniformly from rest to a speed $v$ in a time $T$. The instantaneous power delivered to the body as a function time is given by:

Average density of the earth:

If $S$ is stress and $Y$ is Young’s modulus of material of a wire, the energy stored in the wire per unit volume is:

A $20\text{ cm}$ long capillary tube is dipped in water. The water rises up to $8\text{ cm}$. If the entire arrangement is put in a freely falling elevator the length of water column in the capillary tube will be:

A mass ‘$m$ ’ moves with a velocity $v$ and collides inelastically with another identical mass. After collision the 1st mass moves with velocity $v/\sqrt{3}$ in a direction perpendicular to the initial direction of motion. Find the speed of the 2nd mass after collision:

Two simple harmonic motions are represented by the equation $y_1=0.1\sin (100\pi t+\pi /3)$ and $y_2=0.1\cos \pi t$. The phase difference of the velocity of particle $1$ w.r.t. the velocity of the particle $2$ is:

A particle of mass $0.3\text{ kg}$ is subjected to a force $F=-kx$ with $k=15\text{ N/m}$. What will be its initial acceleration if it is released from a point $20\text{ cm}$ away from the origin?

The moment of inertia of a uniform semicircular disc of mass $M$ and radius $r$ about a line perpendicular to the plane of the disc through the centre is:

A body A of mass $M$ while falling vertically downwards under gravity breaks into two parts; a body B of mass $1/3M$ and a body C of mass $2/3M$. The centre of mass of bodies B and C taken together shifts compared to that of body A towards:

A spherical ball of mass $20\text{ kg}$ is stationary at the top of a hill of height $100\text{ m}$. It rolls down a smooth surface to the ground, then climbs up another hill of height $30\text{ m}$ and finally rolls down to a horizontal base at a height of $20\text{ m}$ above the ground. The velocity attained by the ball is:

A block is kept on a frictionless inclined surface with angle of inclination $\alpha$. The incline is given an acceleration $a$ to keep the block stationary. Then $a$ is equal to:

A bullet fired into a fixed target loses half of its velocity after penetrating $3\text{ cm}$. How much further it will penetrate before coming to rest assuming that it faces constant resistance to motion?

The upper half of an inclined plane with inclination $\varphi$ is perfectly smooth while the lower half is rough. A body starting from rest at the top will again come to rest at the bottom if the coefficient of friction for the lower half is given by:

A smooth block is released at rest on a $45^{\circ }$ incline and then slides a distance $d$. The time taken to slide is $n$ times as much to slide on a rough incline than on a smooth incline. The coefficient of friction is:

A light spring balance hangs from the hook of the other light spring balance and a block of mass $M\text{kg}$ hangs from the former one. Then the true statement about scale reading is:

A block of mass $M$ is pulled along a horizontal friction surface by a rope of mass $m$. If a force $P$ is applied at the free end of the rope, the force exerted by the rope on the block is:

A body is moved along a straight line by a machine delivering a constant power. The distance moved by the body in time $t$ is proportional to:

A wire suspended vertically from one of its ends stretched by attaching weight of $200\text{N}$ to the lower end. The weight stretches the wire by $1\text{mm}$. Then the elastic energy stored in the wire is:

A horizontal force of $10\text{N}$ is necessary to just hold a block stationary against a wall. The coefficient of friction between the block and the wall is $0.2$. The weight of the block is:

A spring balance is attached to the ceiling of a lift. A man hangs his bag on the spring and the spring reads $49\text{N}$, when the lift is stationary. If the lift moves downward with an acceleration of $5{\text{m/s}}^2$, the reading of the spring balance will be:

A car moving with a speed of $50\text{km/hr}$, can be stopped by brakes after at least $6\text{m}$. If the same car is moving at a speed of $100\text{km/hr}$, the minimum stopping distance is:

Consider the following two statements:(1) Linear momentum of a system of particles is zero.(2) Kinetic energy of system of particles is zero.

Three forces start acting simultaneously on a particle moving with velocity $\vec{v}$. These forces are represented in magnitude and direction by the three sides of a triangle $ABC$ taken in order. The particle will now move with velocity:

Let $\vec{F}$ be the force acting on a particle having position vector $\vec{r}$ and $\vec{\tau }$ be the torque of this force about the origin. Then:

An annular ring with inner and outer radii $R_1$ and $R_2$ is rolling without slipping with a uniform angular speed. The ratio of the forces experienced by the two particles situated on the inner and outer parts of the ring, $F_1/F_2$ is:

A body travels a distance $s$ in $t$ seconds. It starts from rest and ends at rest. In the first part of the journey, it moves with constant acceleration $f$ and in the second part with constant retardation $r$. The value of $t$ is given by:

Two stones are projected from the top of a cliff $h$ meters high, with the same speed $u$ so as to hit the ground at the same spot. If one of the stones is projected horizontally and the other is projected at an angle $\theta$ to the horizontal then $\tan \theta$ equals:

Two particles start simultaneously from the same point and move along two straight lines, one with uniform velocity $u$ and the other from rest with uniform acceleration $f$. Let $\alpha$ be the angle between their directions of motion. The relative velocity of the second particle with respect to the first is least after a time:

A couple is of moment $\vec{G}$ and the force forming the couple is $\vec{P}$. If $\vec{P}$ is turned through a right angle, the moment of the couple thus formed is $\vec{H}$. If instead, the forces $\vec{P}$ are turned through an angle $\alpha$, then the moment of couple becomes:

Let $R_1$ and $R_2$ respectively be the maximum ranges up and down an inclined plane and $R$ be the maximum range on the horizontal plane. Then $R_1,R,R_2$ are in:

The resultant of forces $\vec{P}$ and $\vec{Q}$ is $\vec{R}$. If $\vec{Q}$ is doubled then $\vec{R}$ is doubled. If the direction of $\vec{Q}$ is reversed, then $\vec{R}$ is again doubled. Then $P^2:Q^2:R^2$ is:

The bob of a simple pendulum is a spherical hollow ball filled with water. A plugged hole near the bottom of the oscillation bob gets suddenly unplugged. During observation, till water is coming out, the time period of oscillation would:

If a simple harmonic motion is represented by $\frac{d^{2}x}{d{t}^2}+\alpha x=0$, its time period is: