Mechanics Questions

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:

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 $n$ ($n\ge 0$)?

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 small block slides down on a smooth inclined plane, starting from rest at time $t=0$. Let $S_n$ be the distance travelled by the block in the interval $t=n-1$ to $t=n$. Then, the ratio $\frac{S_n}{S_{n+1}}$ 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 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 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 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 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 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:

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 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 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:

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:

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

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:

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 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:

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:

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 $0.50\text{kg}$ 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:

An electric lift with a maximum load of $2000\text{kg}$ (lift + passengers) is moving up with a constant speed of $1.5{\text{ms}}^{-1}$. The frictional force opposing the motion is $3000\text{N}$. The minimum power delivered by the motor to the lift in watts is: ($g=10{\text{ms}}^{-2}$)

The $x$ and $y$ coordinates of the particle at any time are $x=5t-2t^2$ and $y=10t$ respectively, where $x$ and $y$ are in meters and $t$ in seconds. The acceleration of the particle at $t=2\text{s}$ is:

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:

A spring of force constant $k$ is cut into lengths of ratio $1:2:3$. They are connected in series and the new force constant is $k^{\prime }$. Then they are connected in parallel and force constant is $k^{\prime \prime }$. Then $k^{\prime }:k^{\prime \prime }$ is:

The potential energy of a long spring when stretched by $2\text{cm}$ is $U$. If the spring is stretched by $8\text{cm}$, the potential energy stored in it will be:

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:

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:

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:

Consider a drop of rain water having mass $1\text{g}$ falling from a height of $1\text{km}$. It hits the ground with a speed of $50\text{m/s}$. Take $g$ constant with a value $10{\text{m/s}}^2$. The work done by the (i) gravitational force and the (ii) resistive force of air is:

The amplitude of a damped oscillator decreases to $0.9$ times its original magnitude in $5\text{s}$. In another $10\text{s}$ it will decrease to $\alpha$ times its original magnitude, where $\alpha$ equals:

Two discs of same moment of inertia rotating about their regular axis passing through centre and perpendicular to the plane of disc with angular velocities ${\omega }_1$ and ${\omega }_2$. They are brought into contact face to face coinciding the axis of rotation. The expression for loss of energy during this process is:

A particle moving with uniform speed in a circular path maintains:

A rope is wound around a hollow cylinder of mass $3\text{kg}$ and radius $40\text{cm}$. What is the angular acceleration of the cylinder if the rope is pulled with a force of $30\text{N}$?

Two bodies A and B of same mass undergo completely inelastic one dimensional collision. The body A moves with velocity $v_1$ while body B is at rest before collision. The velocity of the system after collision is $v_2$. The ratio $v_1:v_2$ is:

Which one of the following statements is incorrect?

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:

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:

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:

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

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 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 [$G=6.67\times 10^{-11}{\text{ Nm}}^2/{\text{kg}}^2$ ]:

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 $C$:

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:

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 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 bob of heavy mass $m$ is suspended by a light string of length $l$. The bob is given a horizontal velocity $v_0$ as shown in figure. If the string gets slack at some point $P$ making an angle $\theta$ from the horizontal, the ratio of the speed $v$ of the bob at point $P$ to its initial speed $v_0$ is:

A ball of mass $0.5\text{kg}$ is dropped from a height of $40\text{m}$. The ball hits the ground and rises to a height of $10\text{m}$. The impulse imparted to the ball during its collision with the ground is (Take $g=9.8{\text{m/s}}^2$):

There are two inclined surfaces of equal length ($L$) and same angle of inclination $45^{\circ }$ with the horizontal. One of them is rough and the other is perfectly smooth. A given body takes 2 times as much time to slide down on rough surface than on the smooth surface. The coefficient of kinetic friction (${\mu }_k$) between the object and the rough surface is close to:

A ball is thrown vertically upward. It has a speed of 10 m/s when it has reached one half of its maximum height. How high does the ball rise ?

A bomb of mass 30 kg at rest explodes into two pieces of masses 18 kg and 12 kg. The velocity of 18 kg mass is 6 ms^-1. The kinetic energy of the other mass is :

A force F acting on an object varies with distance x as shown here. The force is in N and x is in m. The work done by the force in moving the object from x = 0 to x = 6 m is :

A stone tied to the end of a string of 1 m long is whirled in a horizontal circle with a constant speed. If the stone makes 22 revolutions in 44 s, what is the magnitude and direction of acceleration of the stone ?

A drum of radius R and mass M, rolls down without slipping along an inclined plane of angle θ. The frictional force :

Two boys are standing at the ends A and B of a ground, where AB = a. The boy at B starts running in a direction perpendicular to AB with velocity v1. The boy at A starts running simultaneously with velocity v and catches the other boy in a time t, where t is :

A particle executing simple harmonic motion of amplitude 5 cm has maximum speed of 31.4 cm/s. The frequency of its oscillation is :

The circular motion of a particle with constant speed is :

The displacement x of a particle varies with time t as x = ae^(-αt) + be^(βt), where a, b, α and β are positive constants. The velocity of the particle will :

For a satellite moving in an orbit around the earth, the ratio of kinetic energy to potential energy is :

The radii of circular orbits of two satellites $A$ and $B$ of the earth, are $4R$ and $R$, respectively. If the speed of satellite $A$ is $3V$, then the speed of satellite $B$ will be:

A body of mass $m$ is attached to the lower end of a spring whose upper end is fixed. When the mass $m$ is slightly pulled down and released, it oscillates with a time period of $3\text{s}$. When the mass $m$ is increased by $1\text{kg}$, the time period of oscillations becomes $5\text{s}$. The value of $m$ in $\text{kg}$ is:

A wheel has an angular acceleration of $3.0{\text{rad/s}}^2$ and an initial angular speed of $2.00\text{rad/s}$. In a time of $2\text{s}$, it has rotated through an angle (in radian) of:

A particle of mass $M$ is situated at the centre of a spherical shell of same mass and radius $a$. The gravitational potential at a point situated at $\frac{a}{2}$ distance from the centre, will be:

A ball is dropped from a high rise platform at $t=0$ starting from rest. After $6\text{ seconds}$ another ball is thrown downwards from the same platform with a speed $v$. The two balls meet at $t=18\text{ s}$. What is the value of $v$? (Take $g=10{\text{ m/s}}^2$)

A uniform rod $AB$ of length $l$ and mass $m$ is free to rotate about point $A$. The rod is released from rest in the horizontal position. Given that the moment of inertia of the rod about $A$ is $\frac{m{l}^2}{3}$, the initial angular acceleration of the rod will be:

A mass $m$ moving with velocity $v$ (along $x$ -axis) collides and sticks to a mass $3m$ moving with velocity $2v$ (along $y$ -axis). The final velocity of the combination is:

A belt moves at $2\text{m/s}$. A box is dropped on it ($\mu =0.5$). Distance the box moves relative to belt before coming to rest on it is:

A car moves from $X$ to $Y$ with a uniform speed $v_u$ and returns to $X$ with a uniform speed $v_d$. The average speed for this round trip is:

A mass on a string rotates on a table. If radius is decreased by factor of $2$ by pulling the string, the kinetic energy will:

The displacement x of a particle varies with time t as x = ae-αt + beβt, where a, b, α and β are positive constants. The velocity of the particle will :

Three masses are placed on the x-axis: $300\text{g}$ at origin, $500\text{g}$ at $x=40\text{cm}$ and $400\text{g}$ at $x=70\text{cm}$. The distance of the centre of mass from the origin is:

A particle executes simple harmonic oscillation with an amplitude $a$. The period of oscillation is $T$. The minimum time taken by the particle to travel half of the amplitude from the equilibrium position is:

A car of mass $m$ is moving on a level circular track of radius $R$. If ${\mu }_s$ is the static friction, the maximum speed is:

A solid cylinder of mass $3\text{kg}$ is rolling on a horizontal surface with velocity $4{\text{ms}}^{-1}$. It collides with a horizontal spring of force constant $200{\text{Nm}}^{-1}$. The maximum compression produced in the spring will be:

A block $B$ is pushed momentarily along a horizontal surface with an initial velocity $v$. If $\mu$ is the coefficient of sliding friction between $B$ and the surface, block $B$ will come to rest after a time:

The horizontal range and the maximum height of a projectile are equal. The angle of projection of the projectile is:

Two spheres $A$ and $B$ of masses $m_1$ and $m_2$ respectively collide. $A$ is at rest initially and $B$ is moving with velocity $v$ along x-axis. After collision $B$ has a velocity $v/2$ in a direction perpendicular to the original direction. The mass $A$ moves after collision in the direction:

A particle of mass $m$ moves in the $XY$ plane with a velocity $v$ along the straight line $AB$. If the angular momentum of the particle with respect to origin $O$ is $L_A$ when it is at $A$ and $L_B$ when it is at $B$, then:

A particle has initial velocity $(2\hat{i}+3\hat{j})$ and acceleration $(0.3\hat{i}+0.2\hat{j})$. The magnitude of velocity after $10$ seconds will be:

The height at which the weight of a body becomes $1/16$ th its weight on the surface of earth (radius $R$) is:

The particle executing simple harmonic motion has a kinetic energy $K_0{\cos }^2\omega t$. The maximum values of the potential energy and the total energy are respectively:

A geostationary satellite is orbiting the earth at a height of $5R$ above the surface, $R$ being the radius of the earth. The time period of another satellite in hours at a height of $2R$ from the surface of the earth is:

$ABC$ is an equilateral triangle with $O$ as its centre. ${\vec{F}}_1,{\vec{F}}_2$ and ${\vec{F}}_3$ represent three forces acting along the sides $AB,BC$ and $AC$ respectively. If the total torque about $O$ is zero then the magnitude of $F_3$ is:

A bomb of mass 30 kg at rest explodes into two pieces of masses 18 kg and 12 kg. The velocity of 18 kg mass is 6 ms-1. The kinetic energy of the other mass is :

A force F acting on an object varies with distance x as shown here. The force is in N and x is in m. The work done by the force in moving the object from x = 0 to x = 6 m is :

In the figure given, the position-time graph of a particle of mass $0.1\text{kg}$ is shown. The impulse at $t=2\text{sec}$ is:

A mass of $2.0\text{kg}$ is put on a flat pan attached to a vertical spring. The spring constant is $200\text{N/m}$. What should be the minimum amplitude of the motion, so that the mass gets detached from the pan? ($g=10{\text{m/s}}^2$)

When a mass is rotating in a plane about a fixed point, its angular momentum is directed along:

A car of mass $1000\text{kg}$ negotiates a banked curve of radius $90\text{m}$ on a frictionless road. If the banking angle is $45^{\circ }$, the speed of the car is:

Two satellites of earth, $S_1$ and $S_2$, are moving in the same orbit. The mass of $S_1$ is four times the mass of $S_2$. Which one of the following statements is true?

A spherical planet has a mass $M_p$ and diameter $D_p$. A particle of mass $m$ falling freely near the surface of this planet will experience an acceleration due to gravity, equal to:

The potential energy of a particle in a force field is $U=A/r^2-B/r$, where $A$ and $B$ are positive constants and $r$ is the distance of particle from the centre. For stable equilibrium the distance of the particle is:

A block of mass $10\text{kg}$ is moving in $x$ -direction with a constant speed of $10\text{m/sec}$. It is subjected to a retarding force $F=-0.1x\text{joule/meter}$ during its travel from $x=20\text{meter}$ to $x=30\text{meter}$. Its final kinetic energy will be:

The phase difference between the instantaneous velocity and acceleration of a particle executing simple harmonic motion is:

Two persons of masses $55\text{kg}$ and $65\text{kg}$ respectively, are at the opposite ends of a boat. The length of the boat is $3.0\text{m}$ and weighs $100\text{kg}$. The $55\text{kg}$ man walks up to the $65\text{kg}$ man and sits with him. If the boat is in still water the centre of mass of the system shifts by:

The motion of a particle along a straight line is described by equation: $x=8+12t-t^3$ where $x$ is in metre and $t$ in second. The retardation of the particle when its velocity becomes zero, is:

A vertical spring with force constant $k$ is fixed on a table. A ball of mass $m$ at a height $h$ above the free upper end of the spring falls vertically on the spring so that the spring is compressed by a distance $d$. The net work done in the process is:

The damping force on an oscillator is directly proportional to the velocity. The units of the constant of proportionality are:

The upper half of an inclined plane of inclination $\theta$ is perfectly smooth while the lower half is rough. A block starting from rest at the top of the plane will again come to rest at the bottom, if the coefficient of friction between the block and lower half of the plane is given by:

A candle of diameter $d$ is floating on a liquid in a cylindrical container of diameter $D(D>>d)$ as shown in figure. If it is burning at the rate of $2\text{cm/hour}$ then the top of the candle will: