$A$ uniform spherical object of mass $M$ and radius $R$ has a moment of inertia $I$. It rolls down an inclined plane of angle $\theta$ without slipping. What is its acceleration?

$A$ solid sphere rolls purely on an inclined plane of inclination $\theta$. Which of the following statements are correct?
$(1)$ The friction force acting on the sphere is $f = \mu mg \cos \theta$.
$(2)$ Friction is a dissipative force.
$(3)$ Friction increases angular velocity and decreases linear velocity.
$(4)$ If $\theta$ decreases,friction decreases.

$A$ solid cylinder of diameter $30 \ cm$ is rolled down an inclined plane from a height of $2 \ m$. If there is no energy loss due to friction,the angular velocity at the base of the plane is ....... $rad/s$. (Take $g = 10 \ m/s^2$)

The ratio of the accelerations for a solid sphere (mass $m$ and radius $R$) rolling down an incline of angle $\theta$ without slipping and slipping down the incline without rolling is

$A$ solid sphere of mass $m$ rolls down an inclined plane without slipping,starting from rest at the top of an inclined plane. The linear speed of the sphere at the bottom of the inclined plane is $v$. The kinetic energy of the sphere at the bottom is

$A$ rolling wheel of $12 \, kg$ is on an inclined plane at position $P$ and connected to a mass of $3 \, kg$ through a string of fixed length and a pulley as shown in the figure. Consider $PR$ as a friction-free surface. The velocity of the centre of mass of the wheel when it reaches the bottom $Q$ of the inclined plane $PQ$ will be $\frac{1}{2} \sqrt{xgh} \, m/s$. The value of $x$ is.............