System of Parti...

$$\mathrm{A}$$ particle moves in x-y under the action of the force $$\vec{\mathrm{F}}$$ Such that $$\mathrm{x}$$ and $$\mathrm{y}$$ components of linear momentum, $$\vec{\mathrm{P}}$$ at any time $$\mathrm{t}$$ are  $$  2\cos \mathrm{t}$$ and  $$ 2 \sin \mathrm{t}.\ \mathrm{F}$$ind the angle between $$\vec{\mathrm{F}}\&\vec{\mathrm{P}}$$ at a given time

Identify the correct order in which the ratio of radius of gyration to radius increases for the following bodies.
I) rolling solid sphere
II) rolling solid cylinder
III) rolling hollow cylinder
IV) rolling hollow sphere

$$I$$ is moment of inertia of a thin rod of uniform thickness about an axis perpendicular to its length and passing through its centre. The rod at one of two sides of axis is cut and removed. The moment of inertia of remaining rod about same axis is :

If $$R$$ is radius and $$K$$ is radius of gyration then in which of the following cases of rolling bodies match the ratio $$K^2:R^2$$.

Choose the correct statement about the centre
of mass (CM) of system of two particles.
a. The C.M. lies on the line joining the two particles midway between them
b. The C.M. lies on the line joining them at a point whose distance from each particle is inversely proportional to the mass of that particle
c. The C.M. lies on the line joining them at a point whose distance from each particle is proportional to the square of the mass of the particle
d. The C.M. is on the line joining them at a point whose distance from each particle is proportional to the mass of that particle.

A solid cylinder of mass $$M$$ and radius $$R$$ rolls without slipping down an inclined plane of length $$L$$ and height $$h$$. What is the speed of its centre of mass when the cylinder reaches its bottom -

A rod rests on a friction less surface. Two forces each of magnitude F are applied in the opposite direction on the edges of the rod as shown in the figure below. Which of the following quantities are nonzero and constant:
(i) angular momentum (ii) angular acceleration (iii) Total force (iv) total torque (v) total linear momentum (vi) total kinetic energy (vii) moment of inertia (viii) translation kinetic energy

The ratio of the radii of gyration of a circular disc about a tangential axis in the plane of the disc and of a circular ring of the same radius about a tangential axis in the plane of the ring is

A boat of mass 60kg is floating in still water. A boy of mass 20kg walks from one end to the other end. If the length of the boat is 3m, the distance through which the boat moves is: