System of particles practice · Medium

Two equal and opposite forces act on a rigid body at a certain distance. Then

A projectile of mass $1 \mathrm{~kg}$ is projected with a speed of $10 \mathrm{~m} / \mathrm{s}$ at an angle of $60^{\circ}$ from the horizontal when projectile is at its highest point, its magnitude of angular momentum (about point of projection)

A car wheel is rotated to uniform angular acceleration about its axis. Initially its angular velocity is zero. It rotates through an angle ${\mathrm{\theta }_1}$ in the first 2 s , in the next 2 s, it rotates through an additional angle ${\mathrm{\theta }_2}$, the ratio of $\frac{{{\mathrm{\theta }_2}}}{{{\mathrm{\theta }_1}}}$ is

Assume the earth's orbit around the sun as circular and the distance between their centre as $\mathrm{D}$. Mass of the earth is $\mathrm{M}$ and its radius is $\mathrm{R}$. If earth has an angular velocity $\omega_0$ with respect to its centre and $\omega$ with respect to the centre of the sun, the total kinetic energy of earth is

STATEMENT-1: If there is no external torque on a body about its center of mass, then the velocity of the center of mass remains constant.

STATEMENT-2: The linear momentum of an isolated system remains constant.

Two uniform rods $A$ and $B$ of lengths $5 {~m}$ and $3{~m}$ are placed end to end. If their linear densities are $3 {~kg} / {m}$ and $2 {~kg} / {m}$, the position of their centre of mass from their interface is

A flywheel rotates with uniform angular acceleration of $2 \pi {rad} / {s}^2$. Its initial angular velocity is $20 \pi {rad} / {s}$. What are the total number of rotations made in $20 {~s}$ ?

Statement-1: If the centre of mass of a system is the origin then the total mass to the right of origin is same as total mass to left of origin.
Statement-2: $x_{\mathrm{cm}}=\frac{\Sigma_{m_i x_i}}{\Sigma_{m_i}}$

(A): If body or a system has neither translational nor rotational motion then it is in complete mechanical equilibrium.
(R): A body or system is said to be in mechanical equilibrium when net force and net torque acting on the body are zero.

Given that force $(5 \hat{i}+7 \hat{j}-3 \hat{k}) \mathrm{N}$ acts on a particle at position $(\hat{i}+\hat{j}-\hat{k}) \mathrm{m}$. Find torque of this force on the particle about origin.

A non-zero external force acts on a systems of particles. The velocity and acceleration of the centre of mass are found to be ${v_0}\,and\,{a_0}$ at an instant t. It is possible that

Two particles of equal mass have velocities ${v_1} = 4\hat i\,and\,{v_2} = 4\hat j\,m{s^{ - 1}}$. First particle has an acceleration ${a_1} = \left( {5\hat i + 5\hat j} \right)m{s^{ - 2}}$ , while the acceleration of the other particle is zero. The centre of mass of the two particles moves in a path of

When no external torque acts on a rotating system,

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 :

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.