If the length of a thin uniform rod is ' $L$ ' and the radius of gyration of the rod about an axis perpendicular to its length and passing through one end is $K$, then $K: L=$

A thin uniform wire of mass ' $m$ ' and linear density ' $\rho$ ' is bent in the form of a circular ring. The moment of inertia of the ring about a tangent parallel to its diameter is ' $m$ '

A solid sphere and a thin uniform circular disc of same radius are rolling down an inclined plane without slipping. If the acceleration of the sphere is $3 \mathrm{~ms}^{-2}$, then the acceleration of the disc is

A balance is made using a uniform metre scale of mass 100 g and two plates each of mass 200 g fixed at the two ends of the scale and the balance is pivoted at 45 cm mark of the scale. The error when 300 g weight is placed in the plate at 0 cm to weigh vegetables placed in the plate at 100 cm is

The ratio of radii of gyration of a thin circular ring and a circular disc of same radius about a tangential axis in their own planes is $\sqrt{12}: \sqrt{K}$. The value of $K$ is

A thin uniform circular disc of mass $\frac{10}{\pi^2} \mathrm{~kg}$ and radius 2 m is rotating about an axis passing through its centre and perpendicular to its plane. The work done to increase the angular speed of the disc from $90 \mathrm{rev} / \mathrm{min}$ to $120 \mathrm{rev} / \mathrm{min}$ is

Due to global warming, if the ice in the polar region melts and some of this water flows to the equatorial region, then

If the moment of inertia of a uniform solid cylinder about the axis of the cylinder is $\frac{1}{n}$ times its moment of inertia about an axis passing through its midpoint and perpendicular to its length, then the ratio of the length and radius of the cylinder is

A body of mass $m$ and radius $r$ rolling horizontally $m$ an inclined plane to a vertical

a velocity $v$ rolls up an height $\frac{v^{2}}{g}$. The body is

A constant torque acting on a uniform circular wheel changes its angular momentum from $A_0$ to $4 A_0$ in 4 seconds. The magnitude of the torque is

A particle performs uniform circular motion with an angular momentum $L$. If the frequency of the particle's motion is doubled and its kinetic energy is halved, then its angular momentum becomes

The ratio of the radii of two solid spheres of same mass is $2: 3$. The ratio of the moments of inertia of the spheres about their diameter is

A particle of mass $m$ is moving along a line $y=x+a$ with a constant velocity $v$. The angular momentum of the particle about the origin is

A body is rolling without slipping on a horizontal plane. If the rotational kinetic energy of the body is $50 \%$ of its total kinetic energy, then the body is

Moon revolves around the earth in an orbit of radius $R$ with time period of revolution $T$. It also rotates about its own axis with a time period $T$. If mass of the moon is $M$ and its radius is $r$, the total kinetic energy of the moon is

The spinning of the Diwali cracker 'ground chakkar' involves the concept of

A solid spherical ball is rolled up an inclined plane of angle of inclination $30^{\circ}$ with an initial speed of $4 \mathrm{~m} / \mathrm{s}$ at the bottom of the inclination. How far will the ball go up the plane?

(Use, $g=10 \mathrm{~m} / \mathrm{s}^2$ )

A metre stick is balanced on the knife edge at its centre. When four coins, each of mass 2 g are put one on top of the other at 10.0 cm mark, the stick it found to be balanced at 46.0 cm mark. The mass of the metre stick is

A wheel of radius with 0.5 m and a moment of inertia of $10 \mathrm{~kg}-\mathrm{m}^2$ is rotating freely at an angular speed of 70 $\mathrm{rev} / \mathrm{min}$. The wheel can be stopped in 5.0 s by pressing a wet cloth against the rim and exerting a radially inward force of 88 N . The coefficient of kinetic friction between the wheel and wet cloth is

A solid cylinder of mass $m$ and radius $R$ rolls down on an inclined plane of height 30 m without slipping. The speed of its centre of mass, when the cylinder reaches the bottom is

[use $g=10 \mathrm{~m} / \mathrm{s}^2$ ]

Consider a thin metal strip of mass l kg and length 5 m . Calculate its moment of inertia about an axis perpendicular to strip and located at 100 cm on strip from one its end. (Assume the breadth as the strip is negligible)

A solid cylinder is released from rest from the top of an inclined plane of inclination $30^{\circ}$ and length 60 cm . If the cylinder rolls without slipping, then the speed when it reaches the bottom is

A straight rod of length $L$ is made of a material having mass per unit length $m(x)=\lambda|x|$, where $x$ is measured from the centre of rod. The moment of inertia about an axis perpendicular to the rod and passing through one end of the rod will be $L=1 \mathrm{~m}$ and $\lambda=16 \mathrm{~kg} / \mathrm{m}^2$.

Consider a uniform horizontal solid cylinder of mass 10 kg such that its length is 9 times its radius. Let the radius be 40 cm . Calculate the moment of inertia of the cylinder about a line passing through its edge and perpendicular to its axis.

A solid sphere and a solid cylinder, each of mass $M$ and radius $R$ are rolling with a linear speed on a flat surface without slipping. Let $L_1$ be magnitude of the angular momentum of the sphere with respect to a fixed point along the path of the sphere. Likewise $L_2$ be the magnitude of angular momentum of the cylinder with respect to the same fixed point along its path. The ratio $L_1 / L_2$ is

An object of mass 2 kg is hanging from a rope that is wrapped around a pulley of radius 25 cm . The mass of pulley is 2 kg . Find the acceleration of the object. (Assume, pulley to be a solid disk $g=10 \mathrm{~m} / \mathrm{s}^2$ )