System of particles practice · Medium

A hollow sphere and a solid sphere of same mass and radius are rolling on a rough inclined planes such that pure rolling is sustained for each. The ratio of acceleration of centre of mass of hollow sphere to the solid sphere is

Assertion: The centre of mass of a body may lie where there is no mass.
Reason: Centre of mass of a body is a point, where the whole mass of the body is supposed to be concentrated.

Statements:
a) The motion of the centre of mass can be studied using kepler's laws of motion.
b) The internal forces will not effect the motion of the centre of mass

Assertion (A): In case of rolling without sliding, friction force can act in forward and backward direction both.
Reason (R): The angular momentum of a system will be conserved only about that point about which external angular impulse is zero.

A large tray of mass ${M}$ has in it a cubical block of ice of mass ${m}$ and edge length ${L}$. The ice starts melting. The shift in the height of centre of mass of system (ice + tray) is

S-I: Moment of inertia of a particle changes, when the axis of rotation changes.
S-II: Moment of inertia depends on mass and distance of the particles.

The ratio of the radii of gyration of a hollow sphere and a solid sphere of the same radii about a tangential axis is

Assertion: The rate of change of angular momentum is directly proportional to applied external torque.
Reason: It corresponds to Newton's second law of rotational motion.

Which of the following is true about the
angular momentum of a cylinder down a slope
without slipping:

A cylinder full. Of water, is rotating about its own axis with uniform angular velocity $\omega$. The shape of free surface of water will be

The motion of the centre of mass of system of two particles is not affected by the internal forces

If the earth is a point mass of $6 \times {10^{24}}\;kg$ revolving around the sun at a distance of $1.5\; \times {10^8}\;km$ and in time $T = 3.14 \times {10^7}s$, then the angular momentum of the earth around the sun is

The position of centre of mass along x, y, z axes are 3 cm, 4 cm, 5 cm respectively. Find the position of centre of mass in space is

A particle moves in a circle of radius 25 cm at two revolutions per second. The acceleration of the particle in $m/{s^2}$ is

The radius of gyration (K) of a rigid body changes with change of:-

Assertion: A couple does not exert a net force on an object even though it exerts a torque.
Reason: Couple is a pair of two forces with equal magnitude but opposite directions acting simultaneously on a body in different lines of action.

A kid of mass $M$ stands at the edge of a platform of radius $R$ which can be freely rotated about its axis. The moment of inertia of the platform is $I$. The system is at rest when a friend throws a ball of mass $m$ and the kid catches it. If the velocity of the ball is $v$ horizontally along the tangent to the edge of the platform when it was caught by the kid, find the angular speed of the platform after the event.

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 the centre of mass of the system will be

S-I: A wheel moving down a perfectly frictionless inclined plane will undergo slipping (not rolling motion).
S-II: For perfect rolling motion, work done against the friction is zero.

A particle undergoes uniform circular motion. About which point on the plane of the circle, will the angular momentum of the particle remain conserved?

Two discs $P$ and $Q$ having equal masses and thickness but densities ' $\rho_1$ ' and ' $\rho_2$ ' respectively, are such that $\rho_1 < \rho_2$. The moment of inertia of the disc $P$ and $Q$ are related as

Assertion: According to the conservation of Angular momentum, When inertia gets doubled kinetic energy will be halved;

Reason: Rotational Kinetic energy for the system is $\frac{1}{2} I \omega$

A ring of diameter $0.4 \mathrm{~m}$ and of mass $10 \mathrm{~kg}$ is rotating about its axis at the rate of $2100 \mathrm{~rpm}$. Find rotational K.E. of the ring.

Angular velocity of each particle of a rotating rigid body about axis of rotation is:

A cylinder of mass $m$ and radius $R$ rolls purely over an inclined surface while moving up on it. Correct statement regarding friction acting on it is

The motion of a rigid body which is not pivoted or fixed in some way is either a pure ___A___ or a combination of translation and rotation. The motion of a rigid body which is pivoted or fixed in some way is ___B___ Here, A and B refer to:

A circular plate is rotating about its own axis at an angular velocity 100 revolution per minute. The linear velocity of a particle ‘P’ of plate at a distance 4.2 cm from axis of rotation (in m/s) is

If a rigid body is rotating about an axis with a
constant velocity, then

Moment of inertia of a circular loop of radius R about the axis of rotation parallel to horizontal diameter at a distance R/2 from it is

Two uniform solid spheres having unequal masses and unequal radii are released from rest from the same height on a rough incline. If the spheres roll without slipping,