System of particles practice · Mixed difficulty

Two particles of masses $m_1$ and $m_2\left(m_1 > m_2\right)$, initially at rest, moves towards each other under an inverse square law force of attraction. Pick out the correct statement about the centre of mass (CM) of the system

If a particle moves in the $X-Y$ plane, the resultant angular momentum has

The moment of inertia of a uniform semi-circular disc of mass $M$ and radius $R$ about a line perpendicular to the plane of the disc through the centre is

If $R$ is the radius of gyration of a body of mass $M$ and radius $r$, then the ratio of its rotational to translational kinetic energy in the rolling condition is

Inertia is that property of a body by virtue of
which the body is

A flywheel is attached to an engine to

A particle is moving in a circular path. The correct statement out of the following is

Moment of inertia of a ring of mass $m = 4\,gm$ and radius $r = 1\,cm\,$ about an axis passing through its edge and parallel to its natural axis is

A train is moving with a speed of $12 \mathrm{~m} / \mathrm{s}$ on rails which are $1.5 \mathrm{~m}$ apart. To negotiate a curve of radius $400 \mathrm{~m},$ the height by which the outer rail should be raised with respect to the inner rail is $\text{(Given } g=10 \mathrm{~m} / \mathrm{s}^2)$

Which one of the following has maximum inertia?

The moment of inertia of a disc of mass $\mathrm{M}$ and radius $\mathrm{R}$ about an axis which is tangent to the circumference of the disc and parallel to its diameter, is

If a force acts on a body at a point away from the centre of mass, then

A ring of mass $0.3 \mathrm{~kg}$ and radius $0.1 \mathrm{~m}$ and a solid cylinder of mass $0.4 \mathrm{~kg}$ and of the same radius are given the same KE and released simultaneously on a flat horizontal surface such that they begin to roll as soon as released towards a wall which is at the same distance from the ring and cylinder. Which will reach the wall first?

Position vector of centre of mass of two particles system is given by:

Assertion: A person standing on a rotating platform suddenly stretched his arms, the platform slows down.
Reason: A person by stretching his arms increases the moment of inertia and decreases angular velocity.

Assertion: Toppling is the application of the centre of mass.
Reason: Toppling stability determines the maximum angle that an object can be tilted before it will topple over.

A body of mass $1.5 \mathrm{~kg}$ rotating about an axis with angular velocity of $0.3 \mathrm{rad} \mathrm{s}^{-1}$ has the angular momentum of $1.8 \mathrm{~kg}$ $\mathrm{m}^2 \mathrm{~s}^{-1}$. The radius of gyration of the body about an axis is

The moment of inertia of a straight thin rod of mass M and length L about an axis perpendicular to its length and passing through one end is

A fan is rotating with an angular speed 300 rpm. The fan is switched off, and it takes $80 \mathrm{~s}$ to come to rest. Assuming constant angular deceleration, the number of revolutions made by the fan before it comes to rest is

Calculate the angular momentum of a body whose rotational energy is 10 Joule. If the angular momentum vector coincides with the axis of rotation and its moment of inertia about this axis is $8 \times {10^{ - 7}}\;kg\;{m^2}$

If a solid sphere, disc and cylinder are allowed to roll down an inclined plane from the same height

A circular turn table has a block of ice placed at its centre. The system rotates with an angular speed $\omega$ about an axis passing through the centre of the table. If the ice melts on its own without any evaporation, the speed of rotation of the system

A mass of 10 kg connected at the end of a rod of negligible mass is rotating in a circle of radius 30 cm with an angular velocity of $10\;rad/sec$. If this mass is brought to rest in 10 sec by a brake, what is the magnitude of the torque applied

Assertion: The angular momentum of a body is equal to twice the product of its mass and areal velocity.
Reason: When no external torque acts, the motion of the body takes place in a fixed plane.

From an inclined plane a sphere, a disc, a ring and a spherical shell are rolled without slipping. The order of their reaching at the base will be

Two particles whose masses are $10 \mathrm{~kg}$ and $30 \mathrm{~kg}$ and their position vectors are $\hat{i}+\hat{j}+\hat{k}$ and $-\hat{i}-\hat{j}-\hat{k}$ respectively would have the centre of mass at

If the origin of co-ordinate system lies at the centre of mass, the sum of the moments of the masses of the system about the centre of mass

In rotational motion of a rigid body about an axis, particles move with:

If the earth is treated as a sphere of radius R and mass M; its angular momentum about axis of rotation with period T is

Identify the correct statement. In the absence of external force,
a) the kinetic energy associated with the motion of centre of mass is conserved
b) Centre of mass is not accelerated
c) The centre of mass always move in curved line
d) Centre of mass must be at rest