Mechanics Questions

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

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:

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 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? ($g=10{\text{m/s}}^2$)

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?

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

$\vec{A}$ and $\vec{B}$ are two vectors and $\theta$ is the angle between them, if $|\vec{A}\times \vec{B}|=\sqrt{3}(\vec{A}\cdot \vec{B})$, the value of $\theta$ is:

A particle moving along $x$ -axis has acceleration $f$ at time $t$ given by $f=f_0(1-t/T)$, where $f_0$ and $T$ are constants. The particle at $t=0$ has zero velocity. In the time interval between $t=0$ and the instant when $f=0$, the particle’s velocity ($v_x$) is:

The position $x$ of a particle with respect to time $t$ along the $x$ -axis is given by $x=9t^2-t^3$ where $x$ is in metre and $t$ in second. What will be the position of this particle when it achieves maximum speed along the $+x$ direction?

A particle starting from the origin $(0,0)$ moves in a straight line in the $(x,y)$ plane. Its coordinates at a later time are $(\sqrt{3},3)$. The path of the particle makes with the $x$ -axis an angle of:

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 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 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 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 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:

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 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:

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:

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:

$ABC$ 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:

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:

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:

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 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:

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:

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

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:

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:

An explosion breaks a rock into three parts in a horizontal plane. Two of them go off at right angles to each other. The first part of mass $1\text{kg}$ moves with a speed of $12\text{m/s}$ and the second part of mass $2\text{kg}$ moves with $8\text{m/s}$ speed. If the third part flies off with $4\text{m/s}$ speed, then its mass is:

The Young’s modulus of steel is twice that of brass. Two wires of same length and of same area of cross section, one of steel and another of brass are suspended from the same roof. If we want the lower ends of the wires to be at the same level, then the weights added to the steel and brass wires must be in the ratio of:

If vectors $\vec{A}=\cos \omega t\hat{i}+\sin \omega t\hat{j}$ and $\vec{B}=\cos \frac{\omega t}{2}\hat{i}+\sin \frac{\omega t}{2}\hat{j}$ are functions of time, then the value of $t$ at which they are orthogonal to each other is:

An automobile moves on a road with a speed of $54{\text{km h}}^{-1}$. The radius of its wheels is $0.45\text{m}$ and the moment of inertia of the wheel about its axis of rotation is $3{\text{kg m}}^2$. If the vehicle is brought to rest in $15\text{s}$, the magnitude of average torque transmitted by its brakes to the wheel is:

A given shaped glass tube having uniform cross section is filled with water and is mounted on a rotatable shaft as shown in figure. If the tube is rotated with a constant angular velocity $\omega$ then:

A roller coaster is designed such that riders experience "weightlessness" as they go round the top of a hill whose radius of curvature is $20\text{m}$. The speed of the car at the top of the hill is between:

A satellite $S$ is moving in an elliptical orbit around the earth. The mass of the satellite is very small compared to the mass of the earth. Then:

A remote-sensing satellite of earth revolves in a circular orbit at a height of $0.25\times 10^6\text{m}$ above the surface of earth. If earth’s radius is $6.38\times 10^6\text{m}$ and $g=9.8{\text{ms}}^{-2}$, then the orbital speed of the satellite is:

A body, under the action of a force $\vec{F}=6\hat{i}-8\hat{j}+10\hat{k}$, acquires an acceleration of $1{\text{m/s}}^2$. The mass of this body must be:

The position vector of a particle $\vec{R}$ as a function of time is given by:
$$\vec{R}=4\sin (2\pi t)\hat{i}+4\cos (2\pi t)\hat{j}$$
Where $R$ is in meters, $t$ is in seconds and $\hat{i}$ and $\hat{j}$ denote unit vectors along $x$ - and $y$ -directions, respectively. Which one of the following statements is wrong for the motion of particle?

Point masses $m_1$ and $m_2$ are placed at the opposite ends of a rigid rod of length $L$, and negligible mass. The rod is to be set rotating about an axis perpendicular to it. The position of point $P$ on this rod through which the axis should pass so that the work required to set the rod rotating with angular velocity ${\omega }_0$ is minimum, is given by:

When a ball is thrown up vertically with velocity $v_0$, it reaches a maximum height of $h$. If one wishes to triple the maximum height then the ball should be thrown with velocity:

If $\vec{F}$ is 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 force $\vec{F}=\alpha \hat{i}+3\hat{j}+6\hat{k}$ is acting at a point $\vec{r}=2\hat{i}-6\hat{j}-12\hat{k}$. The value of $\alpha$ for which angular momentum about origin is conserved is:

Two particles $A$ and $B$ move with constant velocities ${\vec{v}}_1$ and ${\vec{v}}_2$. At the initial moment their position vectors are ${\vec{r}}_1$ and ${\vec{r}}_2$ respectively. The condition for particles $A$ and $B$ for their collision is:

The mass of a lift is $2000\text{kg}$. When the tension in the supporting cable is $28000\text{N}$, then its acceleration is:

A plank with a box on it at one end is gradually raised about the other end. As the angle of inclination with the horizontal reaches $30^{\circ }$, the box starts to slip and slides $4.0\text{m}$ down the plank in $4.0\text{s}$. The coefficients of static and kinetic friction between the box and the plank will be, respectively:

Two stones of masses $m$ and $2m$ are whirled in horizontal circles, the heavier one in a radius $\frac{r}{2}$ and the lighter one in radius $r$. The tangential speed of the lighter stone is $n$ times that of the heavier stone when they experience same centripetal forces. The value of $n$ is:

A solid sphere is rolling on a frictionless surface, shown in figure with a translational velocity $v\text{m/s}$. If it is to climb the inclined surface then $v$ should be:

An engine pumps water continuously through a hose. Water leaves the hose with a velocity $v$ and $m$ is the mass per unit length of the water jet. What is the rate at which kinetic energy is imparted to water?

A ball is thrown vertically downwards from a height of $20\text{m}$ with an initial velocity $v_0$. It collides with the ground, loses $50$ percent of its energy in collision and rebounds to the same height. The initial velocity $v_0$ is: (Take $g=10{\text{m s}}^{-2}$)

On a frictionless surfaces, a block of mass $M$ moving at speed $v$ collides elastically with another block of same mass $M$ which is initially at rest. After collision the first block moves at an angle $\theta$ to its initial direction and has a speed $\frac{v}{3}$. The second block’s speed after the collision is:

A bus is moving with a speed of $10{\text{ms}}^{-1}$ on a straight road. A scooterist wishes to overtake the bus in $100\text{s}$. If the bus is at a distance of $1\text{km}$ from the scooterist, with what speed should the scooterist chase the bus?

The heart of a man pumps $5\text{litres}$ of blood through the arteries per minute at a pressure of $150\text{mm}$ of mercury. If the density of mercury be $13.6\times 10^3{\text{kg/m}}^3$ and $g=10{\text{m s}}^2$ then the power of heart in watt is:

Two non-mixing liquids of densities $\rho$ and $n\rho (n>1)$ are put in a container. The height of each liquid is $h$. A solid cylinder of length $L$ and density $d$ is put in this container. The cylinder floats with its axis vertical and length $pL(p<1)$ in the denser liquid. The density $d$ is equal to:

A horizontal platform is rotating with uniform angular velocity around the vertical axis passing through its centre. At some instant of time a viscous fluid of mass $m$ is dropped at the centre and is allowed to spread out and finally fall. The angular velocity during this period:

An explosion blows a rock into three parts. Two parts go off at right angles to each other: a $1\text{kg}$ part moving at $12{\text{ms}}^{-1}$ and a $2\text{kg}$ part moving at $8{\text{ms}}^{-1}$. If the third part flies off with a velocity of $4{\text{ms}}^{-1}$, its mass would be:

At what height from the surface of earth the gravitation potential and the value of $g$ are $-5.4\times 10^7\text{J/kg}$ and $6.0{\text{m/s}}^2$ respectively? Take the radius of earth as $6400\text{km}$:

The ratio of escape velocity at earth $(v_e)$ to the escape velocity at a planet $(v_p)$ whose radius and mean density are twice as that of earth is:

A particle starts its motion from rest under the action of a constant force. If the distance covered in the first $10$ seconds is $S_1$ and that covered in the first $20$ seconds is $S_2$, then:

A car is negotiating a curved road of radius $R$. The road is banked at an angle $\theta$. The coefficient of friction between the tyres of the car and the road is ${\mu }_s$. The maximum safe velocity on this road is:

A body of mass $1\text{kg}$ begins to move under the action of a time dependent force $\vec{F}=(2t\hat{i}+3t^2\hat{j})\text{N}$, where $\hat{i}$ and $\hat{j}$ are unit vectors along $x$ and $y$ axis. What power will be developed by the force at the time $t$?

A ladder is leaned against a smooth wall and it is allowed to slip on a frictionless floor. Which figure represents trace of its centre of mass?

A body of mass $1\text{kg}$ is thrown upwards with a velocity $20\text{m/s}$. It momentarily comes to rest after attaining a height of $18\text{m}$. How much energy is lost due to air friction? ($g=10{\text{m/s}}^2$)

From a disc of radius $R$ and mass $M$, a circular hole of diameter $R$, whose rim passes through the centre is cut. What is the moment of inertia of the remaining part of the disc about a perpendicular axis, passing through the centre?

If the magnitude of sum of two vectors is equal to the magnitude of difference of the two vectors, the angle between these vectors is:

Two bodies of mass $1\text{kg}$ and $3\text{kg}$ have position vectors $\hat{i}+2\hat{j}+\hat{k}$ and $-3\hat{i}-2\hat{j}+\hat{k}$ respectively. The centre of mass of this system has a position vector:

If the velocity of a particle is $v=At+B{t}^2$, where $A$ and $B$ are constants, then the distance travelled by it between $1\text{s}$ and $2\text{s}$ is:

A disk and a sphere of same radius but different masses roll off on two inclined planes of the same altitude and length. Which one of the two objects gets to the bottom of the plane first?

A person is standing in an elevator. In which situation he finds his weight less?

A particle moves a distance $x$ in time $t$ according to equation $x=(t+5{)}^{-1}$. The acceleration of the particle is proportional to:

A particle moves so that its position vector is given by $\vec{r}=\cos \omega t\hat{x}+\sin \omega t\hat{y}$ where $\omega$ is a constant. Which of the following is true?

A uniform circular disc of radius $50\text{cm}$ at rest is free to turn about an axis which is perpendicular to its plane and passes through its centre. It is subjected to a torque which produces a constant angular acceleration of $2.0{\text{rad s}}^{-2}$. Its net acceleration in ${\text{ms}}^{-2}$ at the end of $2.0\text{s}$ is approximately:

Six vectors, $\vec{a}$ through $\vec{f}$ have the magnitudes and directions indicated in the figure. Which of the following statements is true?

What is the minimum velocity with which a body of mass $m$ must enter a vertical loop of radius $R$ so that it can complete the loop?

A particle of mass $10\text{g}$ moves along a circle of radius $6.4\text{cm}$ with a constant tangential acceleration. What is the magnitude of this acceleration if the kinetic energy of the particle becomes equal to $8\times 10^{-4}\text{J}$ by the end of the second revolution after the beginning of the motion?

Assertion: Frictional forces are conservative forces.Reason: Potential energy can be associated with frictional forces.

A particle has initial velocity $(3\hat{i}+4\hat{j})$ and has acceleration $(0.4\hat{i}+0.3\hat{j})$. Its speed after $10\text{ s}$ is:

Two cars $P$ and $Q$ start from a point at the same time in a straight line and their positions are represented by $x_P(t)=at+b{t}^2$ and $x_Q(t)=ft-t^2$. At what time do the cars have the same velocity?

In the given figure, $a=15{\text{m/s}}^2$ represents the total acceleration of a particle moving in the clockwise direction in a circle of radius $R=2.5\text{m}$ at a given instant of time. The speed of the particle is:

A block of mass $m$ is in contact with the cart $C$ as shown in the figure. The coefficient of static friction between the block and the cart is $\mu$. The acceleration $\alpha$ of the cart that will prevent the block from falling satisfies:

A rigid ball of mass $m$ strikes a rigid wall at $60^{\circ }$ and gets reflected without loss of speed. The value of impulse imparted by the wall in the ball will be:

A bullet of mass $10\text{g}$ moving horizontally with a velocity of $400{\text{ms}}^{-1}$ strikes a wooden block of mass $2\text{kg}$ which is suspended by a light inextensible string of length $5\text{m}$. As a result the centre of gravity of the block is found to rise a vertical distance of $10\text{cm}$. The speed of the bullet after it emerges out horizontally from the block will be:

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 :