Moment of inertia of a ring is $3{\mathrm{kgm}}^2$ .It is rotated for $30 \mathrm{s}$ by a force of  $6 \mathrm{Nm}$, Calculate the work done.

A disc is rolling (without slipping) on a horizontal surface. C is its centre and Q and P are two points equidistant from C. Let ${\mathrm{V}}_{\mathrm{p}, }{\mathrm{V}}_{\mathrm{q}} \mathrm{and} {\mathrm{V}}_{\mathrm{c}}$c be the magnitude of velocities of points P,Q and C respectively, then

string

A small mass m is attached to a massless string whose other end is fixed at P as shown in the figure. The mass is undergoing circular motion in the x−y plane with centre at O and constant angular speed ω. lf the angular momentum of the system, calculated about O and P are denoted by$\overrightarrow{{\mathrm{J}}_{\mathrm{O}}} \mathrm{and} \overrightarrow{{\mathrm{J}}_{\mathrm{P}}}$​​, respectively, then:

A hoop of radius r and mass m rotating with an angular velocity ${\omega }_0$ is placed on a rough horizontal surface. The initial velocity of the centre of the hoop is zero. What will be the velocity of the centre of the hoop when it ceases to slip?

A uniform rod of length (L) and mass (M) is pivoted at the centre. Its two ends are attached to two springs of equal spring constants $\left(\mathrm{k}\right)$ The springs are fixed to rigid supports as shown in the figure, and the rod is free to oscillate in the horizontal plane. The rod is free to oscillate in the horizontal plane. The rod is gently pushed through a small angle (theta) in one direction and released. The frequency of oscillation is. ?

If the resultant of all the external forces acting on a system of particles is zero, then from an inertial frame, one can surely say that:

A thin uniform rod, pivoted at O, is rotating in the horizontal plane with constant angular speed ω, as shown in the figure. At time $\mathrm{t}=0$ a small insect starts from O and moves with constant speed $\mathrm{v}$ with respect to the rod towards the other end. It reaches the end of the rod at$\mathrm{t}=\mathrm{T}$ and stops. The angular speed of the system remains $\mathrm{\omega }$ throughout. The magnitude of the torque ∣τ∣ on the system about O, as a function of time is best represented by which plot?