System of particles practice · Mixed difficulty

Where will be the centre of mass on combining two masses m and (M > m)

Assertion: Practically a body rolling on a surface will loose its complete K.E.
Reason: Friction will convert its K.E. into P.E. of atoms.

A rod of mass $m$ and length $l$ rotates about an axis passing through its end and perpendicular to its length. If its angular velocity it decreased from $4\omega$ to $2\omega$ in 3 seconds the torque applied on it

Two solid cylinders P and Q of same mass and same radius start rolling down a fixed inclined plane from the same height at the same time. Cylinder P has most of its mass concentrated near its surface, while Q has most of its mass concentrated near the axis. Which statement (s) is (are) correct

(A): When a rigid body rotates there is always a fixed axis or a fix point about which it has to perform the rotation.
(R): In a pure rotation different particle in a rigid body has different linear velocity.

The distance b/w the carbon atom and the oxygen atom in a carbon monoxide molecule is $1.1{A^0}$. Given mass of carbon atom is 12.a.m.u., and mass of oxygen atom is 16.a.m.u., calculate the position of the centre of mass of the carbon monoxide molecule

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

Four identical spheres each of mass M and radius 10 cm each are placed on a horizontal surface touching one another so that their centres are located at the corners of a square of side 20 cm. What is the distance of their centre of mass from centre of any sphere?

Assertion (A): Angular speed, linear speed as Kinetic energy change with time but angular momentum remains constant for a planet orbiting the sun.

Reason (R): Angular momentum is constant as no torque acts on the planet.

Assertion: At the centre of earth, a body has centre of mass, but no centre of gravity.
Reason: Acceleration due to gravity is zero at the centre of earth.

The moment of inertia of a body is I and its coefficient of linear expansion is $\alpha$ if temperature of body rises by a small amount $\Delta \mathrm{T}$. Then change in moment of inertia about the same axis

Angular momentum $L$ of body with mass moment of inertia $I$ and angular velocity $\omega$ rad / sec is equal to

Assertion (A): If the moment of inertia of a non-uniform thin circular ring is same about two different axes parallel to each other and lying in the plane of ring, then both the axis can be at same distance from geometrical centre of the ring.
Reason (R): From parallel axis theorem $I=I_{\mathrm{cm}}+{md}^2,$ (where terms have usual meaning). Moment of inertia of a body about two axes parallel to each other and at a same distance from centre of mass of the body is same.

Let $I_A$ and $I_B$ be the moments of inertia of two solid cylinders of identical geometrical shape and size about their axes, the first made of aluminium and the second of iron.

S-I: Centre of mass is that fixed point of a system of particles or a rigid body with in the boundaries or a rigid body with in the boundaries of the system where the entire mass is concentrated.
S-II: Center of gravity of a body is the point through which the whole weight of the body acts.

A thin circular disc of mass $M$ and radius $R$ is rotating about an axis passing through centre of mass and perpendicular to plane of disc with constant angular velocity $\omega$. The angular momentum of the disc about same axis is :

What are the coordinates of centre of mass of a point mass of $2 {~kg}$ with coordinates $(2,3)$ ?

Assertion: A sharp needle when balanced on a edge falls even under a small disturbance.
Reason: Small disturbance causes torque in the same direction as that of angular displacement.

A uniform solid sphere of mass M and radius R is pulled on a horizontal plane by a horizontal force F acting through its mass centre. Assuming rolling without slipping what is the angular acceleration of the sphere?

Assertion: Moment of inertia depends on the axis
of rotation and the nature of distribution of the
mass of the body.
Reason: Moment of inertia is the rotational inertia
of the body.

The torque on a body is zero. Which of the following should not change?

Assertion: The centre of mass of system of $n$ particles is the weighted average of the position vector of the $n$ particles making up the system.
Reason: The position of the centre of mass of a system is independent of coordinate system.

A swimmer while jumping into water from a height easily forms a loop in the air, if

Assertion: The torque due to a force always acts perpendicular to the force vector.
Reason: $\vec{\tau}=\vec{r} \times \overrightarrow{{F}}$

A uniform thin rod of mass $M$ and length $L$ is standing vertically along the $Y$-axis on a smooth horizontal surface with its lower end at origin $(0,0)$. A slight disturbance at $t=0$ causes the lower end to slip on smooth surface along the positive X-axis and the rod starts falling. The path followed by the centre of mass of the rod during its fall is a/an.

Torque applied on a particle is zero, then its angular momentum will be

Angular momentum of a body is defined as the product of

From a circular disc of radius R and mass 4m, a small disc of radius $\frac{R}{2}$ is removed concentrically. The moment of inertia of the remaining disc about an axis perpendicular to the plane of the disc and passing through its centre is

Assertion: A man sitting on a rolling table, when he stretches his arms horizontally, his speed is reduced.
Reason: Principle of conservation of angular momentum is applicable in this situation.

Two atoms of oxygen are located at $r_1$ and $r_2$. Their centre of mass is at