A particle moves in a force field given by $\overrightarrow{\mathrm{F}} = \hat{\mathrm{r}} \mathrm{F} \left(\mathrm{r}\right)$, where r is the unit vector along with the position vector $\overrightarrow{\mathrm{r}}$, then which is true? 

Two perfectly elastic particles A and B of equal mass traveling along the line joining them with velocities 15 m/s and 10 m/s. After the collision, their velocities will be 

Four masses are fixed on a massless rod as shown in the figure. The moment of inertia about the axis P is about

A circular plate of diameter d is kept in contact with a square plate of an edge as shown in the figure. The density of the material and the thickness are the same everywhere. The center of mass of the composite system will be

From a circular disc of radius R, a square is cut out with a radius as its diagonal.  The centre of mass of remainder is at a distance (from the centre)

A particle of mass m moves along line PC with velocity v as shown.  What is the angular momentum of the particle about P?

Two bodies with moment of inertia $I_1 and I_2$ $(I_1>I_2)$ and have equal angular momenta.  If KE of rotation are $E_1 and E_2$, then

A mass m with velocity u strikes a wall normally and returns at the same speed. What is the magnitude of the change in momentum of the body when it returns

Three indentical spheres, each of mass 1 kg are kept as shown in figure, touching each other, with their centres on a straight line. If their centres are marked P, Q, R respectively, the distance of centre of mass of the system from P is

$$\begin{gathered} 1. \frac{\mathrm{PQ} + \mathrm{PR} + \mathrm{QR}}{3} \\ 2. \frac{\mathrm{PQ} +\hspace{0.17em}\mathrm{PR}}{3} \\ 3. \frac{\mathrm{PQ} +\hspace{0.17em}\mathrm{QR}}{3} \\ 4. \frac{\mathrm{PR} + \mathrm{QR}}{3} \end{gathered}$$

Two particles of equal mass have coordinates (2m,4m,6m) and (6m,2m,8m). Of these one particle has velocity ${\mathrm{v}}_1=\left(2\hat{\mathrm{i}}\right)$ m/s and another particle has velocity ${\mathrm{v}}_2=\left(2\hat{j}\right)$ m/s at time t = 0. The coordinate of their centre of mass at time t = 1 s will be

The M.I. of a body about the given axis is 1.2 kg x ${\mathrm{m}}^2$ initially the body at rest. In order to the rotational kinetic energy of 1500 J, the angular acceleration of 25 rad/${\sec }^2$ must be about that axis for the duration of

A loop rolls down on the inclined plane. The fraction of its total kinetic energy that is associated with rotational motion is

A ball kept in a closed box moves in the box making collisions with the walls. The box is kept on a smooth surface. The velocity of the centre of mass

 A body of mass ${\mathrm{m}}_1$ moving with a velocity $3 {\mathrm{ms}}^{-1}$ collides elastically with another body  at rest of mass ${\mathrm{m}}_2$. After collision the velocities of the two bodies are $2 {\mathrm{ms}}^{-1}$ and $5 {\mathrm{ms}}^{-1}$ respectively along the direction of motion of ${\mathrm{m}}_2$. The ratio $\frac{{\mathrm{m}}_1}{{\mathrm{m}}_2}$ is

Two solid cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which statement (s) is (are) correct ?

A solid sphere is rolling on a frictionless surface, shown in figure with a translational velocity v m/s. If sphere climbs up to height h then value of v should be -

A soldier is firing 20 bullets per second from his gun having a muzzle speed of 150 m/s. The mass of each bullet is 50 g. If they strike the wall and rebound with the same speed, then the force on the wall is

A projectile is fired at angle $\mathrm{\theta }$ with horizontal with initial speed u. If it breaks into two halves such that one half retraces its path, then the distance of hitting point of the other half from starting point is

The value of M, as shown, for which the rod will be in equilibrium is (rod is massless)

The minimum coefficient of friction for a solid sphere to roll without slipping on an inclined plane of inclination 45$°$ is

A wheel is rolling along a horizontal ground with a speed o 10 m/s. The magnitude of the velocity of the points at the extremities of the horizontal diameter of the wheel is equal to

A projectile of mass 50 kg is shot vertically upwards with an initial velocity of 100ms^-1. After 5 seconds it explodes into two fragments, one of which having mass 20 kg, travels vertically up with a velocity of 150 ms^-1. The velocity of the other fragment at that instant is -          [Take g= 9.8 m/$s^2$]

A sphere of mass m moving with constant velocity hits another sphere of the same mass at rest. If e is the coefficient of restitution. The ratio of their velocities after the collision is

A mass M is moving with constant velocity parallel to the X-axis. Its angular momentum w.r.t. origin is:

If I is the moment of inertia of a solid sphere about an axis parallel to diameter of a shpere and at a distance x from it. Which of the following graph represents the variation of I with X?

A particle is rotating under the central force, in which one of the following quantities is constant:

A small object of uniform density rolls up a curved surface with an initial velocity v. It is reached upto a maximum height of $\frac{3{\mathrm{v}}^2}{4\mathrm{g}}$ w.r.t. the initial position. The object is:

A point is at (x, y, z). What is its moment of inertia about x-axis:

The disc is placed on the rough horizontal surface given only angular velocity ${\omega }_0$. The velocity of the centre of mass when it starts pure rolling is:

A force 2N is along line y = 3x + 4. The torque about the origin is: