$A$ rod is hinged at its centre and rotated by applying a constant torque starting from rest. The power developed by the external torque as a function of time is :

Consider two masses with $m_1 > m_2$ connected by a light inextensible string that passes over a pulley of radius $R$ and moment of inertia $I$ about its axis of rotation. The string does not slip on the pulley and  the pulley turns without friction. The two masses are released from rest separated by a vertical distance $2 h$. When the two masses pass each other, the speed of the masses is proportional to

A solid cylinder $P$ rolls without slipping from rest down an inclined plane attaining a speed $v_p$ at the bottom. Another smooth solid cylinder $Q$ of same mass and dimensions slides without friction from rest down the inclined plane attaining a speed $v_q$ at the bottom. The ratio of the speeds $\frac{v_q}{v_p}$ is

A body of moment of inertia of $3\ kg – {m^2}$ rotating with an angular speed of $2\ rad/sec$ has the same kinetic energy as a mass of $12\ kg$ moving with a speed of ……… $m/s$.

A thin and uniform rod of mass $M$ and length $L$ is held vertical on a floor with large friction. The rod is released from rest so that it falls by rotating about its contact-point with the floor without slipping. Which of the following statement($s$) is/are correct, when the rod makes an angle $60^{\circ}$ with vertical ? [ $g$ is the acceleration due to gravity]

$(1)$ The radial acceleration of the rod's center of mass will be $\frac{3 g }{4}$

$(2)$ The angular acceleration of the rod will be $\frac{2 g }{ L }$

$(3)$ The angular speed of the rod will be $\sqrt{\frac{3 g}{2 L}}$

$(4)$ The normal reaction force from the floor on the rod will be $\frac{ Mg }{16}$