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The centre of a wheel rolling on a plane surface moves with a speed \( v_0 \). A particle on the rim of the wheel at the same level as the centre will be moving at a speed \( \sqrt{x}\, v_0 \). Then the value of \( x \) is _______.

A solid disc of radius \( 20 \, \text{cm} \) and mass \( 10 \, \text{kg} \) is rotating with an angular velocity of \( 600 \, \text{rpm} \), about an axis normal to its circular plane and passing through its centre of mass. The retarding torque required to bring the disc at rest in \( 10 \, \text{s} \) is __________ \( \pi \times 10^{-1} \, \text{Nm} \).

Moment of inertia of a square plate of side \( l \) about the axis passing through one of the corner and perpendicular to the plane of square plate is given by:

A force \( \vec{F} = 4\hat{i} + 3\hat{j} + 4\hat{k} \) is applied on a intersection point of \( x = 2 \) plane and x-axis. The magnitude of torque of this force about a point \( (2, 3, 4) \) is __________. (Round off to the Nearest Integer)

A particle of mass '\( m \)' is moving in time '\( t \)' on a trajectory given by
\( \vec r = 10 \alpha t^{2}\,\hat{i} + 5\beta (t-5)\,\hat{j} \)
Where \( \alpha \) and \( \beta \) are dimensional constants.
The angular momentum of the particle becomes the same as it was for \( t = 0 \) at time \( t = \) __________ seconds.

Consider a uniform wire of mass \( M \) and length \( L \). It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the centre is:

A solid cylinder of mass \( m \) is wrapped with an inextensible light string and, is placed on a rough inclined plane as shown in the figure. The frictional force acting between the cylinder and the inclined plane is:

[The coefficient of static friction, \( \mu_s \) is \( 0.4 \).]

Consider a badminton racket with length scales as shown in the figure.

If the mass of the linear and circular portions of the badminton racket are same \( (M) \) and the mass of the threads are negligible, the moment of inertia of the racket about an axis perpendicular to the handle and in the plane of the ring at, \( \dfrac{r}{2} \) distance from the end \( A \) of the handle will be ________ \( M r^2 \).

A circular disc reaches from top to bottom of an inclined plane of length '\( L \)'. When it slips down the plane, it takes time '\( t_1 \)'. When it rolls down the plane, it takes time \( t_2 \). The value of \( \dfrac{t_2}{t_1} \) is \( \sqrt{\dfrac{3}{x}} \). The value of \( x \) will be _______.

A thin circular ring of mass \( M \) and radius \( r \) is rotating about its axis with an angular speed \( \omega \). Two particles having mass \( m \) each are now attached at diametrically opposite points. The angular speed of the ring will become:

The solid cylinder of length \( 80 \, \text{cm} \) and mass \( M \) has a radius of \( 20 \, \text{cm} \). Calculate the density of the material used if the moment of inertia of the cylinder about an axis \( CD \) parallel to \( AB \) as shown in figure is \( 2.7 \, \text{kg m}^{2} \).

Angular momentum of a single particle moving with constant speed along circular path:

A sphere of mass \( 2\,\text{kg} \) and radius \( 0.5\,\text{m} \) is rolling with an initial speed of \( 1\,\text{m}\,\text{s}^{-1} \) goes up an inclined plane which makes an angle of \( 30^\circ \) with the horizontal plane, without slipping. How long will the sphere take to return to the starting point \( A \)?

A triangular plate is shown. A force \( \vec{F} = 4\hat{\imath} - 3\hat{\jmath} \) is applied at point \( P \). The torque at point \( P \) with respect to point \( 'O' \) and \( 'Q' \) are:

A system consists of two identical spheres each of mass \( 1.5 \, \text{kg} \) and radius \( 50 \, \text{cm} \) at the ends of a light rod. The distance between the centres of the two spheres is \( 5 \, \text{m} \). What will be the moment of inertia of the system about an axis perpendicular to the rod passing through its midpoint ?

A solid disc of radius '\( a \)' and mass '\( m \)' rolls down without slipping on an inclined plane making an angle \( \theta \) with the horizontal. The acceleration of the disc will be \( \dfrac{2}{b} g \sin \theta \) where \( b \) is __________ . (Round off to the Nearest Integer)
(\( g \) = acceleration due to gravity \( \theta =\) angle as shown in figure)

Two discs have moment of inertia \( I_1 \) and \( I_2 \) about their respective axes perpendicular to the plane and passing through the centre. They are rotating with angular speeds, \( \omega_1 \) and \( \omega_2 \) respectively and are brought into contact face to face with their axes of rotation coaxial. The loss in kinetic energy of the system in the process is given by:

A mass \( M \) hangs on a massless rod of length \( l \) which rotates at a constant angular frequency. The mass \( M \) moves with steady speed in a circular path of constant radius. Assume that the system is in steady circular motion with constant angular velocity \( \omega \). The angular momentum of \( M \) about point \( A \) is \( L_A \) which lies in the positive \( z \) direction and the angular momentum of \( M \) about point \( B \) is \( L_B \). The correct statement for this system is:

Two bodies, a ring and a solid cylinder of same material are rolling down without slipping an inclined plane. The radii of the bodies are same. The ratio of velocity of the centre of mass at the bottom of the inclined plane of the ring to that of the cylinder is \( \dfrac{\sqrt{x}}{2} \). Then, the value of \( x \) is __________.

The following bodies,
(\( 1 \)) a ring (\( 2 \)) a disc (\( 3 \)) a solid cylinder (\( 4 \)) a solid sphere of same mass ‘\( m \)’ and radius ‘\( R \)’ are allowed to roll down without slipping simultaneously from the top of the inclined plane. The body which will reach first at the bottom of inclined plane is _______.
[Mark the body as per their respective numbering given in the question]

In the given figure, two wheels \( P \) and \( Q \) are connected by a belt \( B \). The radius of \( P \) is three times as that of \( Q \). In case of same rotational kinetic energy, the ratio of rotational inertias \( \left( \dfrac{I_{1}}{I_{2}} \right) \) will be \( x : 1 \). The value of \( x \) will be ________.

A \( 2 \, \text{kg} \) steel rod of length \( 0.6 \, \text{m} \) is clamped on a table vertically at its lower end and is free to rotate in vertical plane. The upper end is pushed so that the rod falls under gravity. Ignoring the friction due to clamping at its lower end, the speed of the free end of rod when it passes through its lowest position is _____ \( \text{ms}^{-1} \). (Take \( g = 10 \, \text{ms}^{-2} \))

Consider a situation in which a ring, a solid cylinder and a solid sphere roll down on the same inclined plane without slipping. Assume that they start rolling from rest and having identical diameter.
The correct statement for this situation is:

Choose the correct answer from the options given below:

A body rolls down an inclined plane without slipping. The kinetic energy of rotation is \( 50\% \) of its translational kinetic energy. The body is:

The figure shows two solid discs with radius \( R \) and \( r \) respectively. If mass per unit area is same for the both, what is the ratio of MI of bigger disc around \( AB \) (which is \( \perp \) to the plane of the disc and passing through its centre) to MI of smaller disc around one of its diameters lying on its plane? Given 'M' is the mass of the larger disc. (MI stands for moment of inertia)

Given below are two statements: one is labelled as Assertion A and the other is labelled as Reason R.
Assertion A: Moment of inertia of a circular disc of mass '\( M \)' and radius '\( R \)' about \( X \), \( Y \) axes (passing through its plane) and \( Z \)-axis which is perpendicular to its plane were found to be \( I_x \), \( I_y \) and \( I_z \) respectively. The respective radii of gyration about all the three axes will be the same.
Reason R: A rigid body making rotational motion has fixed mass and shape.
In the light of the above statements, choose the most appropriate answer from the options given below :

A uniform thin bar of mass \( 6 \, \text{kg} \) and length \( 2.4 \, \text{meter} \) is bent to make an equilateral hexagon. The moment of inertia about an axis passing through the centre of mass and perpendicular to the plane of hexagon is ______ \( \times 10^{-1} \, \text{kg m}^{2} \).

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JEE Main 2021 18th March Evening Shift

Consider a uniform wire of mass \( M \) and length \( L \). It is bent into a semicircle. Its moment of inertia about a line perpendicular to the plane of the wire passing through the centre is: