Class XI Physics Important Qs on System of Particles and Rotational Motion

A uniform thin rod AB of length L has linear mass density, where x is measured from A. If the CM of the rod lies at a distance of from A, then a and b are related as:

A thin bar of length L has a mass per unit length λ , that increases linearly with distance from one end. If its total mass is M and its mass per unit length at the lighter end is λ_o ;then the distance of the centre of mass from the lighter end is -

A block of mass m=0.1 kg is connected to a spring of unknown spring constant k. It is compressed to a distance x from its equilibrium position and released from rest. After approaching half the distance from equilibrium position, it hits another block and comes to rest momentarily, while the other block moves with a velocity 3 ms^-1. The total initial energy of the spring is:

A 4 g bullet is fired horizontally with a speed of 300 m/s into 0.8 kg block of wood at rest on a table. If the coefficient of friction between the block and the table is 0.3, how far will the block slide approximately? Assume

The moment of inertia of a uniform cylinder of length  and radius R about its perpendicular bisector is I. What is the ratio  such that the moment of inertia is minimum?

Consider a cylinder of mass M resting on a rough horizontal plank that is pulled out from under it with an acceleration ‘a’ perpendicular to the axis of the cylinder. What is at point P? It is assumed that the cylinder does not slip.

A particle is released on a vertical smooth semicircular track from point X so that OX makes angle from the vertical (see figure). The normal reaction of the track on the particle vanishes at point Y where OY makes angle with the horizontal. Then