mechanics Questions

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 :

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

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

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:

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:

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

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

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

$ABC$ 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:

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:

Two particles which are initially at rest, move towards each other under the action of their internal attraction. If their speeds are $v$ and $2v$ at any instant, then the speed of centre of mass of the system will be:

A gramophone record is revolving with an angular velocity $\omega$. A coin is placed at a distance $r$ from the centre of the record. The static coefficient of friction is $\mu$. The coin will revolve with the record if:

A ball moving with velocity $2\text{ m/s}$ collides head on with another stationary ball of double the mass. If the coefficient of restitution is $0.5$ then their velocities (in $\text{m/s}$) after collision will be:

An engine pumps water through a hose pipe. Water passes through the pipe and leaves it with a velocity of $2\text{ m/s}$. The mass per unit length of water in the pipe is $100\text{ kg/m}$. What is the power of the engine?

A man of $50\text{ kg}$ mass is standing in a gravity free space at a height of $10\text{ m}$ above the floor. He throws a stone of $0.5\text{ kg}$ mass downwards with a speed $2\text{ m/s}$. When the stone reaches the floor, the distance of the man above the floor will be:

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 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:

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

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

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: