System of Parti...

A spool ( initially at rest ) is pulled on a smooth horizontal surface by constant horizontal forceF. Such that the spool comes into a state of sustained rolling motion. Kinetic energy of the spool, after centre of mass of the spool has moved a distance d, is

$$AB$$ and $$CD$$ are two identical rods each of length $$l$$ and  mass $$m$$ joined to from a cross is fixed inside a ring of mass $$m$$ and radius $$l/2$$. Moment of intertia of the system about a bisector of the angle between the rods (xy) is

The centre of mass of a system of three particles of masses 1 g, 2 g and 3 g is taken as the origin of a coordinate system. The position vector of a fourth particle of madd 4 g such that the centre of mass of the four particle system lies at   the point (1,2,3) is  $$\alpha $$ $$(\overset { \wedge  }{ i } +2\overset { \wedge  }{ j } +3\overset { \wedge  }{ k } )$$, where $$\alpha $$ is a constant. The value of $$\alpha $$ is 

ABC is a right - angle triangular plate of uniform thickness. The sides are such that AB > BC as shown in figure.$$I_{1}, I_{2} and I_{3}$$ are moments of inertia about AB, BC and AC respectively. Then which of the following relations is correct?

A solid sphere is thrown on a horizontal rough surface with initial velocity of center of mass $$v_o$$ without rolling. Velocity of its center of mass when it starts pure rolling is

A sphere of mass M and radius r shown in figure slip on a rough horizontal plane.At some instant it has translational  velocity $${v_0}$$ and rotational velocity about centre $$\dfrac{{{v_0}}}{{2l}}$$.Find the translational velocity after the sphere starts pure rolling.

If moment of inertia of a body of mass $$m$$ is double than moment of inertia of rod of same mass $$m$$ about transverse axis passing through its centre then radius of gyration of given body is

Two particles $$A$$ and $$B$$ initially at rest move towards each other under a mutual force of attraction. At the instant when velocity of $$A$$ is $$v$$ and that of $$B$$ is $$2v$$ the velocity of centre of mass of the system: 

In the figure shown a hole of radius 2 cm is made in a semi-circular disc of the radius $$6\pi $$at a distance 8 cm from center C of the disc. The distance of the center of mass of this system from C is :

A slender uniform rod of mass $$M$$ and length $$l$$ is pivoted at one end so that it can rotate in a vertical plane (see figure). There is negligible friction at the pivot. The free end is held vertically above the pivot and then released. The angular acceleration of the rod when it makes an angle $$\theta$$ with the vertical is: