System of particles practice · Medium

The moment of inertia of a circular ring of radius $r$ and mass $M$ about diameter is

Two spheres, one solid and another hollow are made up of same material and are of same radius. Both the spheres are heated to same temperature $\mathrm{T}_0$. Now, both spheres are left to cool in an atmosphere having temperature $\mathrm{T_A}\left(\mathrm{T_A \ll T_0}\right)$. After time $\mathrm{t}$ the temperature of sphere becomes $\mathrm{T}_1$ and that of hollow sphere becomes $\mathrm{T}_2$. Then,

A body of mass $m$ slides down an incline and reaches the bottom with a velocity $v$. If the same mass were in the form of a ring which rolls down this incline, the velocity of the ring at bottom would have been

Assertion: For a body to be in equilibrium it must satisfy both the conditions stated simultaneously.
Reason: (a) the vector sum of all the external forces acting on the body should vanish.
(b) the vector sum of all the external torque acting on the body should vanish.

If $\omega_{\mathrm{E}}$ and $\omega_{\mathrm{H}}$ are the angular velocities of the earth rotating about its own axis and the hour hand of the clock respectively, then

Consider the following statements:

If the net external torque acting on an object is switched off, then

1. linear momentum will remain unchanged

2. angular momentum will remain unchanged

Which of the statements given above is/are correct?

A loaded spring gun of mass $M$ fires a bullet of mass $m$ with a velocity $v$ at an angle of elevation $\theta$. The gun is initially at rest on a horizontal frictionless surface. After firing, the center of mass of the gun-bullet system

Assertion: When a sphere rolls down an inclined plane without slipping, its kinetic energy is partly translational and partly rotational in nature.
Reason: When a sphere rolls down an inclined plane without slipping, it possesses both translational and rotational motions.

A constant torque acting on a uniform circular disc changes its angular momentum from L to $\frac{{4L}}{3}$ in 2 seconds. Then magnitude of the torque applied is

A diver in a swimming pool bends his head before diving, because it

Assertion (A): Angular velocity of the seconds hand of a watch is $\frac{\pi}{30} \mathrm{rad} / \mathrm{s}.$
Reason (R): Angular velocity is equal to $\frac{2 \pi}{T}$ where $T$ is the time period.

A constant torque is acting on a wheel. If starting from rest, the wheel makes $n$ rotations in $t$ second, the angular acceleration $\alpha=$

Assertion: Angular velocity of moon revolving about earth is more than angular velocity of earth revolving around sun.
Reason: Time taken by moon to revolve around earth is less than time taken by earth to revolve around sun.

A sphere of mass $2 \mathrm{~kg}$ and radius $5 \mathrm{~cm}$ rolls without slipping along a horizontal plane. The velocity of its centre of mass is $10 \mathrm{~cm} / \mathrm{s}$. The kinetic energy of the sphere is

The velocities of three particles of masses $20 \mathrm{~g}$, $30 \mathrm{~g}$ and $50 \mathrm{~g}$ are $10 \hat{{i}}, 10 \hat{{j}}$ and $10 \hat{{k}}$ respectively. The velocity of the centre of mass of the three particles is

In the absence of external torque for a body revolving about any axis, the quantity that remains constant is

S-I: Motion of centre of mass depends on external forces applied on it
S-II: The motion of the centre of mass of the body is called the translational motion of the body

The centre of mass of a system cannot change its state of motion, unless there is external force acting on it. Yet the internal force of the brakes can bring a car to rest. Then

The principle of conservation of angular momentum, states that angular momentum

I. Angular momentum of a particle moving in a straight line is always constant with respect to fixed point.

II. Moment of inertia of a body remains same irrespective of position of axis of rotation.

Which of the statement(s) is/are correct?

If the rotational kinetic energy of a body is increased by $300 \%$ then the percentage increase in its angular momentum :-

The angular momentum of planet of mass ${M}$ moving around the sun in as elliptical orbit is $\vec{L}$. Then torque acting on the planet is

Two particles of masses 4 kg and 6 kg are separated by a distance of 20 m and are moving towards each other under mutual force of attraction, the position of the point where they meet is

A uniform $\operatorname{rod} A B$ of length $2 l$ and mass $m$ is rotating in a horizontal plane about a vertical axis through $A$, with angular velocity $\omega$, when the mid-point of the rod strikes a fixed nail and is brought immediately to rest. Find the impulse exerted by the nail.

A body rotating at $20 \mathrm{~rad} / \mathrm{s}$ is acted upon by a constant torque providing it a deceleration of $2 \mathrm{~rad} / \mathrm{s}^2$. At what time will the body have kinetic energy same as the initial value if the torque continues to act?

A uniform circular disc of mass $200 \mathrm{~g}$ and radius $4.0 \mathrm{~cm}$ is rotated about one of its diameter at an angular speed of $20 \mathrm{rad} / \mathrm{s}$. Find its angular momentum ( in $\mathrm{kg}-\mathrm{m} / \mathrm{s}$ ) about the axis of rotation.

At any instant, a rolling body may be considered to be in pure rotation about an axis through the point of contact. This axis is translating forward with speed

A transistor has an $\alpha = 0.95,\,$ then $\beta$ is equal to

A wheel has angular acceleration of $6\,rad/{\sec ^2}$ and an initial angular speed of 4 rad/sec. In a time of 4 sec it has rotated through an angle (in radian) of

A child is standing at one end of a long trolley moving with a speed $v$ on a smooth horizontal
track. If the child starts running towards the other end of the trolley with a speed $u,$ the centre of
mass of the system (trolley + child) will move with a speed